Part I. A Formal Expansion Theory for Functions of One or More Variables. Part II. The Number of Representations Function for Binary Quadratic Forms Citation Hall, Newman Arnold (1938) Part I. A Formal Expansion Theory for Functions of One or More Variables. Part II. The Number of Representations Function for Binary Quadratic Forms. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/k80p-sp52. https://resolver.caltech.edu/CaltechTHESIS:05292024-201308777 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Bateman, Harry Thesis Committee: Unknown, Unknown Defense Date: 1 January 1938 Record Number: CaltechTHESIS:05292024-201308777 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05292024-201308777 DOI: 10.7907/k80p-sp52 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16433 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 18 Jun 2024 21:29 Last Modified: 18 Jun 2024 21:30 Thesis Files PDF - Final Version See Usage Policy. 32MB Repository Staff Only: item control page

Part I A Formal Expansion Theory for Functions of One or More Variables Part II The Number of Representations Function for Binary Quadratic Forms Thesis by Newman Arnold Hall In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1938 Part I A Formal Expansion Theory for Functions of One or More Variables Table of Contents 4 Int rodu ct ion Formal Expansion Theory 8 1. Functions of a Single Variable 12 2. Functions of Several Variables 20 3. Relation to Tchebycheff Polynomials 25 Chapter I. Chapter II. Applications. Functions of A Single Variable 28 1. Hypergeometric Series 28 2. Basic Sets of Associate Functions 31 3. Classical Expansions 33 4. Non-hypergeometric Series 39 Chapter III. Applications. t Functions of Two Variables 41 1. Hypergeometric Series 41 2. Expansions in Products 42 3. Expansions in Two-variable Hypergeometric Functions 4. Non-hypergeometric Types 44 48 Introduction While pursuing an investigation of the general properties of the confluent hypergeometric functions of two variables at the suggestion of Harry Bateman the question of the expansion of arbitrary functions in terms of these arose. Efforts to obtain results by extending the theory of orthogonal Tchebycheff polynomials or by some other integral method proved quite unsuccessful. Finally the formal algebraic method suggested by the Neumann expansions in Bessel functions was devised ana. developed. to solve the problem at hand. We shall give an extended discussion of this method, together with sufficient examples to illustrate its application. The idea of representing the power series by the exponential eax, using the umbral notation an= an, is apparently due to Blissard1 who used it to obtain very elegantly the properties of the Bernoulli and related numbers. In recent years this umbral notation has been extended by a great many investigators. It has been placed on a firm alge- braic basis by E.T. Be112 to whom the author is greatly indebted for constant counsel and encouragement. I should also like to state my appreciation to Harry Bateman, without whose original guidance the work would never have been begun. For convenience in writing formulae, we use certain 1John Blissard, Theory of Generic Equations, IV-VI (1861-1864). 2E. T. Bell, Algebraic Arithmetic {New York, 1927), pp. 146-159. notation conVcentions. In place of a repeated swnmation: a ,,, I, we shall write merely, Unless explicitly stated, all summations are to extend over the range o to ~ . Often we shall include polynomials under such a connection by using coefficients vanishing for all values of the index larger than a certain value. Such cases will be self-evident and will be left to the observation of the reader. Since we shall be dealing with hypergeometric functions, a great variety of Gamma functions, factorials, and binomi•al coefficients might be introduced. Rather, we shall use uniformily the symbol introduced by Appell / / A -,-_/4 ) ; - ( ;\ ) Some simple properties of this symbol derived from its relation to the Gamma function will be used. (a,k) =- a(a, 1) ... (a -t- k - 1), k integral (a,o) = 1, (l,k) = k!, (a -k) ' (~ n,k) a = k integral (-)k k integral (1-a,kf' O, n, k integral k;, n (a,n + k) • (a,k) (a+ k,n) b c- J r Chapter I Forma.l Expansion Theory The expansion of known functions in terms of sequences of polynomials or other functions has been considered from many angles. It forms the basis of the theory of orthog- onal functions and of polynomial approximations. The extensive work of Neumann and Schl6milch series in Bessel functions is of this nature. Although the methods of analysis are more power- ful and more- inclusive, yet a purely formal algebraic treatment of this problem exhibits a very attractive . elegance. Partial treatments of this nature have been discussed by previous workers, 1 but the completely general method seems to have been overlooked. In the work to follow we shall unlformly deal with functions defined by the power series 1) or equally well by the matrix of coefficients 2) a ", I Preliminary to a consideration of the general case, 1 N. Nielsen, Fonctions Metaspherique, Chap. IV. N. Nielsen, Recherches sur le developpement d 1 une fonction analytiques en series de fonctions hypergeometriques, Ann. Scientifiques d 1 Ee. Norm., (3) XXX {1913), 12. s. Pincherle, Alcuni teoremi sopra gli sviluppi en serie per funzioni analitiche, Lombardo Rendiconti, {2) XV (1882) 224. J.M. Whittaker, Interpalotory Function Theory, (Cambridge, 1937) 'j__ let us illustrate the theory by determining in terms of the an the coefficients bn of 3) /(x) Since -?f 4) ---- X) ( I - ( I, 0 ) ,,,,_ I-:·, ( -,'YI -r) we have l_/, 5) l .~ . . X ( I ' r) ,- lo,,,, (1-;x.)'1 - r- - h ,..,, j ~ /1 ( I, ' -'1) --r-) -; I , ·Y-) - ,,,.. )'. r- , .,., { I, ,,,, ) ( - r>1' n-,_ ,.,., I_~ X l .: ( I, ,..,_, ) ,r· so that 6) 0.---r-- -- L. (-A, -( ( - "'1, ?-') { I/ --->c ) lo "') -"' lo r) h r>t I, -·>-1 ) -) ~ ) C --- n,-f .-?- ( I, , .,,, ) /?1. "1 To obtain bs' then, we sum over r 7) Hence 8) I I (x) --· ) .L.--- - - { I - ;<.) '! {-)\ -✓ -( I' '() a,..,...,.'"'1. ) / {1 , --r} r'" n, More generally, if we consider the polynomials, 9) (-_.,.,,,_ .,-,) { I' r- ) x r- IO Following the above argument we have, if r -,.,,,r ) 10) /"h , r h,,,, ) ?' J<,,_ (1,,-,,,)(1,r ) ,1 11) and to obtain bn we sum as before over r in; 12) So that 13) --r-- In particular, if )?-- =- ( , we have the formal Taylor• s ex- pansion of a fWlction about an arbitrary point: 14) It is thus possible to proceed to develop our entire theory. The awkwardness of hand.ling the algebra, however, would make the labor prohibitive and hide the elegance of the results. To avoid this, the umbral notation of Bl1ssard2 is introduced and will be used throughout. We consider 15) r1 ,L__ ,, r>t ? as defined by the 1nf1n1 te one- ,owecl matrix 2 Bl1ssard,10c. cit. I/ 16) 1 «,.,, , , and ado.pt the convention 17) I a ::c so that ux 18) e Hence, on this basis: a ;;- ax 19) I (x) -- e :::- C( - (1-).x ) ;\ e 0. E,,,, =: {)_ az ) ✓I. >; )~ ( I, JI,,, (,.) ;: ) (-> ·>r ( 1 ->.-x ) ,,, -( I , /1'( ) "' l ,,,.. _; ;\ e, C,,(...,,.-)-1 + ,-,,_ /\.,.,.,,~ ( I , -r ) giving the previously derived expansion very elegantly. Of course such a derivation is only suggestive, and the other is necessary for completeness. Such a formal derivation of an expansion by either of the methods used above gives no information regard.ing the range of validity, nor even whether the original and deduced series converge in the same range of the variable. In the discussion to follow this difficulty will be even more in evidence. Inasmuch as this forms a large part of the theory of any one particular sequence of functions, it would carry us quite off our track to deal with the problem extensively for any large class of functions. Rather, we shall assume that the arbitrary function and the sequence have some common region of convergence and we shall be satisfied with the conclusions to be drawn from Weierstrass• 3 . Theorem 2.1 Le f 1° { x ) 0 / P (x ) / / be an ordered, infinite sequence of power series in x, each containing both positive and negative powers of the variable with arbitrary coefficients. Let it be possible to choose two real quantities R, R1 , with R1 > R 2- o, such that for all values of x satisfying 21) R < /xi < R' oC> each of the series /;_ (:x. ) converges as well as the sum E', !J. ( x.} and further that the latter converges uniformly for all values of x having the same absolute value. Then, if ""' L= v/ ,J 0 iv) A,,, / (v) A_,,,,_ is the coefficient of Xfa in / ~ f?:. J the sum has a definite finite value for every value of n, designated by A,,u. , , and it may be shown that for every value of x satisfying 21) the series E, A,,,, x _µ converges and the ,?-= - "" equation 22) p (x ) ,I holds. 1. Function of~ Single Variable We shall call two sequences of functions Pn(x), Q.n(Y), defined by the matrices ;o,,,, ~ / p_,_ -~/, L, =- I t "'/~ associate 3 K. Weierstrass, Werke, Bd. 2, s 205. / J if they are such that P,,, ( x J 9 , , _ (-;; J 23) (I,,,,_) Under this conVLention and the above assumption as to convergence, our basic theorem on formal expansion can be stated conveniently Theorem 2.2 If f (x) is given by a matrix u.. ::: /~ /, and Pn (x), ~ (y) are associate sequences of functions, then 24) == e CZ :X Using the umbral notation, this is immediately evident. However, we shall present a more vigorous deriva- tion since we have not developed sufficiently the properties of the matrices. Since Pn (x) and ~ (y) are associate, we hs.ve: l,:(-x:) (),,,,_(J) 25) (1,/?t.) r,,,, o,,,, ~ bs - - - - -~ ~ --- --- ( I, /?1.) so that 26) ( 1, -r ) , r--= S being the essential condition interrelating the coefficients of the associate functions. I , ' •. ' Now we e.ssume that ther e exists for f (x) an expansion: I (t) 27) - .. >- l o( _/ ( I , ,n r>t. ; -:> ) ,..,.,_ ( .( ) ffl.. 1= -- /}t, -- -· ;o,., X - -1-- )'- ) ( I, ,n ) ( I, -,-, ,r- ">t ..,,, o( r "'< ·1 _ / ( I , T ) 1 _: .,.,.,_ -r d...,, r: (I,= ) Hence with 28) ) 29) To solve for /?'>( we multiply by ;y..,.., , to be determined, and ~ sum over r in: ,,,,,, ;»f ?f -r L 30) a .,,,. 1-~' --- (I,-,-) - -r 1 {I,,,,.,· ) ~ cf,.. cf .,., _/ /7) (I , r ·r- r>t if -~-·-~-- ( I, .,.. ) /'l ; ,'I 1 ,,,, ;O r cf r (1, -,, ) n,; -,- O("t 1-: c<.,.,, --r- o( ,,,.,.,_ ) is such that 31) ~-: -r- r ( I, y ) 6 r- /n't n-t c'.) rn-t -1- / 11_ r .,,..,,, _--:~ (1 , .-,.,, ) But if this 1s true, we ce..n deduce by multiplying by and summing over n, that "'1 32) z: />"( - -- J i : I l ../ r- l <5 7' (I, -,... ) 2 . _/ .m Therefore \ ( I, 'l ,.,, I°.,.. u by 26) relating 33) ]_~ _: ,- ,.,, I°.,.. r- ) ,.,,., ~-), "'1 5_~ ,.,, ) ( I, , -n 0 ..,.. ' L , .- 1 (I, .,.-,, ) d" "' ' J !( and 34) ( I, ,,_,,,) The two sets o:f equivalent conditions: 26) />L r ' "ts ( I, ,n 0 "'- /J,., -r -I -S -- -- ,n) (I, / :_-: s r -J and r 35) J -r ? ,-,._ C) 'h,,,_ ' ;I:- / s - ( I, ,,--,,__ ) ,,_,_ (I, r) , r =s :form a basis on which we can build several important theorems on the respective form of Pn and~• Let I?/ represent the discrimine.nt of the coefficients of the l°o 36) /;vi ;/_ I r\ /Jo r,° P, ' /> 0 c) /> .,_ I J ' .i_ ,. '.,_ /0 -- In general this will be an infinite order determinant and the question of convergence 1s best considered in the particular cases which arise. With this notation we have: Theorem 2.3 Let Pn(x) be any arbitrary sequence of fwictions 37) r (x) ,.,,_ 16 If there exists an associate sequence of functions ~(y), the coefficients of the matrix qn representing O"n(x), are given by: ' l°o 0 , >f - 1 P,o 38) -,, ' r f'oo p a-r! ..s ! ,,,, , s (-) - - /p/ a I /'\., I°,. c) ~ - /7) J - I I u,.' I ,,,_, I ,.,, ,, ;:> J - , .,.,., I /'CJ ro I ;-o' I--;) 1 _.,,, I I--;) Q 'f I 10.5._,. , /0 .,, ' In the proof of the previous theorem we showed that when the series ~(y) existed the coefficients were related to the matrix Pn by the equations: ,.,, 26) 2~ ; o,.. a ?f f"- -- { I, ,n) r -1-- -5 (i, y) r -·- ·- s /?<- These can be regarded,as an infinite system of linear equations in (:fat! for all n. Solving these according to the standard methods, we obtain the result stated in the theorem. The same theory would be secured by starting with the equations 35) I - J _I r ;o'°",... (I, tj: r) 0 -- (I, .-;,( ) , 7/»c / n /K -:=- /J'Yt and solving for Utilizing this theorem, we have others as direct consequences. Theorem 2.4 If P0 (x) and Qn(y) are associate sequences of functions and )- =- I ),d, ;,, - - , ,\,,; I is an arbitrary matrix, then ~ (;1. x ) and 9"" (-J-) are associate sequences. If / /~ / is the determinant of the coefficients of the /;,, (?-xj I 7 then, since the matrix of the coefficients is ?,., A =- / )._. _. p: / , 39) Then replacing the j?: in the expression for <i: , by ),,.. ;o: we obtain the new 40) · Hence corresponding to Pn (,.l_x) we have _, - -~ l 41) <t"" ~ J - - ~ _ _J __ --~ --·--- L s 5 ..5 ( I, -> ) l ~ u / (1,.s ) /\., ) A ~ ( -2- Thie very important theorem enables us to obtain from a given set of associate functions Pn(x), ~(y)an almost endless variety of new sets and in our later applications we shall see its usefulness in obtaining some of the classical expansions. Occasionally it happens that the expansion of an arbitrary function in a series of functions is derived by some other method aside from that discussed here. we can thereby deduce useful information from In these cases I I ? Theorem 2.5 If we have given ax 42) ;O= ( x ) - e V~ (a_ ) ( I, r'>1. ) then, subject to convergence conditions, we have also 43) / fx ) rn When the first expansion is given, the coefficient of Pn and~ must certainly satisfy the conditions 0 26) , --r ::/= S which are perfectly symetrical in the p's and q's. Hence the second expansion follows by our previous argument. If the Pn(x) have some peculiar form, Theorem 2.3 often enables us to infer directly certain properties of the ~(y). In view of its import we state Theorem 2.6 If Pn(x) and ~(y) are associate functions, and if Pn(x) is a polynomial of the n 1 th degree in x, then O~(y) contains y to no power less than the n'th. Under the assumptions here, ,P.; =o , ..,,.,>/Yl , and without loss of generality, we can set the leading coefficient of Pn(x), ,., = 1, so that /?/ ::: rr,.,, Hence for ?,., =I . f',.,, s < rn. 'f I Po I,:; 0 () ~" 0 r. 0 c) 0 C ,- ' ' 1-::: 44) ,,,+s I fs -= ( - ) rrt, .s I 0 - J,' I° 0 p, " /0 0 0 0 /:) > . ' () 0 a 0 '71-, I p s., ,,,, . ' ,s+/ /\,, - = o /o, ,,,, _, "rl !°,, _, ?, _, ,,,, ,, p " F', /~ ~- , ,,, - ' 0 ,,' s J . ' ,,, s J - ' f> . ?_j ~/ ;o >,, ~ -• /-:::,,. , /-;) .S7 ,. 0 while for s > r11, 0 I-0 , O 0 j>,,, ' ?,.,, 0 0 o ~,,,,, 0 0 - - C) ? :: : - •• I°;~: 0 0 0 () ,.,., -1 ,,,,, . , , ,,,, , 45) f ,,,,,, ,., - ,,,, J .= ( - ) /"1: /J -~.5 0 cl,' ,$ ~-, 1 ) ana_ clearly j Q - -- P,,, /J -> ·- ' • • j)u -1 f ;::_ = I Similar theorems can be stated if Pn(x) is a polynomial of degree depending in some other manner upon n. method is obvious and we shall not consi der them. The Also, if the sequence Pn(x) has any regular properties, corresponding ones can be inferred for the ~(x) by Theorem 2.3. / 2. Functions of Several Variables The formal theory here developed admits an immediate extension to functions of several variables in a manner more elegantly direct than similar extensions of other methods of expansions in functions. In order to provide a convenient way of stating our theorems, we introduce a rather obvious generalization of our matrix notation. Let / (x.,, - ·, -\ , ) -=/ 0.·) be a function of k variables whose series representation is: 46) j ,..,.~:I a. .,.,.,,, -· rl l. / (Y,, Yt.,- ) /1''/, / ,,-71,J. / .,,,,, " / ~/2 X, · ftf .1 :;l /,' ,.,, " .m, ~ '-.. Then said function will be represented by the k-dimensional matrix which we denote: 47) e,t,_ a • L ' /2 I and we ado.pt the convention 48) == ,,,, J.. ' •• • The concept of associate sequences of functions needs to be extended to several variables. of functions 49) defined by the matrices 50) 51} We say that two sequences al I .1 7 With this notation the general expansion theorem fork variable5is: Theorem 2.10 If I ( x:.A ) is a function of k variables represented by the matrix u' a'"!== /a.A;, , •;. / , and /~ (x.) and Q. (1J are associate sequences of functions, Then 48) Again using the matrices and the umbral notation, the theorem is obvious and for the general case we shall be satisfied with this. For two variables the statement is essentially the same. Theorem 2.11 If f(x,y) 1s a function of two variables represented by the matrix u o."" =/a..,,./, and t:...,~ (x., ✓ ) and Q,~ ((, '1) are associate sequences of functions represented by the matrices /? ;,J~ / , / c;; ~,, 7, then 49) (I, ,,,,,_, ) ( I, .,n ) The proof is a direct extension of that given for one variable. From the definition of ~-"" and 9,,.,, , .,_ we have: 50) so that /0,,..,,,,,rl? 51) 0 ri-.t, , ?/ r: -J <t l-l, ti ---------·-- ( / ,,,_,,, (I,_,,,,,, ) ) 1 ( I, Y) ( I, .s) V We assume there exists the expansion for/(x,J.): 52) /tq) ~ = J; d (1,~ tt;;;> ~,"' (x, X ) L: but ,,. I 53) / = 2_; (!---:;- ) Cl,,.._ .s (x,) ) J X ;J ( I, .5) -r, ,5 Therefore 54) -1 -__: u,,,,,.., -.S - ----- (I, ,,,,,,,,. ) ,,,_,,_ /V}f ,,,,., ,,,,,, /0 --,,-,_ '5 c/,,.,.,,_,..,, (I,,,,_;,) I h j To solve for c::x1._ ~L we multiply by 0✓--,r; -<5) to be determined, and sum over r and s in L 55) - / ;,_j ;( h, /, a --r, J r; ,5 - - -- -- ····· ·- - - -- - ( I, r) ( I, -.S) -y ,-5 l~ />rl,/11; ": -r, h, ,,,,,,_,..,, ,5 ?--,,-,__, (I,,,,,,,,,_ ) ( / 1 4) ( I, r ) (/;- ~) J (I, -r) ( I, if ,,.,.,, ,,,, o( cf) £ t,,,, _, is such that 56) L ,- J ;( h, -/2 -r, s - - - - - - -- (/,-,,-) -,,,,, I° --r 0 h ± /1-'1 a--,,J/ 01- A :/:-,,,,,., , J ( I, s) (;,,,,_.,)(l,"'t) /--, .=-,,-,,,,i -/2 = /71.. ~ r"'f '>/ -, If this 1s multiplied by ci.,,,.,J{i.,_;;;) and summed over m,n we obtain: ,,..,,,4 57) l°,,,,,.s _/ ( I, ") /, /2 6 ( !,---»1) ( /,"'1) by 51). ,,<, J ' Therefore 58) For three or more variables the proof would be similar and it is unnecessary to give the details here. The analogy of Theorem 2.4 is extremely important in this case. Theorem 2.12 If ~ , " t (:t:)')and Q,,,. (f y)are associate sequences of functions and AA-x=/;.A~)is an arbitrary matrix, then!;.,,, 0 (1.x.,)})and 9l.,,,,(f,f}are associate sequences. In the proof of Theorem 2.11 it was shown that the conditions that two sequences Pm n and~ n be associate, namely ' , 51) r_) -r c::/= cA " ---,C V ~,At. /> -,,-, .s ,.-nf / ,t 'b u.'v ( ! , ~ ) (/,,n- ) / (l,Y){!,0) / , = U s .=. V was equivalent to the similar 59) ,,,,,,,"'1 L 'l"' / /" - 7 h /,_ %' ,:, --- - -· · - - - ( /, r) r', a / h ( I, ,-,,,, ) (!, "?J ) J *- CZ,,,,. J /or - /J-r1 J P?:t. - - -( I, ..s) '-' / Ii / /4 = /J1, Now let the matrices of~,,, (k"', >)(;,) and <2,.,,,, (f, J )be designated for the moment by fi., ~.,_ and ?'- [.,,_ ~ Then 60) - L -r, • p - ,,,.,,,,, .,- -- ' ,J (I' , ) _, 't 1,,..£ ~ 7;--5 ( I, ..s ) a-,,d/o-,-- by the original assumption. Hence we show that the second pair satisfies the necessary conditions and are associate. This theorem has its greatest use in obtaining associate functions for given two-variable functions by building them up from known one-variable associates. Thus, if P~(x}, 9rl ff} and /: ("'J.) , ~ ( 1) are two known sets of associate functions, then we have immediately that are associate sequences. And if A ;l,."" =I)._.._ j is perfectly general, the two variable functions can certainly not be actually factored into one-variable functions. We shall use this in all its power in later applications. Evidently it would be possible to state theorems for special types of two-variable associate functions as we did for the one-variable cases. Those needed are such obvious extensions that we shall mention them only when they are used. We have indicated in several places the direction of generalization to any number of variables. On the formal basis which we are building these extensions will clearly introduce no new complications. We have restricted ourselves from treating the general case in order to have the results in a form ready to , ) ol 6 apply in our later discussions. 3. Relation to Tchebycheff Polynomi~s Since the Tchebycheff polynomials form a very large class of functions to which this theory is applicable, it is significant to inquire concerning the possible extensions of their theory to this more general case. Let Pn(x) and ~(y) be a set of associated functions, and we shall assume the desired expansions to be in Pn(x). We search for a weight function Wn(x) such that o o0 61) j r: (X } I°_ ( YI;'., ( J = (,, ~~ ' <) X) - X /1-"tF/V/. = "'1 ;n, 00 If such a function exists it will certainly be unique and since Pn and ~ are associate, we would infer directly that 00 62) 0,,, () ) - f X --:J.. C VV,,,, ( X ) I: (-r ) d X oO In case we have 63) where \IV ( x ) is independent of n and Wn is independent of x, then 64) j ~ (x ) I:,,. (x ) {) 00 - vl(x ) d:t 00 so that Pn (x) for; a set of Tchebycheff polynomials, '1= (?(). And conversely, when the Pn(x) are Tchebycheff polynomials, then ~ ( x ) will certainly be of this form. In the theory of the Tchebycheff polynomials there are several integrals and related functions which have significant meaning. We consider the extensions here: The functions of the second kind Pn*(y) are defined .,, by PA. I 65) ( X ) 1°,,, ( ;;) ( /,0.) J - X from this and the definition of the associate function ~(y) we conclude that I 66) J where r n-t_:c /J1 / (matrix notation). Also from 65) and 61) which would also follow symbolically from 66) and 62). We may indicate a treatment of our expansion theory using the Cauchy ,complex variable theorems, in a manner customarily used for the Neumann, Legendre and Bessel function expansions. The question of the range of applicability of this to follow is another problem beyond our present purpose. Let 65) be uniformly convergent with respect toy in some region R including the point z with yon the boundary C of R. Let ( I, ·?1 ) be some function analytic inside and on C. Then, since 65) is uniformly convergent 68) /( z ) ~ -:f-. ! --/ (?) d) .;i, ,,.,,<: ;;. - z_ C ~ut since the expansion in Pn(x) is unique when the series converges, this implies 69) C If, further, the weight function Wn{x) does exist and the infinite integral 67) is uniformly convergent, then ' ff oO ca)f-f:(-x} ~'t ,J,_71.,; 70) C. G - - 00 -'X {T 00 in agreement with the inference above. The two-variable theory outlined can also be treated in this manner. The method is obvious and since it contributes nothing of significance to our theory, we shall be content with the brief discussion of the one-variable case. Chapter II - Application. Functions of a Single Variable We shall develop the application of this theory to the hypergeometric functions of one and two variables. All the classical orthogonal expansions are in this category and will appear as special cases in a logical sequence. 1. Hypergeometric Series f a~ is said to be hyper- A single infinite series M~O geometric if the ratio a=/a~ is a rational function of n. When this is true, we may write: where P(n) and Q(n) are relatively prime polynomials inn. If they are of degrees p and q -r 1 respectively, we me,y exhibit them in the factored form: ( I?'( -1- c( ' ) ( ;74 i- 1) (0 -+- /3 , ) (/)1 7 ot " ) - - /.J.,_) - - - -- ( ,,,,, -r 0( !"_) __~\~ (M -I /3g){,n -I /[Jt,_,J A being the ratio of the coefficients of the highest terms in P and Q. Without loss of generality we set 1, , = / according to established custom. In case any one of the/:3's is a negative integer, say Pi= -k, then obviously ak and all previous terms in the series must be zero. Hence, the series will begin with 8krl' so a new set of ;-3: ' s is obtained with ~. =I!:_ -1- A + I. Thus, again, without any loss in generality we may assume that none of the p ·•s are negative integers . . Applying 1) successively, we deduce ( o:.,,'>r ) ("'-.,ht. ) ( o(p I -· -- - - - ( /3, / ~ ) (;CJ rn ) />I. ) - t{ 0 ( / (jg /J'l) ~ So that For the series occurring in the second member of this equation, the standard notation is: In case any one of the oc 1 s is e, negative integer, the terms become zero after a certain stage and the series is merely a polynomial in ) . In all other cases the series will be non-terminating and from the Weierstrass ratio test we infer the following convergence properties: p < q + 1, p = q + 1, , a. c. / i\ I < a. c. ! J p = q +l, /},,/ ~ /, R (r,cx.-L'rJ ) <OJ p = q + 1, 1 " 1 = 1, 1 ? R r-;: °" - r /13 >> u ) >- 1: 1 , c. c. p = q + 1, 1r ( E p = q + 1, 17 rE P = q + 1, I/\ I > I , p > q + 1, - , a. c. o< -I;a ) J u, - , d. 0( -- 1_; ,✓cl J 2 1, , a. , a. , a. a. c. - Absolutely convergent; c. c. - Conditionally convergent a. - Divergent. / ; _,,. ... To avoid the complication of encounters with the divergent forms above, we shall consider them replaced when they do occur by the series for which the divergent forms represent an asympto_tie expansion according to the treatment of Barne~1 the series being considered as functions of A . The formula, valid for F > g + 1 , is o<. I , - - .I [ 13 , , - o( r., I - / ;3 !;' ., A] .. o(-/1_ / ff g+ ! where the asterisk indicates the omission of the term with equ~l subscripts. When p -: : : g- +- 1, the same formula gives the analytic continuation of the function beyond the unit circle. These formulae are particularly useful on occasion to replace certain polynomials by equivalent forms, the replacement being in these cases merely a reversion of the order of terms. The various analytical properties of these fW1ctions have been considered in detail by a great many writers. Since they have no immediate connection with our formal expansion theory, we merely refer to these sources. 2 1 E. W, Be.rnes, The Asymptotic Expansion of Integral FW1ction defined by Generalized Hypergeometric Series, Proc. London Math. Soc., (2) V, 59-118. 2 Appel et Kampe de F'er1{i-eti Fonctions Hypergeometrique, (Paris, 1926); an extensive bibliography is geven here. W. N. Bailey, Generalized Hypergeometric Series, (Cambridge, 1935). ( \(i 2. Basic Sets of Associate Functions ----- All of the known expansions in hypergeometric functions of a single variable are deducible by our theory from two basic sets of associate functions. We are primarily concerned with such expansions and shall treat them in detail before proceeding to non-hypergeometric series. The first basic pair arises as an extension of the well-known expansion in Hermite polynomials. defined by the generating function e. .'..l X - - ;;z _ ,: I° ,,, \ (_3_ /, ni ) 0n, p fl ~ has the explicit form: p H (X ) /"1 =- ) (x. ) ,.,. t ;v ) ------,, =- The function /-/ I"(/. ) hi c$ X -r- Z t-( ;a- " _(_/ _, 7' ) { I, u ) n-;, " ' . ,P.J .>( ,,J(f'-r, ) -) -s p The associate function, }l () ), is readily determined from the generating function, since: C Hence Expansions derived from these by application of our several theorems will be called Hermite type. The well-known Neumann expansions in Bessel functions are special cases of the expansions obtained from the associate functions: /, /3,,,, (ot,x) ( o<. -f :,,,, r I, r ) ( I, 7 ) We show that these are associate by referring to the basic definition: ), {I,,;,,) ~ ,.,,.,-4.,,. /3~ ( X ) L,,, (;J ) E/2.,T_~ : I-J m,~s (-/>1, / k ,J) X l ,,,-As (cl-,. '"1,"'11,-r){-o,:- ~t-1, .s}(1,-,r)(1,J)(1,'1t) Thus Bn and ½1 fulfill the definition of associate functions. Expansions derived from these will be called Bessel type. 3 W. N. Bailey, loc. cit., p. 25. In the discussion to follow we shall show how these two sets give rise to all the classical expansions by using Theorem 2.4. The two sets here given e..re essentially different in this respect, in that one cannot be derived thus from the other by application of Theorem 2.4. Further basic expansions of more complicated nature doubtless exist, but only provide expansions in functions of greater complexity than concern us. 3. Classical Expansions The several well-known expansions in series of Bessel functions are ihcluded among both the Bessel and the Hermite type expansions above. The Neumann series of the first kind gives expansions in: -✓ ( ~ x) Jv +,,-,,,. (x) -- · - -- - ---- r(-,1+,,,,,,, ,J Hence it is of the Bessel type and the associate function is: The Neumann expansion of the second kind gives expansions in~ 1--!--J {-Jx) "'- J_ ~(x ){~"'(x) 0 ~ .,_ (-,,,)fl/ = r (o/~~ ,,J __,(. /2 /- ( -ol·vf / -f /, + I ) where Related to these is the expansion of even functions in series ~!j /_!__ ~.,2 where x) J y -fv (~ ) (-) ~ /7;;-;-·~ Tl) This is of the Hermite type, so the associate function is: In addition to these well-known expansions, we may construct from our basic set an almost unending variety of expansions in Bessel functions not discussed before. For ex- ample: ..,... Thus we have a Hermite type with the associate function: Again, the fact that suggests a number of expansions. The Jacobi polynomial may be similarly considered. We have the general form: f- (- r>1 X ) o( -f ""'- -.. I • / ==- (-) , ., ( ol. -t ;( ; / r?1 / rJ1 ) L.,,.I ( 0( )_ ;:\'. ) / / Hence expansions in Jacobi polynomials are of the Bessel type. Rather than discuss the Legendre polynomial, we --.J consider the Gegenbauer function ( ,, (;;,;_) which includes the former as the special case J _/ (1- ::2 h X -f \ A") -- 7 By definition J /2 .,,, C ( x. ) "' -\- -- 1 (J _x ) (- J _;:-,-- ; -:: -) ( I,") L-- - ✓ n, 0 Hence J cm ( -J, ,.,., ) (-x) {_ /J,7 ) }__ _/ " ( -J, ,,,._) 7;-_,,,,.) ( .J / .2x ) showing it to be of the Bessel type. here is: - .2 J {-0, :i sJ The associate function d (; It may be observed her e that thus, formally the expansions in Geg enbauer polynomials and the Neumann series of the first kind are, aside from trivial factors, equivalent in the light of our theory. The various expansions in generalized Legendre polynomials are of course related to the expansions in Jacobi polynomials mentioned above. Among the functions expressible in terms of the confluent hypergeometric function: - -i-) we shall illustrate our theory for t h e following : The Hermite polynomial: ..3.:. l. ( X. ) 'r The parabolic cylinder function: J. _3 _ /J rH (-x ) -- c "' :::- t fl,,,, (x) , ...!.. 'I - -;,- 0/,, X ~ ;;·, ,,! (x ) .!.,,.,., fx: ) I - ,,___ -1- The Sonine polynomial: --i f---,,) I e T,,,,.,,,.,, (x ) = 0/rf,,,,,,.,..,,.,.,,J ;>( I ;;, X' \Ill /1'1-;- - I ~ ,,,,..,_ The Laguerre polynomial: I I - ot For the Hermite polynomial we have: \ ..,!.__/ (-,,,,,,;;_") ( I ' J ) X , -r ~··- / °' Hence the associate function is: .-y L ?/ J ._/ ;,_ ~ { I, .,.. ) ?' The relation between the Hermite polynomial and the parabolic cylinder function points the way to obtain expansions in terms of the le.tter. I (x) ~ e <-1.X as a series r; ex,.,, /J_,., (x) --( 1' ,;; ) -- If we wish to expand ' we consider the fwi.ction ;;:: ,._ hx e '-I where Then we have ;1/,.,,, ( h ) fl,,,, (A) C (I, ,,-;,z) so that e This device is one which may be profitably applied often to similar situations. The Sonine polynomial: = (I,,,-,,,, ) (I,;;,,) A;; =:::. (;;., -;- /-11().x) "" I I, ; ) so the associate function is: = r -1""(1,,,,-.,)(l,r>t)), r- (/J-'">t _.,.. 1/ ,,,-x .": 7") ( 1,---,.-) nr -I" .-,;;., The Laguerre polynomial: X "' L~"'-;;,.,,(- + ) ,,, " - 1 __ ,,., (o<-t-,,.,,,'!'J - ) (I,,,,,, ) r (-rn ~ ni X I -o<.- .J.a,--rl.,- xi ) I = (-),.,,, L ,.,, I ( 0<, X) so the associate function is: (/,;,,) 1 - /''(d..-r·;,,,, "'1) -r Just to mention one or two further expansions to illustrate the method, we mention first: (o(, ( ,G r I '. ,,,,,-,-,-) (oL /n t ,,.. ) ~, ,,,,-,.,,-) X '>t-,..,.... -- ( I, -r ) (c1., - I /2) (c<"' I /2 ) - - - -- ·- ---- ( 13, ,{ ) which has the associate function: - ( o(I , nt - $ ) { o<_, I /' 1 - J") Again, the Bessel function of the third order introduced by P. Humbert, 4 illustrates the theory nicely. Con- sider expansion in terms of analogous to the Neumann series of the first kind. finition: 4 P. Humbert, C.R., CXC (1930) 59. By de- ,.j 7 ( o( -+ .: V /YI -r I, /3 -f ji..3 0 1 + / • .,I 3-~) a... 7 { I , r) F (or .,. } ,,,., -, , ) ,s These examples will suffice to illustrate the treatment of expansions in various hypergeometr1c functions. Further cases can be given at pleasure. 4. Non-hypergeometric Series The Hermite type expansions considered above lead directly to one elegant application of our theory to functions of a more general nature than the hypergeometric series: The /polynomials of L. M. Milne-Thompson5 are defined by the generating function: It can be shown that if of degree n. v~rJ) f o,, ~(-.!{:><) is a polynomial in x The properties of the polynomials have been con- sidered in detail. 5 i In addition to including as special cases L. M. Milne-Thompson, Proc. London Math. Soc., (2) XXXV, (1933). 'f 0 the generalized Hermite polynomials mentioned above, they also include the well-known Bernoulli and Euler polynomials. ~ (>r. ) makes (-.I) The form of the generating function for the application of our theory immediate. ,,, We have - ,7 (').) ,., C 7 i1 ( )- ) Hence if we introduce the new function -y_ (.j) ( ,.,, t { and 1: () ) are associate and we can proceed to ex_,0 then ) ). (/) ?(_ ) pand any arbitrary function in a series of either function. Other general types of functions, such as powers of polynomials and any well-defined sequence, may be treated since we know from our previous theorems methods whereby the associate function may be constructed, although not always in such convenient form. 't I Chapter III - Applications Functions of Two Variables 1. Hypergeometric Series As in the case with single variable functions, the hypergeometric series are most elegantly suited to illustrate the theory. A double infinite series£ a,..,_,,,,_ is said to be hypergeometric in case the two ratios: 0-,..,,,,,,.,/0.,,,,.,,,,,,. and 0-,__,,,,, rational functions of m and n. are both In such a case it can be shown, as in the one variable case, that ratios can be expressed: a.,-,m .,...,, ff{ a,..,., , ,,,. 0-,..., ,,,,., ' a,7'>1,/)f -- ;/- ( a: ( o( A' - ..-1'.. h 71 -1. :; ' lo. />7 , :r (?A ff( + O.A•/>'1 T r '.:", l~,,_.,, b_./1'/ a..,;) , c,., ) .,;: -- (,,.,,., fl) /2J X ( ?(,· -t- c:.- -""1 --; e_, /11 , C'_.) ( M-f- ( D(A -t- a;,,,.,., -r / I) where aA, Iv,,,, c,,,~ e.,: are positive, negative integers) or zero. The general term of the series can then be written so we have finally the complete expression: CA.,. • e· • ..A- I This latter notation we suggest as being completely descriptive, compact, and readily extensible to functions of many variables. The convergence properties have been considered in detail by Horn. 1 We omit them as unessential here. 1 J. Horn, Uber die Convergens der Hypergeometrisch Reihen zweier und dreier VerAnderlichen, Math. Annalen, XXXIV, (1889), 544. The analogue of the Barnes asymptotic expansion the formula has not been given due to the uncertainty as to Aexact meaning of asymptotic expansions of two variables. It is pos- sible, however, to proceed by a method of double complex integration extending the Barnes integrals to obtain certain relations between series of an asymptotic appearance. These relations furthermore give immediately the formulae for reversion of series in case any of the series occurring are merely polynomials. As these are the only applications of the asymptotic formulae we need and since they may be obtained directly, we shall omit further discussion. 2. Expansions in Products As with the one-variable functions, we refer our various expansions back to a few basic sets. Here, however, we shall be satisfied to illustrate with use of only combinations of the one-variable sets. Others of a more complicated nature not derivable from these doubtless can be exhibited. The simplest two-variable expansions are those in products of one-variable functions: For example the Neumann Series of Bessel functions can be extended. then _j ( -1: )() ( / 1 ~ ) Thus, if: - "'- (i ;:) { I, m) JJ+.,.,,, { x ) J ,<A-_,. ,,,, 7' v where: o( ""'-"1 m ;=,-~--;7;-t« 1,,,.,, ,) where we have here made use of the associate functions for the Bessel functions previously obtained and applied Theorem 2.11. Again, we can have expansions in mixed products. Thus for f(x,y) above, we have: II~ (x) T,;,rt(~) ( / , /M'C- ) ( I, rn,, ) where: ( -Jr>t ( I, /'3,,,,,.,,,.,, ( I, /11 ) /, ) ,,,.... ' J giving the expansions in products of Hermite and Sonine polynomials. Again we have made use of the associate functions given before. Such examples could be multiplied indefinitely, but these will suffice to illustrate the principle involved. Among the special functions which might be treated in this manner the following pseudo-addition formulae are very interesting: /2 - -~ ~ (x f _, ) 7) where: - - r _--- 1 -~ / JJ ,,_ /2 ( J J (,,, f -r-7J )) )/ =,"f (1 -: =3) r_-r, ,... ; ( _ $ , .,,, ) r foe__, __,_ , , J r ( /3 + s_: , J r- -J, __,. ,_ ., ) 7 ( 0------:-----::~--~--; : T);' _; -j ) where: -==-- - I' J ) ;,,_,,.., -) _J ,,,,,,, ' ,,, where: )I ,,,,.,,,.. Again such examples could be extended at will, but these will suffice. 3. Expansions in Two-Variable HyPergeometric Functions In addition to expansions in products, there can be derived expansions in distinctly two-variable polynomials and functions by application of Theorem 2.12. P. Humbert 2 introduced the set of confluent hypergeometric functions of two variables, special cases of which can be variously interpreted as extensions of the Sonine and Jacobi polynomials as well as the Bessel functions. Several of these give particularly compact expansions of two-variable functions. 2 P. Humbert, Proc. Roy. Soc. Edin., XLI, (1921), 73. 't 6 The polynomial: - r; ✓\ ~ ) _ _ /7 <-=-~, ,,._) X (1,--r) ( -a-,, ( J) <5 ~ I . .s) ~ cl l5 ---,- I ~as the associate function: (-) (if', r -f s +,..,,., -- - - --- --~- - ( i, r) (1,0) So that {;, rht) ( l,"11) The polynomial: _/ ( -/"1 , -/1"1 ; ~ / X d / -:;_ ) X J. - (1,-r-)(J,u) (-/Y/,s) (-,..,,r) X ( 'if, 0 ) ( I, -1" - 6) ( I, r) -r ;t 6 { {I, ,,- - Hence the associate function is: ( I , ,,- - ' (-;' ---,-, ,S i ,,,_., - ,,, ) ( -1-.,.,) ( I, '7" l O, ..I 7_-n__~ (/, ,S) ,S The polynomial: ( - /J'11 , - ;-V{ _J ( 'a', -r ( I, -r ) r .s ) Mence the associate function is: 1-)""'""f (/3 f/Y'I,,,,,.,, ) \ - £___./ ,.,.., .j I { I , cl ) I, ..! ) .j ) ( if, .j ) 7' / T,he semi-polynomial .: (o!-,,,,.,, -r--ts) { · ,,-,,.,,--,..) ( o' / ,.,.. ) ( 0 I t· ;;_ ,,, .,. I I ,S ) { ol.' --,.- r/YT .,. .s ) ( - _.,,,.,, ...,. ) X - -- ~-~- ~ { 0 , ,.,,--) (c5 1 -f - ,J,.,.,,,, -rl, <.S) I Hence the associate function is: The semi-polynomial:: l""'>'n-r4 0( d. .,.. I, u;, I, 0 ,· - 0, I ( c(-tr>-z f / 1 .5) (l,-r) ( /,.s) 1, / - {1 1 -r-)(1,.sj Hence the associate function is: ( oi, ~ -;"! J~ d)_ ( KJ L ~J_ <;{'; :!- ) ( ex I The Appel hypergeometric functions of two variables will give a similar extended list, as well as more complicated cases of the hypergeometric functions. As remarked before, further basic pairs of associate functions doubtlessly exist which could be used to give additional applications of the theory. 4. Non-hypergeometric Types As in the case of one-variable functions, the taeory is applicable to expansions in functions of a more general nature still than those considered above. Let us consider the polynomials ~,ni { ;,;., ) . ) defined by the generating function: /, hr( ,t_ 4 ( I , ,n,z) ( I, "'t ) As in the one-variable case, the l ,, , ( x, ~) can be shown in _'f '1_ general to be polynomials in x and y of degree m,n. The associate function is directly deducible si nce: I<,,. (3-~_i) / I f, ~) --- - ( 1,,,,,,,) (1,4r) t'/Yl., '? ~ whence: -,f(""l) e - --·--· ~--~ - / / fr, /J'J J f,,,,, ,,,,( x, }- ) is the associate of As an example of this, we mention the two-variable Hermite polynomials //,,..,_ ✓><- f-x, ;. ) defined by the expansion: h (a x + b; } -1- -h ( /;;o< + c ;'/. ) h ,,,,,,, - ~ ( a h .,_ r .z 6 Ii J -t c / ) ,C""- -h r-::~ ) (1, /}'(_) If we write r1- -;J I )( > C f-= /1a +lab, ~ = h b 1-J c. \---, :=::- L _ / )I/>Of , ,., ( x, J-_) r1,,,.,,,, J !1, 0 J h ,,,,.,). then ± fa_ /,,_t-2h /2 i-1- cJ."') rlf c_ ,.,.,,,,0. so that the fun ct ion a s sociate to fl,,.,, "1 ( ?<,) ) is where The several two-variable extensions of the Bernoulli and Euler polynomials can be similarly treated, using t he generating function. 3 Ch. Hermite, Oeuvres, II, 293-308. One could proceed to build up any number of examples in expansions of functions of three or more variables. The results so obtained would be in the large obvious generalizations of those given above. The many variable problems will introduce no complications in theory and only the manipulation of the al~ebra will be cause for any difficulties. 5 I Part II The Number of Representations Function for Binary Quadratic Forms Table of Contents Introduction 53 Chapter I Discriminants with a Single Class to a Genus 58 Reau cti on Formulae 72 Chapter II Chapter III The Number of Representations Function 93 Appendix A Tables of Reduced Forms and Characters 103 Appendix B Numerical Examples 11? Introduction The problem of finding the number of representations of an arbitrary integer by a given binary quadratic form has yet to be solved in complete generality. In the two centuries that have followed the first general investigation by J. L. 1 Lagrange ) of any part of the problem, the investigations have proceeded in two directions. A great number of specific forms have been considered individually for which more or less general solutions have been given. Again, certain general investigations have reduced the problem to more simple and direct questions. The early investigations of Dirichlet 2 ) and more recently those of Pa11 3 ) are of this nature. In the discussion to follow we offer as a contribution to the general problem, the general explicit expressions for the number of representations function for all forms whose discriminant is such that there is a single class in each genus. For the original suggestion of this problem and for continued advice and encouragement, the author wishes to express his deep appreciation to Professor E.T. Bell. 1) J. L. Lagrange, Recherches a'Arithmetique, Oeuvres t III p 693 - ?85. 2) G. L. Dirichlet, Zahlentheorie, ed. 4 (1894), p 229 3) G. Pall, Mathematische Zeitschrift, v 36, p 321-343 (1932) 0 7 In our terminology and notation we shall follow closely those of Dickson used in his -&eduction to the Theory of Numbers. We are concerned with binary quadratic forms nated by [u,h , c] , of discriminant - Ll=h' - ta.c , ax'+ bx;; +- c,/, desig- and shall examine the form of the number of representations function this being the number of solutions in integers, x and y, of If a, b, and c have the common factor A, 0 ..,,. according as mis or is not divisible by exists, the form is primitive. If A. If no such factor L.l < o, N l,,,,,, ~ 17- :x.' t bx) -re;/] is :fini te, and the form is called indefinite. When .LI ? o, in- a..-x.~-rhA -:; -,- c-:;.."- is either always greater than zero, or always less than zero, i.e., either positive definite or negative definite, there is no loss in generality if we restrict ourselves to positive definite, primitive forms. All forms hereinafter mentioned will be assumed as such. Let J be a transformation whose determinant has unit absolute value, which replaces the form C ( ,__ . Two such forms are called equivalent and all equivalent forms are said to form a class. both cases. Clearly the discriminant is the same in Because of the restriction on the transformation, we have further .:: In view of this it 1s convenient to choose one form representation of the class, the reduced form. if -a_ d.s: a . / c.2a. with 620 A form ,.f c. ::a. l a, h, c J is reduced It is unique. We shall base our investigation on the following theorem of Dirichlet: Theorem 1.1 Let m be positive and prime to L\ . The number of representations of m by all the reduced forms of discriminant - LI is "'"'Lt (-IJ/ 1 ), "I/= where w ::: ;;, .ii ;, 'I, w ~ ti LJ.:: 'I, "" = t LJ::: 3 and (-JJ /'i ) is Kronecker's symbol. If there existed invariants whereby we could determine whether a certain integer could be represented by a certain reduced form, we could proceed with the use of them and Theorem 1.1 to obtain the complete solution of our problem. The only known quantities associated with a particular form, invariant in that they are equal for all integers represented by said form, separate the forms given discriminant into genera which may or may not coincide with the several classes. These invariants, the so-called characters, are defined by the following: Theorem 1,2 If p_ p_ . v are the distinct odd prime factors of ~, then ,I ' (,,, I 1~) has the same value for all integers n prime to represented by a form La..;,, c J of discriminant - LI . -7. Ll , When Ll. is even, Ll-=-l"JJ, the same is true of -f (,,,., ~,) 5 = (-1} ;j 0 - 0 c = (-!) ,j ;J 3 0 ;J ,0 V,,,, -,) ~ E / = 0 0,. ,3 ( "',d ,,;) 0 ,- ;;; {rndd ({) OY 6 { mod I{) We shall use the notation for a certain form, excluding acters. The notations ,,,,c,.,. of the characters ~ t L;__ for the set of characters if both S and € are char- and ( (,-,.,J are to represent the value (, form or for the form representing m. All forms of a given discriminant whose characters have the same value are said to form a genus. Since equivalent forms represent the same numbers, all forms in the same class are in the same genus. When there is a single class in each genus we may proceed using these characters to give the explic~t form for the number of representations function for integers m prime to [ a., h,c] represents the integer s, Theorem 1.2 states :i L1 • If c~ f,,,,,, J -= c~ (.s) as a necessary condition that m be represented at all by r~, h,c ]. Since we a£sume a single class in each genus, each reduced form has different values for the characters. we obtain Hence by Theorem 1.1 L.··' Theorem 1,3. Let [a,6,c ] - LI<-'/ be a form of discriminant~,such that there is a single class of forms in each genus, with character If m is an integer prime to + [, A • :2 Ll {,~ {a) ( (.,.,.,., }J J -~{- iJ / ,y ) -'( ; ..,.,, Before we proceed to derive reduction formulae for the number of representations of integers not prime to double the discriminant so as to apply Theorem 1.3 to the complete solution of our problem, we shall make some general remarks on the set of discriminants of forms of which there is a single class in each genus. Since we are restricting ourselves to forms of sµch a discriminant, these results may prove useful. 0 Chapter I. Discriminants with a Single Class to a Genus It is known that the number of genera is equal to one half the number of characters, discriminant. C,.: , for forms of a particular Further, it is well known that the number of classes is the same in each genus. Hence the question of the number of classes in the genus is closely related to the classical class-number problem. Following the analytic methods used in the study of this latter, s. Chowla has shown recently that the number of discriminate such that each gems has . a specified number of classes is finite. Although this gives a certain qual- itative satisfaction, yet it gives absolutely no light on the question as to just what particular discriminants have the specified number of classes in the genus. Rather extensive specific lists of this nature have been given and all evidence points toward the conclusion that many are complete. In particular, it is an open question whether the known lists of discriminants with a single class in each genus are complete, although if there are others, they certainly must be very large. In this connection we shall extend the known tests for there being a single class in each genus, and also give some conclusions regarding the elimination of discriminants satisfying certain congruences and also those whose factors are of a peculiar nature. 7 As a basis for our investigations we use the Theorem 2.1 A necessary and sufficient condition that there be more than a single class of forms of a given discriminant in each genus is that there exist a reduced form of said discriminant with [ a., h, c ] c / a >h > o The sufficiency follows directly, since if [ a., h, c. J ,, c ? a > b 7 0 is a reduced form, [ ~, -h, c. J is also reduced, represents the same integers and is hence in the same genus. The necessity requires a detailed examination of several cases and as we do not require it now, we postpone its proof. As it stands, this is scarcely convenient as a method to test certain discriminants, inasmuch as it requires a more or less exhaustive consideration of all reduced forms. It is certain in this respect, however, that having definitely listed all reduced forms, one can tell by immediate inspection whether there is more than a single class of forms in the genus. On this account it is essentially the final test that is always required. However, in order to make preliminary steps simpler, we derive from Theorem 2.1 several direct tests. Using the sufficiency in Theorem 2.1, Dickson obtains tests for odd dis crlminants which we state for reference. then for positive integers, k, or, also, T. f -I ~ :r- JI" ,' 0 T J a I a,. J 'I • ,J , etc. ~ Write ~ =[LJ" (;i' ,J]J'I , must be a prime p, its square p 2 , / .:,.= 7;o r>r 7'-- ; Here as elsewhere we assume all b U primes to be positive. The corresponding tests for even discriminants are similarly stated. Write .s~ :: fLJ + l'. k., the 5"" We list below for successive integers, which are eliminated by the corresponding reduced forms satisfying the sufficient conditions of Theorem 2.1 or its equivalent interchanging a and c, and summarize the resulting necessary values for For 5-it • sary values of we give the details of deducing these neces- 5'3 that there be a single class in each genus. Sk We let (r, s,m) denote argument in the other cases is quite similar. the greatest common divisor of r, s, and m s, -- r,5 s, -- J_ == ,)_ S, 4 r I :> j ;) / ( ---;' 5 J J. ) .) /4 .J = ) !i I [ i( s :; [ Y, sJ :2£-a.,._,] J ' ' I 0 , 'I, 7 ] Hence 81 must be a prime p, p2' 2p, 23, or 24. / · <.) s). -y ~ { 7', J s J' s = J.. /4 l. '-I J ;,- / .I r>1 /YI / 'i j 'I) -= I )._ 7 Hence S2 must be a prime p, p2, 2p, 4p, 24 (2) 53 -- (3) S-:i -- f- /'1 (4) s :: 3,:2_4 (5) Sa ( 6) ..5-J s/11 J f L J .J J or 26. C, J J [S, 'I, , -;z-1} [ f-, .:? .I /Y{-;;_ /4 _/ J / [ z( :,; , 3 ,) /4 / '-I ) A-.3 - I ] [ £5-', ;,? ' :J. /2 ? 'I I J ,, I ;_;, ;} 'J /2-J -t ] /'1 ;, C .) ? / I? I 5 ' 2 ' [ r ·, y = )_ le = ) 'J :k J l 3, ,) ' 7' I] S2 === 2. ,5.J -- y , j .., ~,0-1 L J::l, I ~ ' ;J_ fo - 6L V, 'I, t.3 I ] I ? c:' 1, I{' s2 --- 1 (1) s] L' , f, [ J/ .I / { r; .5, ( ) .::.. I Co I oc!J ;, ,.J .I .I The J k-J ;) I (7) :: 5<1 (8) Sa -- (9) .58 f ::2 1, I :; , J A- /2 l s JA J T I J ,,, ;) ·3 [ 7, :?,:?.2] s = c3 [ ( / ; ;;;_, '-17] We consider exhaustively the possible forms which S3 may take and refer to the numbered cases above to eliminate certain forms. Let S3 ~ p.n where pis the smallest prime factor of 83, then n 2 p or n = 1. If p > 5 or p - 5, 53 is eliminated by (1) or (2) respect- ively unless n ~porn~ 1. If p ~ 3, let n = q.m where q is the smallest prime factor of n > 3 if such exists. Then according as q ) 5 or q = 5, we eliminate by (1) or (2) unless m = 1. If no such q exists then S3 = 2k.3J, and when k > 2, j > 1 we use (1). For k = 2 consider (3); k = 1, (6) and (8); k == O, (5); j = 1, (4); j = O, (7) and (9) . If p =- 2 let n ~ Q.m where Q is the smallest odd prime factor of n if such exists. Then according as Q > 5 or Q = 5 we eliminate by (1) or (2) unless m = 1, 2, or 3. we use (3). If m ~ 3 proceed as when p - 3. If m = 2, If no such Q exists, 53 = 2k and we apply (5) • 2 2P, 3p, 6p, 2, 4 3, 3 3, 4 23 .3, or 2 •33• Hence 63 must be a prime p, p, s" -- )' . -5 S1 -- 7 . /?1 s.,. - 5. /YI S" '/ ~ 3/>1 -· • <; y --- - :2_ A / 54 -- 211 -- 210 , 84 [7. S4 --- 29 , l nz-1] Is, ::> />f-JJ 13, :J, ff/ -s] ,/2 ;, I I ' [ ; 1 , rs 'I, 9 ;;_ ] 'I, 11] r' !./ Hence s4 must be a prime p, p2, 2P, 4p, 8p, 2.3 2 , 3.7, 25 , 26, 27, or 2~ 5·. J ..,-. s / > /0 Y,? ,5 {r, s ~.r -- - 7 ,0 Ss -- J /'Y( -n 5 .,- ::-.:::: Z5 /YI ...,,, ocfJ / 7 ,, ~s ·==- 'I /Y/ ' '1 o c-lJ 7 7 I £s ---- S- :Jr,__ Sr ==-- ,}_ /2 £§' = '11 ~.r =- s [7, '-l ,~ -3] /0 ;, 13 , J, [ / ;,, (, />'I [ '1,ol, / - b' ] / '7 - ~J /1-1-( ] ,3 / Es, A /4 > 3 >S ' s- I. ) , = r [ -r, Io, s] I ;, II A - /2. :) , ..) ,,.,,,, 1 ) /0 )-= :J, S , .7A - -:1_3 J L!f , ;) f .2 6-, / A )cl ;J. 1 t - J - 3] ,;J O / ,) , s h - -f- J J L [;; s-, :2 0, S /2 - -r ,J ] ;i_ u [ '6, 'f, 13 ] Hence s 5 must be a prime p, p 2 , 2p, 5p, lOp, 2 4 , 26, 52 , 2 3 .5, 2.53, 2.3 2 , 3.7, 23 .3, 3 3 , 3.11, or 22.7. To prove the necessity in Theorem 2.1 we must exhibit all the reduced forms occurring when there is a single class in each genus and show that when other reduced forms arise, giving more than a single class to a genus, they must be of the form stated. In addition to accomplishing this purpose, we will have available for later reference the explicit reduced forms for the disciminants we shall be considering. The original definition of reduced forms allows, in addition to forms Ca.., h, C J u ,- C a. ? )' - /2 :1 0 only the three possibilities [ a, lo [a.. , d, c ] Ci ] For these we have respectively L] ::: 'la c / J LI - f ,Jo.. - /4 ) (Ja.-f 1:,) 'p b 0 ff Restricting ourselves to these three types, we shall show that except in some cases where it is already known, there are more than a single class in each genus, and that all these possibilities give exactly the same number of reduced forms as there are genera. Hence when there is only a single class in each genus they are all the reduced forms and further when there are two or more reduced forms to a genus, among them must be at least one satisfying the conditions of Theorem 2.1. Let the discriminant be odd, then Ll : .: A"f with ;t 5"1 ; from L! cc -1 (.,.,.,,,J t) There can be no reduced forms Ia, cJ, c) ~ ~ - 1 r~,d 1/). we have since ;t L1 we find the single solution of Ll cc u{'ic_ - a).:: ✓-'IA and 11' ~ / 4. --t./4(; o (-m 0 is odd. Write J,;). For [a, a, c] satisfying A+-"<] with the restriction to give the form [,>--, 11, --;;- Similarly, considering [a., h, u J, we solve c Ju. >o,,, --0.l.J). L'l" ( ;; o.. - h) (:;a. -d) under the conditions a.> h > o to obtain the form [ ~;-'(, ~~ , , ,."l___}::!] with "'1 5 "J ,, • Now, if ~ is a prime or the power of a prime, the only primitive form among these is obtained when tl-=-;,. "'-1::: LI , i.e., the principal form [ If /l I , I contains, say k, distinct prime factors, we obtain primi- tive reduced forms . in addition to the principal form by separating LJ into all possible co-prime pairs A and kt and using 0-Y according as .,A1 -2 J;,. or ..,1 5 J 71. This gives in all c2 ;,_, reduced forms, being the same as the number of genera. The following, relating to oa.d discriminants stated by Dickson, is important for later consideration: Theorem 2.2 When Li -:- -1 ( ,.,,, 1 25 ) th ere is more than a single class in each genus 0 unless L1 ::._· I L et L'..l or IS, = v A - ; , then all but k . . : 1 or 2, i.e . , ;;: = 2 i. and Dickson's tests exclude LI =- 7 o ,- I S The kno~m odd discriminants giving a single class in each genus are listed with their prime factors: 435 3 = 3 .5. 29 7 1 15 = 5.2 3 4 83 = 3 .7. 23 11 12 3 = 3 . 41 555 = 3 . 5 . 37 1 47 = 3 . 72 59!::J = 5. 7.17 627 = 3 .11.19 715 = 5.11.13 15 = 5. 3 1 63 18 7 35 = 5 . 7 = 11.17 1 9 5 = 3 .5.1 3 235 = 5 .47 1155 = 3 .5.7.11 51 = 3.17 267 = 3 . G9 1435 = 5.7.41 67 315 = 3 2 .5.7 1 9 95 = 3. 5 . 7 .19 40 3 = 1 3 . 31 300 3 = 3 . 7.11. 13 427 = 7. 61 3315 = 3.5.13.17 75 = 3 . 5 ? , ;J '7 ....> Let the discriminants be even and fJ = a (mod 'I), so with A s.,y • S LI 3 'I ( = I It) 0 is the single even character. Reduced forms [ a, o, G J will be £,1, C:-"t J. then , Write Lo = For primitive so/,,. f,071 [a.,o.,c] we find the single set, of .il. =- o.{'lc-o..J -=-'l,l'i satisfying and to give the form [,n,-< )., ;..;_-'f] ;,.,_ i ./4f Similarly considering [iJ. lo, a_J , we solve I under the conditions a. ;, Iv ) 0 with ./4f ~ J ;\. • If lJ c_ with the restriction Ll ::. 'I ;1 "1 (:2a.-!o)(,1a..-,-b) to obtain the form [ ~ -:,__"-l 1 >o..;,o ,A,{ _2 J)... ='1.:\/f.f ,~At] .-<t - ;1 / is a prime or the power of a prime, the only primitive forms are obtained when ;\. =1 and _,if== f , give the principle form When d contains k distinct prime factors, we obtain primitive reduced forms in addition to these two forms by separating all possible co-prime pairs a,., 11. and .-A--/ LYi into and using d according as A1 -2 3 11 or A ~ ,1 :\ • This gives in all zk-- reduced forms, being the same as the number of genera. When Ll ~ ~ ( 7>7o j /() 1 the single even character. D 5.:: ~ (rn OJ )") so el ther ~ ;:\--<t [,1.,0, .:>"1) : . (;,,,,_ - lo) Oo 1 ;, ) or ~ 6- is We write ~==~>~and observe that no reduced forms ["', a, c] or [a, Iv, a..] exist by examining and e respectively. g ;I -1 _.,. a. ( t c -a..) This leaves reduced forms and [,,2:-\ , 0,_,,,_,] or [.,,,,,0,.:2.,1] according as "L\<-0 or J.>-?-c/ obtained from [q, o, G J with c 7 a ,, o When LiIv is prime or the • power of a prime the two reduced forms are "'- When 71 j Ll/v contains k distinct prime fa,ctors, we obtain primitive reduced forms in addition to these two forms by separating all possible co-prime pairs [ )._/ ;\ () and _, ,-<-f L1/v into and using .L,,l{J a,-, cl according as ;;. 11 <A or ; _ ;;i ? -"f • This gives in all~ reduced forms, being the same as the number of genera. When LI -= n. (-,,,"J 1t )/ o.::: 'l-/2 r 3 so that s, = 'I ( le -u), there are then more than a single class of forms in each genus for all such discriminants except LI = 1~, J S--: co . The reduced forms for these cases are given in the tables. When Ll -= a ( '7n o J 1,), D = :/1 ( 2/,. 1- 1 J, h 2- 'J. • For /. 2. 'I-/ s,, = 1.t [ ;z. h•~(., -t. -u >_, , ] so there are no discriminants with a single class in each genus except when and ./4. -= o, 1 0 , / 7 • For /. , ='J A = o, 1, :1 / 7 excluding all cases except , hence The reduced forms for • these cases are given in the tables. The case treatment. Write l'.'.l /, = J , that is Here D ~ o (,,.,,,J ,,,, ) =- J ~ ,"\ Af with and [,,_,_, 0/ 6 ) 1 ) or ;\ '5/f. ii ~ J ~ (-,,, J {'I), requires separate so that 0 S and € are the characters. Reduced forms [a , o, c) will be [A, o, fr'A-t J [ ff' .7\, 0 / j1 J according as -"1 ~ is' " or _,/4./_ ? r:, i1 • To find the primitive reduced form [ a., Ci, c J, we solve 3 2. ;t .A-t - o. ( <1 c - ci. ) l, to obtain the roots C / a.. = 1/ A I c = ~ -1- A ✓- a= 8' ).., c: -==__,,,u + J. ) satisfying >a >O and ).~ , whence the forms [ 'I 1\ 'I)./ ~-t)] a,,,J ['i>'A , 5") i/-t J. tl] with the restrictions 'Y'1 > 3 ). and/-' > (). respectively. J Similarly, for [a, 6, a. the equation 3 .J. = ( ::2 a.. - lo) ( 2.. a.. + ,b J ) _/-'( under the condition a> h > O has the roots a= J. A +.,,LA,, U- T ~/ h = ~ - c<.11.; a,-=). r- ~,, lo =:V-, - 'I); a.=,)>- i_/A, 1i = ¥). -<!<{ whence the reduced forms 'l_,,u - ~ A, A -f ~ 1, [.}_ ) -1;/A / ~ - 'I ) I ;,;_ A !:,,"" ] J A0 [;; / 1/). - .,(~.) ,2 A !,A J under the respective conditions J_., ,-,<-1 -< J AI ~) ~ , t ) / __,,u < J. ;t If L\ is a prime or the power of a prime, the only primitive forms are obtained when LI/ = /(3 < There are two or four or /' > I , = 1 according as / ). = I,, / J"'~ ] I / 0 2 /A t../, J~ + I ] ;> I / [ /A, a_, or o r If bjJ.. has k distinct prime factors, we obtain primitive reduced L'.l/3 ~ forms in addition to these four by separating possible co-prime pairs [). [ .,,M, o, &" /\ ] [ '-/ r ). / '-/ ) . 1 __,,LA 1 I A and / c}/ -< ~ 1\ / and using the four forms ~ ] o ,- [ ~ t- ;\ ] / 0 ) 3 A , o r [ 1\. &"' /\ Cl, /-'I f- ~ , 'fA - J ;\ ?J 1\ ' ~,,{_ I . / ' + J. /\ J I /1 / ( ) , 0 ,- [ ;2_ 1\ -f__,A <' r This gives in all J number of genera. ~+/ into all [ d_ /1 f-,/'1 1 I ;2_,,,M - 'I A - ~ I ] y) I 1 / / ~ ). 1 1\ -f ;) ;)_ /\ 1\ ~ ] , ~ <3A 1/ ] / J_ A 0 < ( 1\ -/;fa 1, , /A < ;)_ ;\ reduced forms, being the same as the Thus we have exhaustively considered all types of discriminants and have shown by actual exhibition sufficient reduced forms to account for each genus in every case where it is not shown definitely by the sufficiencies of Theorem 2.1 that there are more than a single class of forms to the genus, which reduced forms include all those not covered by the conditions of Theorem 2.1, thereby proving the necessity of Theorem 2.1. For reference we list the known even discriminants with a single class in each genus according to the above division by congruences. When L\ - 4 (mod 16), 4 = 4.1 20 = 4.5 36 = 4.3.3 52 = 4.13 84 = 4. 3.7 100 = 4.5.5 132 = 4. 3.11 When L1 = J 148 = 4. 37 180 = 4.3.3.5 228 = 4. 3.19 340 = 4.5.17 372 = 4. 3. 31 420 = 4.3.5.7 532 = 4.?.19 8 (mod 16) and has the values: 660 = 4.3.5.11 708 = 4.3.59 1012 = 4.11.23 1092 = 4.3.7.13 1380 = 4. 3.5.23 1428 = 4.3.?.l? 1540 = 4.5.7.11 5460 = 4.3.5.7.13 Ll - • 32 (mod 64), we may set = 8m and 32m respectively, where m takes the values: 1 3 5 9 = 3.3 When Ll 11 15 = 3.5 21 = 3.7 29 35 = 5.7 39 = 3.13 51 = 3.17 65 = 5.13 95 = 5.19 105 = 3.5.7 115 = 5.23 231 = 3.7.11 Ll ::::: 0 (mod 16), there are only the eight values: /\ = 16, 48, 112, 240; /\_ = 64, 192, 448, 960 Finally, when LI Ll 12 (mod 16), there remain the cases: = 12, 28, 60 b 'i With these tables and the reduced forms given above in general form, we can write down explicitly the reduced forms for any known discriminant with a single class in each genus. In addition to an explicit knowledge of the single reduced form in each genus and the factors of discriminant in our later discussion, it is necessary to treat separately the cases where the discriminant contains the square of any odd prime as a factor. In this connection we state: Theorem 2.3 If the discriminant contains as a factor the square of any odd prime, t here is more than a single class of forms in each genus except for the ten known cases LI = 36, ?2, 100, 180, 288; LI = 2?, ?5, In proof, let 99, 14?, 315. be even and contain as a factor the square of the positive odd prime q. Write Ll = 4q 2 (r.s - 1). Then j_\ either r.s = 2hqk, or we may chooser as a positive odd prime distinct from q. In the latter case we consider the primitive form [r,2/cr - q/, sq 2 + c(cr - 2q)], where c ~ 0 1s chosen so that q = er + b, O < b < r/2. The conditions of Theorem 2.1 are clearly satisfied when r < sq 2 + c(cr 2q). If r > sq 2 + c(cr - 2q) then r > q(sq - 2c), and q < r, if sq - 2c > o. by choice of c. But q > er - r/2 Hence sq> scr - sr/2 = scr - 2c - sr/2 + 2c > 2c, c > 1, since ecr - 2c - sr/2 > 0 follows from r ~ 3 > 4c/s(2c 1); and sq~ 3 > 2 = 2c, c = 1. If q < r/2, c = 0 and 2/cr - q I = 2q < sq 2 = sq 2 + c(cr - 2q). If r/2 < q < r, c = 1, r < 2q < sq 2 , 7 0 2/cr - q/= 2r - 2q < sq2 + r - 2q = sq2 + c (er - 2q). Hence the conditions of Theorem 2.1 are always sattsfied. If = 4q 2 (2hqk -I), then [q 2 , 4q, 2hqk + 3] satisfies jj the required conditions when q > 3, k > 1; and h > 1, k = 1. If q = 3 then s 3 = 2h3k+2 , allowing only the cases h = 1, k = l; h = 0, k = 2; h = 0, k = 1; and h = 1, k = 0; or 288, 72, and 36 respectively. Ll = 180, When h = 1, k = 1 we use [q + 4, 24, ·20 2 - 9q + 36], q > 19, and eliminate q = 5, 7, 11, 13, l?, 19 since s 3 = 9.26, S1 = 11.58, s 1 = 31.82, s 3 = 58.?3, s 2 = 31.431 respectively. S1 = 38.251, Similarly for h = 0, k = 1 we use [q + 3, 12, q 2 - 4q + 12], q > ?, eliminating q =5 and 7 by S2 = 8.15 and S1 = 5.59 respectively. Finally, if k = 0, s 1 = q 2 (2h - 1) + 1 = 4(2h- 2 q 2 + (1 - q 2 )/4), h > l; and when h = 1, S1 = q 2 + 1, 52 = q2 + 4, so that 5 divides 52 for q '= :t 1 (mod 10) and 5 divides s 1 for q ==- + 3 (mod 10), thus excluding all oases except q = 5, namely, Again, when Ll. -== - = 100. is odd and contains as a factor the square of the positive odd prime q, we may write since L1 Li 1 (mod 4). Ll = q 2 (4 rs - 1), Then either r.s = qk or we may choose r as a positive prime distinct from q. In the latter case we consider the primitive form [r, 12 er q~ sq 2 + c(cr - q)], where c > 0 is c h osen so that q = 2cr + b, 0 < b < r. of Theorem 2.1 are clearly satisfied when r If r c < sq 2 + c(cr - q). > sq 2 + c(cr - q), then r > q(sq - c), and r > q if sq - c > o. But q > 2cr - r by choice of c. - The conditions Hence sq> 2scr - sr = 2scr sr + c > c, c > 0, since 2scr - c - sr > 0 follows from 7/ r ~ 2 > c/s(2c - 1). When r > q, c = 0 so that / 2cr - q / = q < sq 2 = sq 2 + c(cr - q). Hence the conditions of Theorem 2.1 are again satisfied. If Ll = q 2 (4qk - 1), then [q2, 3q, qk + 2] satisfies the conditions of Theorem 2.1 when q > 3 and k > 1. If q = 3 then T1 = 3k+2 , and hence by Dickson's test there are allowed only k = O, 1, 2, nam e ly, = 27, 99, 315. /j When k = 1 all remaining cases are eliminated by [q + 2, q - 4, q2 Finally, when k q = 6m + e where Ll = 36m ( 3m + e) = o, namely, !l 2q + 2]. = 3q2' we write is + 1 or - 1 as the case may be, so that e + 3. Then To = 9m ( 3m + e) + 1 =- 0 (mod 7) when rn = 4e or 5e (mod ? ) ; Ti = 9m ( 3m + e) + Z :=: 0 (mod ? ) when m = 3e or 6e (mod ? ) ; T2 = 9m ( 3m + e) + 7 ==:. 0 (mod ? ) when m -:::: 0 or 2e (mod ?) ; and T 3 = 9rn(3rn + e) + 13 -'= 0 (mod 49) when m = e (mod ?) • When e = 1 Dickson's tests eliminate all cases except m = 1, namely, q = 5, since T1 = 3.7 and is allowed. When e = + 1 all cases are eliminated except rn = 1, namely, q = ?, since T3 = 49 and is allowed. final two allowable odd discriminants ~ These g ive the = 75, Ll = 147. Chapter II Reduction Formulae In stating Theorem 1.3 it was added that in order to apply it to the complete solution of the number of representations problem it would be necessary, among other things, to derive formulae whereby the number of representations for integers not prime to the discriminant could be expressed in terms of the number of representations function for integers prime to the discriminant. The results needed here have been given by Pall, who derived them and expressed them using the terminology and methods of the theory of composition of binary quadratic forms. In order that our treat- ment may present a unified completeness, we propose to deduce these theorems with some extensions by more elementary and direct methods. The essential part of our development will be based on certain directly derived properties of the quadratic characters defined in Theorem 1.2. In most cases, if I'/ l f ,--,,,, -= ax,_+ Iv x ) 1- c;; L {=, LJ J ~ 1 J = /'(' and i I Ll, then N z,,,,,,, : "' x 1 I L x J- I:, -r "- ';y • ] where K ls some constant, fa, L, c J has discriminant -Ll, and [a'.6'.c'J has a discriminant derived from .LJ by some prescribed method. Before giving theorems of this nature, we require the result on transformations of quadratic forms stated in Lemma 1 If the prim1 tive quadratic form is a prime / J ' is such that b :>. - </a_ C. 3 u. x. Cl 2 -1- (r-. 0 j /2 ::t: ;; 1 A J) / ~ c) , where ,/4 = I O r /\. ~,) /3 then there exists a transformation such that where If all of a, b, c are ==- o ('/?(()cl)), the form is not pr1mi ti ve and so not considered. /o a._ '/a_ C If a::= o {,n od J..) and C == o {-m 0 d )), 6 ::=- 0 {mo J )_) ~ 0 {,noc/ ;\} then since / reducing to the case mentioned. Hence for all other cases we may, without loss of generality, choose C,, '-f O {-rno d A~ Now if a.='. 0 {mod A) ' then 1/·= 0 (mod ✓1) and fJ E O {-n,od ✓·u so ' Hence, since /J 2- 'lac ~ o (r,-,..ocl ;\ ~ ) and If h == () (,-;, o / ) ), b ;i .": O {-mod ✓1')and a =0 {n,dd ;1~). a = O(-mod 1\~). Therefore, if either a or b=O(m e1cl;\ ), a..::=::0(7>10J C 7- a (mod A), Ak)and fo :.=:0 (7>7od,l) in which case we use the identity transformation. If a, b, c are all =/ o {-mod J.), then we must have If 0- o< 'fa ' cl..< -r + :\ h o< er r,,, o J ex --t lo~/ -= O ( Tnocf ,'\) since '/ a. c - b :..E o (nr d 1'\) , but since 0 J a. c1. Conversely, if .;l) -r 'I a c. ;!;;.. = ( J a. c1._ r /o ~) ~ -1- {'I a c - lo<)~ oz i a. b ci. 0 = ( Jc, -r c tf;; = o Jao< .,.. + h o' A is prime == o {nio cl ;\ ) h o == O {moJ ;\ ), 7 "I A is a prime > J. and since and a ;E o (muJ )), so we wish a solution of Take y ::: I ~ o< then since the ; and )_ are co-prime, o< may be determined. Then /3 == o( S -- I • /3 is fixed. So for all Thus we have a solution and these values give a transformation which reduces the case to that where a =- o {m d ,\) considered above. 0 When ) is an odd prime dividing takes two essen- tially distinct forms according as A~/ Li and /\ \/LI . These results are incorporated in the following: Theorem 3.1 If ). I LI , an odd prime, and /l a./ L1 , then /\ Iv I x 4,,,,,,, -1 a_ x ~ f h J ,_ c ;z"'- J -_: )I l /7'>1. l( T C) ·J NL C: C ;'>rl ==a + iC I X ol. / I /-!J X ), I -f-C;);t _lJ where By lemma l !V[ A:ir>-r -::ax 'tlo~ ) 1-c. ) .;.j 1V [ >- .,_ /11. -- af -7 1- c- "7 ., J l :-= 0 ( -mu d 1\)/ c- f O {-r11od ,-\ ) . Hence ~ ·-c: a (,-.,,o J )), but a. ~~==o ( rn °d tl ~) so f ~o{-,-,.,.o d ;\) . Therefore 7-= ,\;,_ , f =,"\y and IV l A = X -t /:; :t ) C) J where Q.. :::' ,/>f C( l -t L =~- )/ f 0 Also by lemma 1 _ a x L r h ~-;;._ -r c ;;t ' ] ~ 1 70 where a = l : -_cc O { 771 a j ) ), CI- 0 (nio j ;l) . Hence ~ =-0 (-rn j )). O ~ ' - 2 where i e2' Ac - J- // = 1 a-: .- -/, = fa c - ./2 " • If we put The theorem follows by suc- cessive application of these results: Theorem 3.2 il If A/iJ , where an odd prime, then 1/ o , c ' - ;oI ' , = LI/,"' ,l_ By lemma 1 // [;./\x __ ax'- r h ,'Cl = : which requires x =~ 1 ; IV [ /\ l, c:: ) I ) a +c;'-] /7'1 ? . fi! d == ?(' f 6 ,( /j f- c- ',J i] Hence so 11" ,/4 =- 1 and /1'/ = /1-?'( is prime to ), , there can be no representations and· when le - Also '/ a I C - h ;z I ==- ~ a i r <- f- bl( "77 z. a, 'I ) ' f- C h,_ A -· --- Ll / \ <,_ 7 0 Hence the theorem follows. As might be anticipated, the number of representations for 2n reauires special treatment. Before proceeding, we recall that our development is being restricted to cases with a single w class of forms to a genus. Ll -= - 1 In theorem 2.2 we sh~ed that if (mod 8), i.e., if the discriminants were odd and yet some of the coefficients of the form were even, there was more than a LI. = 7 or 15. single clas.s to a genus unless These cases we consider separately in Theorem 3. 3 If [a, b, c] has a iscriminant - il N [ 2km = ax 2 + = ( k b x y + = - + cy 2 , m odd] 1 ) N [ If [a, b, c] has discriminant - ,1 N [ 7 m = ax = + bxy + cy 2 2 ] 15 2km = ax 2 + bxy + cy 2 , m odd] = ( k + 1 ) N [ m = a' x 2 + b ' xy + c 'y 2 ] where [a',b',c'] is in the same class with [a,b,c] or not according ask is even or odd. When !.I = 7 the single reduced form is [l, 1, 2]. 2n = x 2 + xy + 2y 2 , then x(x + y) :x = O (mod 0 (mod 2) so that 2), x + y -,= 0 (mod 2), or both. replacements according to the scheme: x + y even, x 7 - Hence, if If we make x even, x --, 2x, y --, y; x + y, y -7 x + y; x and :x + y even, x __, 2x, y ~ 2y, then we observe that 17 = X N!:1/"I ,. 1 cl.)'] f- , -- /V [ -;- N !/'YI cc: X - Jv[rn ;::: ,;;_ (-x.' + j setting m : = ; ,_ ,m1 X). f- /rrl. = X r'1 X.) +- ::l J' ~ -f ,. e ve '1 :x' t-.2;;/] ;1. -f X J x;; + ~;, ')} a--,, cl odd and repeating the procedure, the result follows. When L'1 = is-the only reduced forms are [,, 1, ?"] and [.), 1, .:2J. If and X is even, or 1s even, or both are even. -x ,- -;;... even x -, ~"' ' , even, ;,: -t "'"'-,)--'; 2;;., we observe: replacements according to the scheme, even, x - -;> -J.x -d , ;.x ->,;; /V/Jm ; x a,,,J '<'. -1-- If we make ;; -X"rct;+'I; '} X eve n --1-!l[,,-n --=c2.x" 1-.:r ;;r -;ti 1/11 = J.(x_",- () If, however, 2/'r[ = ~X or ;x -fXd t- X-t-'). I Y;t 2 ) ] / f :.2J\ then is even, or both x and placements: ) 1 1-;z;r'-J ~ X a~J XJ- cc'=O ('F>,vd~) are even. even .:x -t-;t and X c-.Ve.-'1 is even, So, if we make re- x X and even,x,1_-:x:,;:7.1..), we observe: /ii [d.. r'1 == d.. X .. -f X:; __ JV [ t ;Y{ IV [ - ;V I t- ;;; I'] ~ ~ '1-x) X I;,] -t r,-t_ - X 2 f /\ ) -f ~1 ") /Y[ -- ~ ~ X ,_ I X ~ { /4 ;- cJ l , ) J' C.VC'( C'. V C )' Combining these results wilh /11 -== 2 ,✓,-,,r, rJ-n. odd, the theorem follows. '· When L1 = 3 ( ni o d '6), that is for all other odd discriminants, all the coefficients must be odd, so the reduction theorem takes the form: Theorem 3.4 If LI ===-= c = o J &') 2 + C) ::::: ., 0 Since LI :~, 3 (7>1 °d 't!) , a, b, and care odd; therefore 2. so that if -~ and ) .), />1, 0.. ·,: ··- then ,\'. or ( t ·1,1- )' h1::;. f 2 l So -x -,- }. and are such that and ; 'I:, J --j- \.) I x , - (_) ( c.; ·n,,J j ,.2 ) (--,.,.," J a<.) are both odd or both even; but if 1J are both odd and x even, so that a cl , \'.) ·r · and ) --1- ~ is even. are both even x )J Hence ,'\: :J. and is odd, x -1 ) x are x -= ; )_ f, ~ :-: ;;r . This requires and by a successive reduction the theorem follows. For even discriminants it is convenient to make a separation into two types according as: . ., --,. 79 In which connection we state: Lemma 2 A..· quadratic form [a,2b,c]has discriminant -<l, Ll if, and only if, ac = 2 ===:: 4 or 8 (mod or ac = 2b + 1 (mod 4); or equivalently if', and only if', a ==' 1 (mod 2), c =-~ 2 (mod 4); a :==: 2 (mod 4), c ~= 1 (mod 2) = 1 (mod 2), c - 1 (mod 2) or a a(a + 2b + c) ~,,. 2 When L.l - b2 4 (mod 16), ac . (mod 4) - 3 (mod 4) ac - b2 + 1 When LL - -- 1 === 2 ac 2b + 1, -- b2 - ac -:? 2 8 (mod 16), :=::: -- ii = 4ac ac 2 (mod : -.:. 4 <l = 4ac - (mod 4) 2b + 1, - 3 - 4b2 - 8 ac even b odd ' - 8 When ' b b2 + 2 "-= 2 Conversely, when 16) :::==: 4b b even b odd 4) 4b 2 (mod 16) , b ' even b odd 2b + 1 (mod 4) 2 =- - 4b 2 + 8b + 4 (mod 4 , b even =8 , b odd - 16) Thus proving the first part of the lemma. Then, i f or ac = 2 ( mod 4) a .=::: 1 ( mod 2), C ::::c· 2 a :'c-: 2 ( mod 4), C cc:': (mod 4) 1 (mod 2) and the converse is obvious. ac ~ 2b + 1 (mod 4) Again, i f a ==' 1 (mod 2L c =- 1 (mod 2) so that a + c c::.-=: a c + 1 (mod 4 ) hence a + c = 2b + c (mod 4) or a + 2b + c =--= 2 (mod 4) and finally a(a + 2b + c) ==- 2 (mod 4) Conversely, i f a -- 1 (mod 2), c c:c: 1 (mod 2) then a ( a + 2b + c) ", 2 (mod 4) implies a + 2b + c - - 2 (mod 4) or a + c , : 2b + 2 (mod 4) but since a and care odd, this 1s equivalent to a c -::: 2b + 1 (mod 4 ) Theorem 3.5 Let [a,2h,c] have even discriminant a A ' 1- l b X ) + C) ' ] - -- -- - = : - Ll • If ;1/ / ~ = a . :-c' -f 0 - A::..: 4 or 8 (mod 16) ,) .0 ·..- ;; T C. ;;t •] 3 (J 1.1. (-n, o J It) where [a ,';.,; c: 'J has discriminant - o/y . We consider separately the two cases: I. One of a and c odd, the other even II. Both a and c odd Case I Without any loss of generality we choose a = 1 (m cl .:2 ), O C =- 0 ( -171 C) d .2. ). so that :.:- .: 2 f . Then Hence -+ Now if Ll-= 'l o,- ti' { 1>t o d so that ) = .J ,.-7 • I? ) , ( ) that is by lemma 2, if c :=:J.(n,dcly~ Hence However, if 11 ,.·:: 0 ,.» /,) ( n,o cl I{, ) , that is by lemma 2, if C ~.:i O (n,_ c,J Y), then '-/ 'f -; C / c.. :, · ] (T and where I I - Q_ 'I a (.. I.,(_ I lo' - a c- l--.:i ~ I::> c. - -7' C. LI// /2 , Case II. The biunique transformation: replaces a by where u. c2. la .C\'. ;r i ,}_ /.::, l ~ 0-- + X -t /2 a ·r (, = a t- r c. le. ·-··- ·f ·- f +~ I ') .. '7 ? ~_ :1 (mod L C. - ( --7 ,\:' C. ) -1 h Since Thus we reduce Case II to Case I. When LJ .-:: "I OY ~ (--,,,od ;.; ) , then by lemma 2 c-= a. rJ /2 r c ?" J ; Thus in all cases: When LI When .Ll - Q O r / ~ ( ln d cl /( The theorem follows then by mere repetition of the above process. Theorem 3.6 Let la., 2 6, c] have discriminant - L.l, and let m be an odd integer; then: If t./ L\ - 0 ,- [(" 1Y l 2 rni = a {.,,. ~ I 6 } o cl ' -t ;;__ /::, X ;t --t C;; '] I j)/ [ /rV/ = 0. where If a c - h L\ ~ 0 "- -- 0 7 , a / C .2 - ,2/:,\K) -f X c'7'} -f- /2; ,. ( _,.,,,, 0 J I[, ) Consider first the case where a·c is even. of generality we set a odd and c even. Without loss Then so that x = :2 ( and where 1 ce. = J a. Now when c'.'.\ ~'-/ o, CI ~ (n-,ud /V [ ; ;_ ,,.,,, -= a x ' -t- ,;J. h x d = 0 V r / 2 ( C ol. by lemma 2 /(;) so that When Ll L ___:- n--1 0 cl + c ;; ' I,; ) by c. ~ :;_(--,,,u.Jd 'I), J 1 emma 2 ii)' C. .,,. () ( ,.,., ,? cl so that -; 1 o2 /;:, 1 --' t 1 ) -j C rl , 7 -- C _J 1 C ==' 1 (,,,,vcl oZ) When a-c is odd, as in the previous theorem, the transformation x * f + "'? , -;; " '7 reduces the situation to the case a -c even. These theorems give the reduction formulae necessary for the forms we are to consider. In order to identify the several forms occurring in these formulae, we need some extensions on the properties of the characters as given by Dickson. We generalize the statement of Dickson's Theorem 67 to read: Theorem 3.? ;i_ If ;o__,: tinct odd prime factors . of L'.l , _theil !-_he__ characters ( [ a, lJ, c J has the discriminant -Ll , _Ll = t/a c - h . are the disr'1 / pA. ) have the same value for all intege:i::-_s n represented by a form of discriminant - 1-1 , excl1..1:<!~~~ --t~ose char~cters, _( /Pv) , where 1ul0 . 01 i (,, -,) same is true for the characters s -==- {- > , t = c1 -,!;J,, ' - I) 1) , The for odd integers £, c In the proof of Dickson's Theorem defining the characters and giving their properties, it is only necessary to shift the emphasis that n be prime to L1 , hence allowing a consideration of all odd prime factors of ~ , to a restriction to consider only charac- ters corresponding to those odd prime factors of LL not dividing n. The details of proof remain unchanged. In connection with the theorems to follow, we state: Lemma 3 Let Lo, b, c.] have the odd discriminant - LJ =- - ), 1\. If and + 2- C y ). is square free I C,'./ t/' j there exist ,? nz such that and By hypothesis so that ( :J.. Q, X ·- · • { ~ Q, X. -f- b°J- r -r· I:> J) l ..t. LI ) -/ -f i'L., ~\ I J l. and similarly Since A, is square free, this implies f to determine and Hence 1 , but from these CL or (hf;' r ;Jc_ ( ) L -- -; C ( ;;_'-<( /2 {l( -,-- J c_"o/ ) {;;_ c.,r I t 'o/ ) -f f- h'"'l):; f- Lemma 4 Let fa , :2 t, c] have discriminant - Ll "" - tf 11, 1· there exist f and If ) , is square free and n-; such that The proof parallels that of Lemma 3 starting with , 0. ; , fr/ = { a. X C J, a t :::- ( C -r 'J_ + ID I ) +- A, );, ).:,_ hA / ' f- ;\ and using the determination of f and >-., )( ~ "1 • It is winecessary to repeat details. The characters of forms representing certain products of integers occurring in certain of the above theorems are re.}.ated according to the following four theorems; Theorem 3.8 .,.. Let - LI be a dis~iminant such that .t1 = .A/ , (>-,J-<.)= 1. an odd prime and A Then, i f A,n., rn odd)is represented by a form of dis- criminant - LI whose characters are ,\,..c--t. , n 1s represented by a form of discriminant - LI whose characters are ,.,,c.,; , where >- C,. being the value of the characters for the form representing ), . Consider first odd discriminant, -.Ll, let )/>?. Ci. u." -t- KJu.v o.r-"-1-h--r-s -r cv< -rc,.s"- so that where ( I ) :::( == ;)_ a ::::: US l,{ - b u .s r ..,.., r -/· h -r v --r u ".,, cl v' ;;J. c v s V If Ll "' ),.,P , by lemma 3 there exist such that CV' :, Hence -f LI J I But (1) and {2) state for all p prime factors of Ll • Therefore I I But by lemma 3 (>-I 1v ) or ) and ~ -,,, are represented by the same form, hence v / ;-v ) include all >. c.,.., • and the theorem is proved for odd disc1m1nants. For even discriminants the proof begins similarly. = a r f- -t- C c5 '- so that where ),( -- a u. ,- r - -,,- b 1,1 .s 1- lo --,- v v1 ' ' -1- «b Hence - But (3) and (4) state c v s V If - l:J == A/ , by lemma. 4 there exist ,,, "".,, J = a r v1 ' v ' + FJ J, c v ' v ' , such that Let for all p, odd prime factors of D. Therefore ( i\ I ;o) {,-,,Jr)(~ /p) - I / r -=-- ). I ~ But by lemma 4 and / ~ hence ( >--/ ?) or ( Let /J = 3 (mod r). but 0 ;v) / are represented by the same form, include all ,\ C . -( Then A , n, and mare odd, so that whence of this type. , ..\ 1- ,,., - 1 = ,,,,,, ( 77( o c1 )_ rn ,,,,,,,, -., I ( ,,, " j 'I ) Y) so = (-) /, C r"l e_ L If /J ::: J. <7>1." cl f), then x is odd in (3) . According as y is even or odd, >..,.,,,- = .,- / Hence ) ',,,,, cc' /Jn /\ n I, ,\ /Yi ·~ +_ r""f (-,,,,J cl ?J) - 2 o 2 ( J>tu d ;.) ' and n are odd 1\ ' Hence l /M. - - ·- , ,\ r />t -- ( - I) -{;-(- ' - 1 J (-!) - I - 2 n¥1 -f,[0'-1) ( )7( o J n) ,. (,n'-1)] I, enc. e ,J~ (mod 6-'}, For the first alternative For the second so that / ",- /2 " t .!.. but since If D = O{moJ '!-)/ If A;?fa>l ~ x· == I (m,cl :1) "",, J /)=.O(n-,odFr), ">._,,-,,,/m==x'=-/ (,,,oclFr) an cl Theorem 3.9 Let -Ll be an even disciminant. If 2m, m odd, is represented by a form of discriminant, -11, whose characters areJr121C__,<,, m is represented by a form with characters ,,,,,, C ". where ..l C, A. ~ C, .... ~C~· being the value of the characters of the form representing 2. The proof follows the second part of theorem 3.8 for the characters (=/;J_,) • (That 2 is even is immaterial dealing with odd characters). Since m is assumed odd, we require Li= '/ or I f Ll = 'is (mod u), ~-r>rt 1s represented by Fr( rn °J 1 b). (Theorem 2) the same form as 2m by Lemma 4 and may be used in the manner of Theorem 3.8 to complete the proof for even characters. ,._ When IJ='f(,,,,,J u), lJ= 3 { = ,j 'I ) and as before;;,.,,,,,,,,= x ' -1);1, since m, n, 0 Hence and Oare odd, x and y must be odd. so so that I - and Hence ). "1 - J /»-1 -,,,- (-;} ' (- ) ;:> f l) --f (7n 0 /0 -f- I J &--) --; ":l e_ ! "-- ( -I) ... But ~~8 is represented by the same form that represents 2. Therefore S (,;i) S (,.,,,,,) = & (c),,,,,) 6 is the only even character here. Theorem 3.10 Let 4m be represented by a form of discriminant, - Ll , with characters jl,,_,, C..,· • m will be represented by a form of . discriminant - Ll or -~/1/ whose characters for all i for which both exist. For the odd characters, (,...,., / ; J_,_ ) , the proof parallels that of Theorem 3.8 replacing A everywhere by 4 and omitting mention ii is odd or L\ =' 1:2 (7>7oJ 10 there are no of_)l as unnecessary. If even characters. L\ ='Io , &' ( m o d I t ). 1 by When Theorem 3.5 4m and m are represented by the same form, hence all the characters must be identical. When LI = { n1° cl 3;;_), S, c , an d we go to LV ,; cS are all characters, but when t two of these characters will no longer occur, leaving one as follows: ~ / € 6 6 cl 'I) L1 - 0 Ll ·- '1 ( ( ,-,,.; J ;;z&') L'.J - .3:1. ( n-t 0 d Id. (l) d J a.,, J ( 7n 0 {, We assume and as before Now, if L.l =o ( n,o d 6 '1), IJ == o Cm O 1t; C,( s () and 'J · C If J c ve >r so that and If Ll - 0J ( ~ a cl u 1, J X) ( 7n ocf 3 «) so that and Theorem 3.11 Let LI == 7\~u , ). an odd prime, If 1\ .. ,,.,.,,l , m odd 1s represented by a form 0£ discriminant - LI w1 th characters >..",,,,/.:. , m 1s represented by a form of discriminant - ~/\' whose characters for all i for which both exist. where By lemma 1 there exists a primitive form in the same class of the form[,\~,< ;( c J. We may equally well consider this. But and />YI = a I x 2 h + I x~ + c '~ "" Now the form [ c,: I.:,: c 'I representing m has discriminant - o/'i and certainly represents all integers represented by [a.,h,c ] and its equivalent [>.. 'c.1 ', U'. c 'J . An integer prime to 1 LI , and hence also to represented by both. they are identical. .vl; ;;i,. , is Hence all common characters must be equal as i3 Chapter III The Number of Representations Function The explicit formulae for the number of representations function for all integers in the cases summarized above where there is a single class of forms to each genus follow now in logical summary. These are given in the theorems below. We lets uniformly stand for some odd integer prime to the discriminant ✓ -4J represented by the form[a , h, cJ. Unless otherwise specified, k will stand for the number of characters in any particular case as determined according to Theorem 1.2. The connection with this k and the number of prime factors of the discriminant is used. Theorem 4.1 If L.'i = fl, fl. - - /J, ¥= 7 a.,.. 10 ( ,,,, d cl h) and LJ > J where the J?,,· are distinct odd primes, and [a,.b,c ] has discriminant -L'.l, then: By Theorem 3.4 /v [,1 A p , ot, - - • / ~ = or,I,,-,,.,,_ d. :;,;_ • +- /a .::t -;J -t- C J l , By repeated application of Theorem 3.1 = a x ,_ -,- !vx "21 =- a where 'la 'c ' _ 1:,'' =: 'IQ c - /2 .-i. , l I .x and by Theorem 3.8 if q is an odd integer prime to Ll represented by lo : c ' J [a ', Hence by Theorem 1.3 the result follows. Theorem 4.2 If Ll 0 _,c: / $ / 0, , - - /~4. =o 1:2. ( nr a J I c) , "r where ?, ::::. ;;. , 8 = J. o ,- 3 and the remaining /~ ,_ are distinct odd primes, and [a, h, c:] has discriminant - ii , then: ;V [?, °', /> o<,_ " - - ; , ; fi {; - - -- - f- /-1c. r1o J(C. (,) c,r- >J L ' (- 1/J /j - -- - -·----- --·--- - - --- -~------·- - - '-"'_/_ ,...,._ _ _ _ _ By repeated application of Theorems 3.1 and 3.6, _...: a X a 1 where '/ a c. ~- /4' =· "la (.. - lo 1 L I '."\'. l f- /v X d -t- C ) , J J. and by repeated application of Theorems 3.8, 3. 9, and 3.10, if q is an integer prime to Ll represented by [a.', b I C 'j / ' Hence the result follows from Theorem 1.3. Theorem 4.3 J- I f .6 = ;:; / ~ - -- r;, _1 where ;v, = ~ and the r,, •_~r_e_ _~~-s_t_i_n_c_t_ o_d_d_p_r_i_m_e_s_, and La., h, c.] has discriminant - L'l , then: L/ r/ o<, ::::: I 0 where C4 is the character for Ll • not a character for L'.'1/'-I • By repeated application of Theorem 3.1 N [ ;o,<X'___ /~~,-/ where r'o 'c I - = /'1-n. oxl -f ,b,, = 'la C - /a., , and if represented by L a I I / Ji) I I c ' -f c~'] = C) , q being an odd integer 0<, 1 c r" ) :: A /ox;y. fa· I Jz o( ' 1c. (/ J.); ✓ CA ( 0 ~ s ) and by Theorem 1.3, the first part follows. If o</ = I , Theorem 3. 6 indicates If o<,2_ 2 , Theorem 3.5 indicates !Y f /~ where o(, - - i('la ''c" - 0 "'h rm. t:1. x , 1 1o x 7 -f c;; ·] 2 /2" ) Hence this falls wider Theorem 4.2 and by Theorem 3.10 our result follows, noting that (-Lly) = (--#:-)? ). Theorem 4.4 By Theorem 2.3 the odd discriminants containing a square factor and such that there are a single class of forms in each genus, .... are all of the form L1 == ;o, /J.,_ ?a where the ?,._· are all odd primes on one. If [a., h, c] has discriminant - L1 , then N [ ~ r, <>:,/~ A d, /~ = a X:, cx_,O?! -f- h ,t' '). -t C;; ", (/,,,1_, -;; A) ::= ] C) By Theorem 3.4 where ta 'c. '- /2' \ '-la "" - t .,__ and if q is an odd integer prime to LJ represented by [a. ~ lo'. c], by Theorem 3.8. 8 ( · ( ~ ) _- : "l ~ [ CA ( F, )] If r,1_, , = o o, spectively. I c{_' V (_ - ( s f the result follows from Theorem 1.3 and 3.2 re- '1 7 If rx., > 1 , by Theorem 3.2 IV [ r, ot, /?-?? =- a ,X -f- L /2 I X )' + 7 1 C 2 1 - /V f P, "Y'; - ~ = a " x " -1- lo "x ) where ?, '(1,, "c .,_ t ,,') ~ 'IO ~ '- );', and if r c ,,-;x , ] ~ ' ie an odd integer prime to represented by [o ': h ,., c "], by Theorem 3.11. Hence Theorem 1.3 gives the final result. Theorem 4.5 t} If iJ = ;2 /0 ; 0 = 0, ~, 5". & ; p .:: 7, I ..5-; /1 =/~. r .J ,, ;-:), -- ~ , and [ a, /2, c] has discriminant -A, then: ;V [lo, /n,:. "'•;-:: ~.,.,,,,, ,, -, = c,, a X , -t- b X ) -t- By Theorems 3. 6 and 3. 5, if =- j ! /t ( - J )V l;, r ;" I ct ' / 0 \3 ,,,,J /7,?? [o;b : c'J has discriminant - ;;_Cy/. C )' , ; ( rtr>t, -<, ,{__/,, ) = I ] o(, ~ 0 = a A I T tJ ~; -t- C J' ,! ] If q, an integer prime to L1. , is represented by Jc;~ .6, c' ), then by Theorem 3.10 1 If <¥1 2 {9 , by Theorems 3. 6 and 3 .3 ( o( 1 - {;! -;- I ) )/ [ f>.,_ « • ?_, ot_, /J'>'{ .=:- 11 LI where [a': t': c'J has discriminant - ':!; 0 , and if g, to Ll A ,.· + C{ 1 ),/x ) C -f 'j, ~] an integer prime , , is represented by [ 0 11, ti:o 11, C "} I~ By Theorem 3.1, if ["',/3, if ] has discriminant - /.:2;., , then: . 1 Where [c,</J, 1 } has the same discriminant as lot,~ y J_,and if 1 P are integers prime to / -:J and / J and represented respectively by [o<, ,<3'. /] and [a{>< t 'J, Theorem 3.8 indicates Combining these results with the above and applying Theorem 1.3, the statement of the theorems follows. Theorem 4.6 () [a,b,c ] If ti = :1 •3,; <9 ::: u, :2, tf, r; , and, has discriminant - Ll , then: /V f ,;2 ... 3 ,o /h{_ = u x' t- b .: ) 3 "' =;1,-[1+(-,J j - -------···- = /;. [ u- I ,tj ?Y ·r/ ·~ A l! t- + c) .,_ ,, _(.,,,.,, 0) .:- 1 ] c;_(J)4( {s )(,,(,._)]_1~ (-3 // .. . . {1 f ( (, )" C, (o) _.k</ - ······ ~ •-- -- -- - C, (,-,., 1J,L_; (- / .-"''/- By Theorems 3.4, 3.5, 3.6 .. J :,J j,p J , 0( < (9 'J' 7 C,( X 0-oe where [a: I>: c '] has discriminant -Ll = - 0 , if oc 2.. a and .LI= :i. ~ and if is an odd integer prime to 3 if ex'.~ C3- ; :2. LL represented by [a ; k ! c 'J, d C by Theorem 3.10 By Theorem 3.1 N [ 3 t<J r,1_ L " where if f i( / -t- /0 •/ -i'. I I i )' z J is an odd integer prime to ;,_ LI represented by [c:1 ; I:,:· c "} by Theorem 3.8 Hence by Theorens 1.1, 1.3, and these results, the Theorem follows. Similarly, the number of representations formulae for the remaining discriminants to be considered: those even ones according to Theorem 2.3 containing an odd square factor 180, 280; and the even ones LI =- '-I, 1 (, t L/ • ~ = 20 , 7~, ; oo_, We shall om1 t the de- rivations here as evident from Theorems 3.1 - 3.11, 1.1 and 1.3. / 0() L\ :,:-: ; __ V V ..s -- J. ... • 3 , 0. X ·f- ~ {.) ( ( .,,.,, ) ] l_~_,, ( - L.I ~ ) , 1 C o( = /J :: .,,,,,/-..., cJ 0 •• {) Ll :.-=-. ioo 0. 1 ?( -f I, ;;./i ;?,Y"/1 r C.,(o1 )-:{_ (.s ) c_ r.-JJ 2__: (- 4 £ .u) u j ,..... 0 ; 3 -= I /3 _-_ 1 a CJ 10 I /3 C) =I - u. ;,1: ' 7 - . , :-,~1 /1 L p A' J -/ (__ ) / l c,(a)" ( ( , ) c, -f c-~)}) _; ( LJ / " ) , /J :: /1/- I 7 ) ,I ( - / / l< ) _,l;/3 13-~ i l l N I ;2. X /Y-1 -- u_ ,\. t j ,b -l )" i C. cl ' (, , ,., / L)) - I J 0 ( -1 d- \ (- L___ / I II ) (,( - ex. ,,..,,I Ll ) - I J C --"' / ,,,.. o<- -- c; / 1) - I I (- I I . '1. ) D( I I l lo .-:i_ J > J_ ' -f/,-,., il /\I [.2 " r'Wf ax' __ ; cd (- 1/u.) \., (- I / l( ) I L- _• . - ~/ { - ; I/ . ,.,_) I :/1./ r'v>f In these cases [a,b,c] is to be any form of discriminant indicated. / 0 '3 Appendix A Tables of Reduced Forms and Characters Chapter II exhibits in detail the appearance of the several reduced forms when there is a single class of forms in each genus. however, In order to provide complete reference, we append here a complete table of all reduced forms for all known discriminants havi ng a single class to a genus. Further, in order to facilitate the calculation of the number of representations function, we indicate the several characters for each discriminant and indicate the values these characters assume for numbers represented by each of the reduced forms. The reduced forms which represent the several odd prime factors of the discriminant are likewise indicated. An explanation in detail for one case will suffice to show the organization of the table. 280 = 2 3 .5.? l,O,?O] 2, o, 35], 2 [5,0,14],5 [?,0,10],7 f Consider the selection: (n/5) (n/?) 1 1 1 -1 -1 1 € 1 -1 -1 1 -1 The negative discriminant, -280 = -2 3 .5.?, has the reduced forms: -1 (1,0,?0], [2,0,35], [5,0,14], [?,O,lOJ. The prime factors of the di scriminant, 2,5, and 7, are represented by the last three of these respectively. characters, ~ There are three , (n/5), (n/?), which take on the values listed for numbers represented by the form opposite. I a '-I 3 (n/3) (1, 1, 1], 3 4 = 22 ~ [1,0,1],2 1 [1,1,2],? (n/?) 1 ? 8 = 23 s€ [1,0,2],2 1 [l, 1, 3], 11 (n/11) l (1,0,3],3 (n/3) 1 [1,1,4] (2,1,2],3,5 (n/3) 1 -1 11 12 = 22 .3 15 = 3.5 16 = 24 24 = 2 3 . 3 2? = 33 28 = 22.? 32 = 25 35 = 5.? (n/5) 1 -1 ~ € (1,0,4],4 1 [1,l,5],19 (n/19) 1 s (n/5) [1,0,5],5 [ 2, 2, 3], 2 1 -1 1 -1 19 20 = 2 2 .5 1 (: [l,0,6] [2,0, 3],2, 3 1 -1 (n/ 3) 1 -1 (n/ 3) (1,1,?],9 1 [1,0,?],? (n/?) 1 [l,0,8],8 [ 3, 2, 3] [1,1,9] [ 3, 1, 3] , 5, ? s (:- 1 -1 1 -1 (n/5) (n/?) 1 -1 1 -1 I O 6 36 = 22. 32 40 = 2 3 .5 [l,0,9],9 [2, 2, 5], 2 = 2 2 .13 (n/ 43) 1 [1,0,12] [ 3, O, 4], 4, 3 & 1 -1 (n/ 3) 1 1 [1,1,13] [ 3, 3, 5] , 3, 1 7 (n/ 3) 1 -1 (n/17) 1 -1 Ei (1,0,13],13 [2,2,7],2 1 -1 (n/ 13) 1 -1 [1,0,15] [3,0,5],8, 3,5 (n/ 3) l -1 (n/5) 1 -1 b (- [1,0,16] [4,4,5],4 1 1 1 -1 (1,1,17],67 (n/ 67) 1 [l,0,18] [2,0,9],2,9 SE 1 1 (n/ 3) 1 -1 (1,1,19],25 [ 3, 3, 7], 3 (n/ 3) 1 1 (n/5) 1 -1 b 1 (n/3) 1 -1 1 -1 6 67 72 75 84 88 = 23. 32 = 3.5 2 = 2 2 • 3. 7 = 2 3 .11 (n/5) 1 -1 [1,1,11],43 60 = 2 2 . 3.5 64 = 2 s €l -1 48 = 2 4 . 3 52 (n/3) 1 -1 [l,0,10] [2,0,5],2,5 43 51 = 3.17 b 1 1 [1,0,21] (2,2,11],2 [ 3, 0, ? ] , 3, 7 [5,4,5] - -1 -1 1 [l,0,22] [2,0,ll],2,11 1 -1 E: (n/11) 1 -1 (n/7) 1 1 -1 -1 / Ob 91 = 7 .13 96 = 2 5 . 3 99 = 32 .11 100 = 22.52 112 = 2 4 .7 115 = 5.23 120 = 2 3 .3.5 123 = 3.41 132 = 2 2 • 3. 11 147 = 3.7 148 [l,1,23] [ 5, 3, 5], 7, 13 (n/13) 1 -1 s E [l,0,24] [3,0,8],3 [4,4,7],4 [5,2,5] 1 -1 -1 1 1 -1 1 -1 (1,1,25] [5,l,5],9,11 (n/ 3) 1 -1 (n/11) 1 -1 [l,0,25),25 [2,2,12].2. b 1 1 (n/5) 1 -1 ¢ [l,0,28J [4,0,7],4,7 1 -1 (n/7) 1 1 (1,1,29] (5,5,7),5,23 (n/5) 1 -1 (n/23) 1 -1 (: [2,0,15],2 [3,0,10], 3 [5,0,6],5 1 1 -1 -1 (n/ 3) 1 -1 1 -1 [1, 1, 31] [ 3, 3, 11], 3, 41 (n/ 3) 1 -1 (n/ 41) 1 -1 (1, o, 33] (2,2,17],2 [3,0,ll],3,11 (6,6,7] b 1 1 -1 -1 (n/ 3) 1 -1 -1 1 [1,1, 37],49 [ 3, 3, 13], 3 (n/ 3) 1 1 (n/ 7) 1 -1 ~ (n/ 37) 1 -1 [l, O, 30] 2 = 2 2 .37 (n/7) 1 -1 (1, o, 37], 37 [2,2,19],2 1 -1 (n/3) 1 -1 1 -1 (n/5) 1 -1 -1 1 (n/11) 1 -1 1 -1 I U 160 = 25 .5 [l,0,40] [4,4,11],4 [5,0,8],5 [7,6,7] 163 [l,l,41],163 168 = 2 3 .3.7 180 = 22 . 32 .5 187 = 11.17 192 = 26 . 3 195 = 3. 5 .13 228 = 22 • 3. 19 232 = 2 3 .29 235 = 5.47 [l,0,42] [2,0,21],2 [ 3,0, 14], 3 [6,0,7],7 I 6 1 -1 E 1 1 -1 -1 -1 1 (n/5) 1 1 -1 -1 (n/163) 1 SE 1 -1 1 -1 (n/ 3) 1 -1 -1 1 (n/7) 1 1 -1 -1 [l,0,45] [2, 2, 23], 2 [5,0,9],5,9 [7,4,7] b 1 -1 1 (n/ 3) 1 -1 -1 (n/5) 1 -1 1 -1 1 -1 [1,1,47] [7,3,7],11,17 (n/11) 1 -1 (n/17) 1 -1 s <= (n/ 3) [l,0,48] [3,0,16],3 4, 4, 13], 4 7,2,7] t -1 1 -1 1 -1 -1 1 [1,1,49] [ 3, 3, 17], 3 [5,5;11],5 [7,l,7],13 (n/ 3) 1 -1 -1 1 (n/5) 1 -1 1 -1 (n/13) 1 1 -1 -1 s (n/ 3) 1 (n/19) (1,0,57] [2,2,29],2 [3,0,19],3,19 [6,6,11] 1 1 1 -1 -1 -1 1 -1 [l,0,58] [2,0,29],2,29 6f 1 -1 (n/29) 1 -1 [1,1,59] [5,5,13],5,47 (n/5) 1 -1 (n/ 47) 1 -1 1 1 1 1 l -1 -1 1 I O 240 = 2 4 • 3.5 f> = 3.89 280 = 2 3 .5.7 288 = 25. 32 312 = 2 3 • 3.13 2 315 = 3 .5. 7 2 340 = 2 .5.1? 352 372 = 2 5 .11 = 2 2 . 3. 31 (n/ 3) 1 (n/5) 1 -1 1 -1 [l,0,60] [3,0,20], 3 4,0,15],4 5,0,12],5 1 -1 -1 1 [1,1,67] [ 3, 3, 23], 3, 89 (n/ 3) 1 -1 (n/89) 1 -1 G (n/5) 1 -1 1 -1 (n/7) 1 1 -1 -1 t 267 \( -1 1 -1 [1,0,?0] [2,0,35],2 [5,0,14],5 [7,0,10],? 1 -1 -1 1 f t s l,0,?2] 4,4,19],4 [8,0,9],9 [8,8,11] 1 -1 1 -1 1 -1 1 -1 (n/3) 1 1 -1 -1 € [1,0,?8] [2, o, 39], 2 [3,0,26], 3 [6, o, 13J, 13 1 1 -1 -1 (n/ 3) 1 -1 -1 1 (n/13) 1 -1 1 -1 [1,1,?9] [5,5,17],5 [ 7, 7, 13], 7 [9, 9, 11], 9 (n/3) 1 -1 l -1 (n/5) 1 -1 -1 1 (n/?) 1 -1 -1 1 ~ [l,0,85] . [2, 2, 43], 2 [5,0,17],5,17 [10,10,11] 1 -1 1 -1 (n/5) 1 -1 -1 1 (n/17) 1 1 -1 -1 s 6 t l,0,88] 4,4,23],4 [8,0,11],11 [8,8,13] 1 -1 -1 1 1 1 -1 -1 (n/11) 1 1 -1 -1 f:. [l,0,93] [2,2,47],2 [ 3 , o, 31], 3, 31 [6,6,17] 1 -1 (n/ 3) 1 -1 1 -1 (n/ 31) 1 1 -1 -1 -1 1 403 = 13-31 408 420 = 2 3 . 3.17 = 2 2 .3.5.7 [1,1,101] [11, 9, 11], 13, 31 448 480 = 2 6 .7 = 2 5 . 3.5 483 = 3.?.23 (n/ 3) 1 -1 1 -1 (n/17) 1 1 -1 -1 s (n/ 3) 1 -1 -1 -1 1 1 1 -1 (n/5) 1 -1 -1 1 1 -1 -1 1 (n/ 7) t [l,0,105] 2,2,53],2 3,0,35],3 (5,0,21],5 1 l -1 1 -1 -1 1 -1 [6,6,19] = 3.5.29 G- 1 -1 -1 1 [7,0,15],7 [10, 10, 13] [11,8,11] 435 (n/ 31) l -1 · l,0,102] 2,0,51],2 [ 3, o, 36], 3 [17,0,24],17 t 427 = 7.61 (n/13) 1 -1 (1,1,107] [7,7,17],7,61 1 -1 (n/ 61) 1 -1 l,1,109] 3, 3, 37], 3 (5, 5, 23) j 5 [11,7,11 ,29 t (n/ 3) 1 1 -1 -1 (n/5) 1 -1 -1 1 (n/29) 1 -1 1 -1 5 E (1,0,112] [4,4,29],4 [7,0,16),7 [11,6,11) 1 1 -1 -1 1 -1 1 -1 (n/7) 1 1 1 1 £ € [l,0,120] [ 3, o, 40], 3 [ 4, 4, 31), 4 [5, o, 24), 5 [8,0,15] [8,8,17) [11,2,11] [12, 12, 13] 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 1 -1 -1 (n/ 7) [1,1,121] [3,3,41],3 7,7,19]j7 11,1,11 ,23 (n/ 3) 1 -1 1 -1 t 1 -1 -1 1 (n/ 3) l 1 1 -1 -1 -1 -1 1 (n/23) 1 1 -1 -1 (n/7) 1 1 -1 -1 -1 1 -1 1 (n/5) 1 -1 l 1 -1 -1 1 -1 I I U 520 = 2 3 .5.13 2 532 = 2 .7.19 (n/5) 1 -1 1 -1 (n/13) 1 -1 -1 1 1 -1 -1 1 s (n/7) 1 1 -1 -1 (n/19) 1 -1 1 -1 [l, 1,139] (3,3,4?],3 [5,5,29],5 (13, 11, 13], 37 (n/3) 1 -1 -1 1 (n/5) 1 -1 1 -1 (n/ 37) 1 1 -1 -1 (1,1,149] 5, 5, 31], 5 7, 7, 2 3], 7 [ 13, 9, 13], 1? (n/5) 1 1 -1 -1 (n/7) 1 -1 1 -1 (n/17) 1 -1 -1 1 (1,1,15'7] [ 3, 3, 53], 3 [11,11,l?],ll [1 3,7,13],19 (n/ 3) 1 -1 -1 1 (n/ 11) 1 1 -1 -1 (n/19) 1 -1 1 -1 [, 1 -1 -1 1 1 -1 -1 1 (n/ 3) 1 -1 1 -1 -1 1 -1 1 (n/5) 1 -1 -1 -1 1 1 1 -1 (n/ 11) 1 -1 1 1 -1 -1 1 -1 s € (n/ 3) 1 -1 -1 1 -1 1 1 -1 1 -1 1 1 -1 -1 -1 1 (n/ 7) 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 (1, o, 133] (2,2,67],2 7,0,19],?,19 13, 12, 13] t 555 = 3.5. 37 595 = 5.7.1? f 62? = 3.11.19 660 = 2 2 .3.5.11 [l,0,165] [2, 2, 8 3], 2 3,0,55], 3 5, o, 33), 5 (6,6,29] [10,10,19] [1i;o,15J,11 [13, 4, 13] f 672 € (1, o, 130] [2,0,65],2 [5,0,26]j5 (10,0,13 ,13 = 2 5 . 3.? [l,0,168] [ 3, o, 56], 3 [4,4,43J,4 [?,0,24],7 [8,0,21] (8,8,23] [12,12,17] (13, 2, 13] 1 -1 -1 -1 1 -1 1 1 I I I 708 = 2 2 .3.59 [l,0,17?] [2,2,89J,2 3,0,59], 3,59 6, 6, 31] 6 1 1 -1 -1 (n/ 3) 1 -1 -1 1 (n/59) 1 -1 1 -1 [1,1,179] [5, 5, 37], 5 [11,11,19],ll [13, 13, 17], 13 (n/5) 1 -1 1 -1 (n/ 11) 1 1 -1 -1 (n/13) 1 -1 -1 1 6 t 715 = 5.11.13 760 = 2 3 .5.19 795 = 3.5. 53 [l,0,190] [2,0,95],2 [5,0,3SJ,5 [10,0,19],19 1 1 -1 -1 (n/5) 1 -1 -1 1 (n/19) 1 -1 1 -1 [1,1,199] [3,3,67],3 5,5,41],5 15,15,17],53 (n/ 3) 1 1 -1 -1 (n/5) 1 -1 1 -1 (n/53) 1 -1 -1 1 Sf f 840 = 2 3 .3.5.7 [1,0,210] 2,0,105],2 3, o, 70], 3 [5,0,42],5 [6,0,35] [7,0,30],7 [10,0,21] [14,0,15] 1 1 1 -1 1 -1 -1 -1 (n/ 3) 1 -1 1 -1 -1 1 1 -1 (n/5) 1 -1 -1 -1 1 -1 1 1 S € [1,0, 232] [4,4,59],4 [8,0,29J,29 (8, 8, 31] 1 -1 1 -1 1 -1 -1 1 (n/29) 1 1 -1 -1 t 5 928 = 2 .29 6 960 = 2 . 3.5 s [l,0,240] [ 3, o, 80] [4,4,61],4 [5, o, 48], 5 l2,12,23] 15, 0,16] (16,16,19] [17,14,17] t 1 -1 1 1 -1 -1 -1 1 € 1 -1 -1 -1 1 1 -1 1 (n/ 3) 1 -1 1 -1 -1 1 1 -1 (n/7) 1 1 -1 -1 -1 1 -1 1 (n/5) 1 -1 1 -1 -1 1 1 -1 2 1012 = 2 .11.23 2 1082 = 2 .3.7.13 5 1120 = 2 .5. 7 1155 = 3.5.?.ll 1248 = 2 5 . 3.13 6 (n/ 11) l -1 1 -1 (n/23) 1 l -1 -1 (n/ 3) l -1 1 (n/7) 1 1 -1 -1 l -1 1 -1 (n/13) l -1 1 -1 -1 -1 1 1 1 1 -1 1 -1 -1 1 -1 (n/5) 1 1 1 -1 -1 -1 -1 l (n/7) 1 1 -1 -1 1 l -1 -1 (n/ 3) 1 1 -1 1 -1 -1 -1 1 (n/5) 1 -1 1 -1 1 -1 -1 1 (n/7) 1 -1 -1 1 1 1 -1 -1 (n/ 11) l l 1 -1 -1 1 -1 -1 S' 1 -1 -1 -1 1 1 1 -1 € (n/ 3) 1 -1 1 -1 -1 -1 1 1 (n/13) 1 1 1 -1 -1 1 -1 -1 [l,0,253] [2,2,127],2 ll,0,23]jll,23 l?,12,1? t 1 -1 -1 1 [1,0,273] [ 2, 2, 137] , 2 [3,0,91], 3 [6,6,47] [ 7, o, 39] j 7 [13,0,21 jl3 l4,14,23 17,8,17] t b l l -1 -1 -1 1 -1 1 s €- [l,0,280] (4,4,71],4 [5,0,56],5 [7,0,40],7 [8,0, 35] (8,8, 37] [l?,6,17] (19,18,19] 1 -1 1 -1 -1 1 1 -1 l,1,289] 3, 3, 97], 3 (5, 5, 59], 5 [ 7,?, 43], 7 (11,11,28],11 [15, 15, 23] (17,1,1?] (19,17,19] [l,0,312] [3,0,104],3 [4,4,?9],4 [8,0,39] (8,8,41] [12,12,29] [13, o, 24], 13 (19,14,19] t -1 1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 I / <J 1320 = 2 3 . 3. 5 .11 2 1380 = 2 . 3.5.23 b€ (n/ 3) 1 -1 -1 -1 1 1 -1 1 (n/5) 1 -1 -1 1 1 -1 1 -1 (n/ 11) 1 -1 1 1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 (n/ 3) 1 -1 1 -1 -1 1 -1 1 (n/5) 1 -1 -1 1 1 -1 -1 1 (n/23) 1 1 1 -1 1 -1 -1 -1 [ 1, o, 35?] 2,2,179],2 3, o, 119], 3 [6,6,61] [?,0,51],7 [14,14,29] [l?,0,21J,1? [19,4,19] fi (n/ 3) 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 (n/7) 1 1 -1 -1 1 1 -1 -1 (n/ 17) 1 1 -1 -1 -1 -1 1 1 (n/5) 1 -1 -1 1 (n/ 7) [1, 1,359] 5,5,?3],5 7, 7, 53] j 7 [19, 3, 19 , 41 [l, O, 330] [2,0,165],2 [ 3, 0, 110], 3 [5, o, 66], 5 [6,0,55] [10, o, 33J [11, o, 30], 11 [15,0,22] 1 -1 1 -1 -1 1 1 -1 [1, o, 345] [2,2,1?3],2 [ 3, o, 115], 3 [5,0,69],5 6,6,59] 10, 10, 3?] [15, o, 23], 23 [19,8,19J f 1428 2 = 2 . 3.? .17 t 1435 = 5.?.41 t 2 1540 = 2 .5.?.11 s s [1,0,385] [2, 2,193], 2 [5,0,?7],5 [?,0,55],? [10,10,41] [11,0,35],11 [14, 14, 31] [22,22,23] 1 1 1 -1 1 -1 -1 -1 1 -1 1 -1 (n/ 41) 1 1 -1 -1 (n/5) 1 -1 -1 -1 1 1 1 -1 (n/7) 1 1 -1 -1 -1 1 -1 1 (n/11) 1 -1 1 -1 -1 -1 1 1 / I '-/ 1632 = 2 5 . 3.17 [l, O, 403] [ 4, 4,103], 4 [3,0,136],3 [8,0,51] [8, 8, 53] [12,12,37] [17,0,24],17 [23, 22, 23] 3 1844 = 2 .3.7.11 1995 = 3.5.7.19 €- 1 1 -1 -1 -1 -1 1 1 (n/ 3) 1 1 1 -1 -1 1 -1 -1 (n/17) 1 -1 -1 1 1 -1 -1 -1 """'.l 1 1 1 -1 [l,0,462] [ 2, 0, 2 31], 2 [ 3, 0, 154] , 3 [6,0,77] [7,0,66],7 [11,0,42],11 [14, o, 33] [21,0,22] 1 1 -1 -1 1 -1 1 -1 (n/ 3) 1 -1 1 -1 1 -1 -1 1 (n/7) 1 1 -1 -1 -1 1 -1 1 (n/11) 1 -1 1 -1 -1 1 1 -1 [1,1,499] [3, 3,167], 3 5,5,101],5 7, 7, 7 3J, 7 [15,15,37] [19,19,31),19 [21,21,29] [23, 11, 23] (n/ 3) l -1 -1 1 1 1 -1 -1 (n/5) 1 -1 1 -1 -1 1 1 -1 (n/7) 1 -1 -1 -1 1 -1 1 1 (n/19) 1 -1 1 1 -1 -1 -1 1 s f 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 (n/5) 1 1 1 -1 -1 -1 1 -1 (n/13) 1 1 -1 -1 -1 1 -1 1 (n/ 3) (n/ 7) 1 -1 1 1 -1 -1 1 -1 (n/11) 1 1 -1 1 -1 -1 -1 1 (n/13) 1 1 -1 -1 1 -1 1 -1 E t 5 2080 = 2 .5.13 ~ 1 -1 -1 [l,0,520] [ 4, 4, 131] , 4 [5,0,104],5 [8,0,65] [8,8,67] [13, o, 40], 13 [20, 20, 31] [23,6,23] 3003 = 3.7.11.13 [1,1,751] [3, 3,251], 3 [7,7,109] 7 [ll,ll,7lj,11 [13, 13, 61], 13 [21,21,41] (29,19,29] [ 31, 29, 31] 1 -1 1 -1 1 -1 -1 1 I I 0 3315 = 3.5.13.17 (1,1,829] [3, 3,277], 3 [5, 5,167], 5 [13, 13, 67], 13 l5,15,59] 17, 17, 53], 17 [29,7,29] [ 31, 23, 31] t 5 3360 = 2 .3.5.7 [l,0,840] [3,0,280], 3 [ 4, 4,211], 4 [5,0,168],5 [7,0,120],7 [8,0,105] [8,8,107] [12, 12, 73] [15,0,56] [20,20,47] [21,0,40] [24,0,35] [28, 28, 37] [29,2,29] [31,22,31] 5 5280 = 2 . 3.5.11 [1, o, 1320] [ 3, o, 440], 3 [4,4,331],4 [5,0,264],5 [8,0,165] [8,8,16?] [ll,0,120]jll [12,12,113 [15,0,88] [20,20,71] [24,0,55] [24,24,61] [ 33, o, 40] [37,14,37] [ 40, 40, 43] [41, 38,41] 1 1 -1 1 -1 -1 -1 1 (n/5) 1 -1 -1 -1 1 -1 1 1 (n/13) 1 1 -1 -1 -1 1 1 -1 (n/1?) 1 -1 -1 1 1 1 -1 -1 6 (;- 1 -1 -1 1 -1 1 -1 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 1 1 -1 1 1 1 -1 (n/ 3) 1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 (n/5) 1 -1 1 (n/ 3) & 1 -1 -1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 1 e 1 -1 -1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 -1 1 1 1 -1 1 1 (n/ 3) 1 -1 1 -1 -1 -1 -1 -1 1 -1 1 1 1 1 1 -1 (n/5) 1 -1 1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 -1 1 (n/ 7) 1 -1 1 -1 1 1 1 -1 1 -1 -1 -1 1 1 -1 (n/11) 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1 -1 ; I G 2 5460 = 2 .3.5.7.13 [l, O, 1365] [ 2, 2,683], 2 [3,0,455],3 [5,0,273],5 [6,6,229] [?,0,195],? [ 10, 10,139] [13, O, 105], 13 [14,14,101] [15,0,91] [21,0,65] [26,26,59] [ 30, 30, 53] [ 35, G-,; 39) [3?,4, 37] [ 42,42, 43] ?392 = 2 5 .3.?.ll [l,0,1848] [3,0,616], 3 [4,4,463J,4 [ ? , 0, 264], 7 [8, O, 2 31] [8,8, 2 3 3) (11,0,168],ll (12,12,157) [21,0,88] [24,0,77) [24,24,83] [28, 28, 73] [ 33, o, 56] [43,2,43] [ 44, 44, 53) [47, 38,47] b 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 (n/ 3) 1 -1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 s 6 1 -1 1 -1 (n/ 5) 1 (n/7) 1 (n/13) 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 1 1 -1 1 (n/ 3) 1 1 (n/?) 1 -1 1 -1 1 1 1 -1 1 -1 -1 -1 -1 1 1 -1 (n/11) 1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 1 -1 1 1 1 -1 1 1 -1 1 -1 1 1 1. -1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 II 7 Appendix B Numerical Examples In order to make explicit the application of the results obtained, we proceed to calculate in detail the number of representations function for the two forms: , 7x + xy + 7y 2 • 2 These should illustrate sufficiently the methods required. The form 2x2 + 35y 2 has the discriminant -280 = -2 3 .5.7 • and characters, € = <-1 )ff,,,,'-,) The number of {n/5), {n/7). , representations function is given for this case by Theorem 4.2. Referring to the table of Appendix A to obtain the values of the characters for the form, we obtain explicitly: /l[.J."'•s0/~7"'a,,,,_,,,,_ = .l.-x:'+ 0 .s-l', (r'Y"1,7o)= 1] = ~[! - (-1)01,,:•f,,,..))[J- (-,t•'"'r-10-))[1 t- (-,t•f~}=/7)])_ __:(-J!sOj/') /</.,.,..., We have further, ~ ( ,'l,?1 ( frVI) Is) I /1'>( =- I - -- I , =-I {;'VY//7) = :::: I I , - I, - I 0 ,- 7 ( ,.,.,,., d ,,,,.., - ,3 0 ·J- s { /hf - I 0 ,- '-I (,..,.," cl s) r»1. - .(_ 0 , - ,3 ( ~ -- I / -< , rn--t - J, -I - 6- ) ) ) , ' C l } ·- '-/ ( Cl-,-- ~ ( nu, ,j , I 7) ) Hence we may seperate integers, m, (m,70) = 1, into residue classes, modulo 280, with the triplet f (""'1) , (m/5), (m/7), identical for all integers in the class: € (r>ii) 1) 2) + (m/5) (m/7) + + 1, 9, 39, ?1, ?9, 81, 121, 151, 169, 191, 239, 249. + + 11, 29, 51, 99, 109, 141, 149, 179, 211, 219, 261, 221. (; ( n,,r ) 3) (m/5) (m/?) + 4) 5) + 6) ?) + 2 3, 5?, 113, 127, 177, 183, 193, 207, 233, 247, 263, 137. + 37, 53, 67, 93, 107, 123, 163, 19'7, 253, 267, 277, 43. + 31, 41, 111, 89, 129, 159, 199, 201, 209, 241, 271, 279. + 19, 59, 61, 69, 101, 131, 139, 171, 181, 229, 251, 269. 17, 33, 47, 73, 87, 97, 103, 153, 223, 257, 143, 167. + 8) 3, 13, 27, 83, 117, 157, 173, 187, 213, 227, 237, 243. According to the parity of ~, , o<.,_ o<' ,+ o1 ,. o( , + cl., "'• r ctJ even even even odd odd odd even o(., o( J even odd even odd even odd even odd even even odd even odd even odd even odd odd even even odd odd odd odd even odd even even oaa o{ I a) b) c) d) we have the four cases: o1.J Applying these results to the number of representations function as given above, we see that it is possible to state further: c2 X' --r J. ) 1/ r.3 6 ) 2 / { ,-,,,,,r, l Cf ) =I] {- .L",/0/&<-) __µ;,...,, when cl, , o(, oiv and m are paired, according to the divisions above, a),4); b),l); c),7); a),6); otherwise the number of representations is zero. I I 1 The law of quadratic reciprocity shows that: =-l -/fA' - ,)(/1/ s) L/-< /?) (-;)_'i{oj_,,µ.) - t (_,,P, ) ( tAj s ) v / 1 ) Hence£ ( -.z ls l' /;.{) is equal to the excess of the divisors of m in ./A/,.,,, c 1 a s s es 1 ) , 4 ) , 6 ) , 7 ) over tho s e in c1 asses 2 ) , 3) , 5 ) , 8 ) • We illustrate further by verifying the formula for the number of representations of 23,902. If 23,902 = 2x 2 + 35y 2 , we obtain empirically the solutions : x = .±. 11, y = .±. 26; X = .:!: 59, y = .:!:, 22; X = .:!:, 101, y = .:!: 10; with all choices of $ign permissable. representations is 16. s ~ ;;i_ Hence the number of To check this with our formula, we observe that 23,902 = 2.17.19.37, hence /I [ ~ 3/ cJ :2 = ;;_ x -.. -t- J rl ' ] = JV [ ;2 • I 9 .±, 109, y = + 2; X = \ 1 .:!'_ ..../ l • I y · 3 7 = ,)_ X ' -t 3 Jj ' ] f y) /-1 ju v ,;- / Since, referring to the previous notation, l?.19.37 = 11951 == 0 191 (mod 280), we have case b),l). Further- more, for the prime factors of 11951, t (/"}, ( u/ s J- (u I 1) is seen by reference to 4), 6), and 7) to be + 1. for all factors. Hence the same is true In all, 11951 has eight factors: l; 17; 19; 37; 17.19; 17.37; 19.37; 17.19.37. Thus, finally 2 N [23902 = 2x 2 + 35y ] = 2.8 = 16, to agree with the empirical result. The form ?x 2 + xy + ?y 2 has the discriminant -195 = -3.5.13, hence characters, (n/ 3), (n/5), Wl3). The nurrber of representa- tions function is given by Theorem 4.1. Referring to Appendix A to obtain the values of characters for this form, we obtain I J.. D A/ [ IV ).. ol, d . 3 • «., at , _s- • I J ,,-,-,,r ;t ;; = 7 ,-"I: , f- 7 ;r , / f- ( ,,,,,,,,, J 'i O ) --= I ] • ol., .,. al._, - (~ I / <3 ) ( - I ) J - .., l_/ ( - I I s / l,( ) /-'//»! Using the same table as above relating the parity of d_ L (- sJ✓ 1 <t V( ) -4/ ,,,,..., in the cases, (m/ 3) a) b) c) d) 1 -1 -1 1 In all other cases, including ~ odd, (m/5) 1 -1 (m/13) -1 -1 1 1 -1 1 the number of representa- tions is zero. By the law of quadratic reciprocity, we have { - I '1 S l JA ) = C- I) -j ( ,,-/ - I ) (,.u I d ) l L-< I .,,- ) V1 / ;J ) In a similar manner as in our first example we may separate numbers, n, into classes modulo 4.195 according to the values -!;. ( .,,-, ) taken by the set r- 1) , (n/3), (n/5), (n/13), so as to first test by the above whether the number may be represented at all. Then by combining into two classes according as (-195/n) is + 1 or - 1, )~ (-1rs//-1) is obtained as the excess of the divisors __µ; ...,., of m in the first cla.sses over those in the second. details are routine and are omitted. Further