An arithmetic formula of Liouville

Transcription

1 Journal d Théori ds Nombrs d Bordaux (006), 3 39 An arithmtic formula of Liouvill par Erin McAFEE t Knnth S. WILLIAMS Résumé. Un pruv élémntair st donné d un formul arithmétiqu qui fut présnté mais non pas prouvé par Liouvill. Un application d ctt formul donn un formul pour l nombr d rprésntations d un nombr ntir positif comm étant la somm d douz nombrs triangulairs. Abstract. An lmntary proof is givn of an arithmtic formula, which was statd but not provd by Liouvill. An application of this formula yilds a formula for th numbr of rprsntations of a positiv intgr as th sum of twlv triangular numbrs. This papr is ddicatd to th mmory of Josph Liouvill (09-).. Introduction In his famous sris of ightn articls, Liouvill [3] gav without proof numrous arithmtic formula. Ths formula ar summarizd in Dickson s History of th Thory of Numbrs [, Volum, Chaptr XI], whr rfrncs to proofs ar givn. Th formula w ar intrstd in involvs a sum ovr sxtupls (a, b, c, x, y, z) of positiv odd intgrs satisfying ax + by + cz n, whr n is a fixd positiv odd intgr. It was statd without proof by Liouvill in [3, sixth articl, p. 33] and rproducd by Dickson [, Volum, formula (M), p. 33]. Lt n b a positiv odd intgr, and lt F : Z C b an odd function. Thn (F (a + b + c) + F (a b c) F (a + b c) F (a b + c)) (.) (d )F (d) 3 d n ax<n a,x odd σ(o(n ax))f (a). Manuscrit rçu l juin 004. Th scond author was supportd by a rsarch grant from th Natural Scincs and Enginring Rsarch Council of Canada.

2 4 Erin McAf, Knnth S. Williams In (.) o(n) dnots th odd part of th positiv intgr n, that is o(n) n/ s, whr s n. Non of th authors rfrrd to by Dickson giv a proof of this rsult, although Nasimoff [5] indicats how an analytic proof can b givn. Nor hav w bn abl to locat a proof in th mathmatical litratur. In this papr w giv what w bliv to b th first proof of this rsult. Our proof is ntirly lmntary using nothing mor than th rarrangmnt of trms in finit sums. Prliminary rsults ar provd in Sctions -5. Any omittd dtails can b found in [4]. Th proof of (.) is givn in Sction 6. An application of (.) to triangular numbrs is givn in Sction 7.. Th quation kx + y n and th quantitis L, L, L 3, M, M, M 3 Lt N {,, 3,...}. Lt k, n N. In this sction w ar intrstd in th numbr of solutions (x,, y) N 3 of th quation (.) kx + y n, whr k, n ar fixd positiv intgrs and th unknowns x,, y ar rquird to satisfy crtain inqualitits and crtain congruncs modulo. Throughout this papr all congruncs ar takn modulo. Thus, for xampl, th sum <k counts th numbr of tripls (x,, y) N 3 satisfying th quation kx+y n with < k and x a (mod ), b (mod ), y c (mod ), whr a, b, c ar fixd intgrs and k, n ar fixd positiv intgrs. Dfinition.. L (a, b, c) L 3 (a, b, c) M (a, b, c) <k k x>y, L (a, b, c), M (a, b, c), M 3 (a, b, c) >k xy Th following rsult follows immdiatly from Dfinition..,,.

3 Lmma.. An arithmtic formula of Liouvill 5 L (a, b, c) + L (a, b, c) + L 3 (a, b, c) M (a, b, c) + M (a, b, c) + M 3 (a, b, c). Nxt w valuat L 3 (a, b, c). modulo.) Lmma.. L 3 (a, b, c) Proof. From (Rcall that all congruncs ar takn 0, if k n or k n, k b or k n, k b, n k a + c, ( n ) k 4 ( )a ( + ( ) n/k ), othrwis. L 3 (a, b, c) k x+yn/k x a,k b,y c w asily obtain th assrtd valuation of L 3 (a, b, c). M 3 (a, b, c) can b calculatd similarly. Lmma.3. 0, if a c or a c and n a(k + b),, othrwis. M 3 (a, b, c) d n,d>k d k+b n/d a Our nxt lmma givs a usful rlationship btwn L (a, b, c) and M (a, b, c). Lmma.4. Proof. W hav M (a, b, c) M (a, b, c) L (a + c, k + b, c). x>y >k x a+c, k+b, y c k(x+y)+yn x a+c, b, y c, L (a + c, k + b, c), as assrtd. For positiv intgrs n and k w st {, if k n, δ(n, k) 0, if k n. kx+(k+)yn x a+c, b, y c Th nxt rsult is a simpl consqunc of Lmmas.,. and.3.

4 6 Erin McAf, Knnth S. Williams Lmma.5. If k and n ar positiv odd intgrs thn ( ) (n/k) L (,, 0) M (,, 0) L (, 0, 0) L (,, 0) δ(n, k). 3. Th quation kx + y n and th quantitis R, R, R 3, S, S, S 3 In this sction w introduc th quantitis R, R, R 3, S, S, S 3 formd from L, L, L 3, M, M, M 3 rspctivly by taking th sums ovr rathr than. Dfinition 3.. R (a, b, c) R 3 (a, b, c) S (a, b, c) <k k x>y, R (a, b, c), S (a, b, c), S 3 (a, b, c) >k xy W now giv rlationships btwn R, R, R 3, S, S, S 3. Ths rlationships ar vry similar to thos w saw in Sction for L, L, L 3, M, M, M 3. Th following rsult follows immdiatly from Dfinition 3.. Lmma 3.. R (a, b, c) + R (a, b, c) + R 3 (a, b, c) S (a, b, c) + S (a, b, c) + S 3 (a, b, c). Th sums R 3 (a, b, c) and S 3 (a, b, c) ar asily valuatd. Lmma 3.. R 3 (a, b, c) 0, if k n or k n, k b or k n, k b, n k a + c, (n k) k 4 ( )a ( + ( ) n/k ), othrwis. Lmma , if a c or a c and n a(k + b), (d k), if a c and n a(k + b). S 3 (a, b, c) d n,d>k d k+b n/d a Th following lmma givs a rlationship btwn R (a, b, c) and S (a, b, c). Th proof is similar to that of Lmma.4.,,.

5 Lmma 3.4. An arithmtic formula of Liouvill 7 S (a, b, c) R (a + c, k + b, c) kl (a +, k + b, c). Th valus of R 3 (,, 0) and S 3 (,, 0) whn k and n ar both odd follow asily from Lmmas 3. and 3.3. Lmma 3.5. If k and n ar positiv odd intgrs thn ( ) n k R 3 (,, 0) δ(n, k), S 3 (,, 0) Solutions of kx + by + cz n (k, n fixd odd intgrs) Throughout this sction k and n ar rstrictd to b positiv odd intgrs. W ar concrnd with solutions (x, b, y, c, z) N 5 of kx + by + cz n with b, c, x odd and y, z vn. For such solutions w hav b+c k, b c k and x z, so that + and + b c<k, x<z b c<k, x>z + + b+c>k b c>k, x<z b c>k, x>z With this in mind, w find it usful to dfin th following sums. Dfinition 4.. A B B 3 b c<k, x<z b c<k, x>z, A, B, B 4 b+c>k b c>k, x<z b c>k, x>z.,..

6 Erin McAf, Knnth S. Williams Immdiatly from Dfinition 4. w obtain Lmma 4.. A + A B + B + B 3 + B 4. It is also convnint to dfin anothr sum. Dfinition 4.. C <z. W now giv various rlations btwn A, A, B, B, B 3, B 4 and C with th ultimat goal of dtrmining th quantity A B + C, s Lmma 4.7. Lmma 4.. B y<x<z. Using Lmma 4. w obtain Lmma 4.3. C B S (, 0, 0). Lmma 4.4. B 4 y<z<x. Appaling to Lmmas 4. and 4.4 w obtain Lmma 4.5. B + B 4 B 3 S (, 0, 0). Lmma 4.6. A B 3 + R (, 0, 0).

7 Proof. From Dfinition 4. w obtain A b+c>k b+c>k, y>z b+c>k, y>z An arithmtic formula of Liouvill b+c>k, y<z b+c>k, yz +, b+c>k, yz whr w usd th transformation (b, y, c, z) (c, z, b, y) in th last stp. W now xamin th abov two sums. Firstly, w hav b+c>k, yz by Dfinition 3., and scondly, b+c>k, y>z kx+(b+c)yn b+c>k b,c,x odd, y vn vn >k x odd, y vn >k x, 0,y 0 R (, 0, 0), kx+b(z+y )+czn b+c>k b,c,x odd, y,z vn kx+by+(b+c)zn b+c>k kx+by+zn b+c,>k b,c,x odd,,y,z vn (y z + y ) (y y )

8 30 Erin McAf, Knnth S. Williams kx+by+zn >b,>k b,x odd,,y,z vn kx+by+(k+c )zn k+c >b b,c,x odd, y,z vn k(x+z)+by+czn b c<k kx +by+czn b c<k,x >z b,c,x odd, y,z vn B 3, ( k + c ) (c c ) (x x + z) by Dfinition 4.. This complts th proof of th lmma. Appaling to Lmmas 4.3, 4., 4.5 and 4.6, w obtain Lmma 4.7. A B + C R (, 0, 0) S (, 0, 0) S (, 0, 0). 5. Solutions of ax + by + cz n (n fixd odd intgr) Throughout this sction k and n ar positiv odd intgrs. W ar intrstd in th solutions (a, x, b, y, c, z) N 6 of th quation ax + by + cz n with a, b, c, x, y, z odd and a + b + c k or a + b c k or a b c k. W dnot th numbr of such solutions by U, V and W rspctivly. Dfinition 5.. U a+b+ck, V a+b ck, W a b ck In th nxt thr lmmas w rlat U, V, W to quantitis of th prvious thr sctions. Lmma 5.. U (k ) δ(n, k) + 3 (kl (,, 0) R (,, 0)) + 3A..

9 Proof. W hav U a+b+ck An arithmtic formula of Liouvill 3 a+b+ck x y,y z,z x + a+b+ck xactly of x,y,z qual + a+b+ck xyz W bgin by considring th first sum on th right hand sid of th abov xprssion. Th condition x y, y z, z x, compriss six possibilitis, namly, x < y < z, x < z < y, y < x < z, y < z < x, z < x < y and z < y < x. Simpl transformations of th summation variabls show that ths six cass yild th sam contribution to th sum. Thus a+b+ck x y,y z,z x a+b+ck <z (k b c)x+by+czn <z b,c,x,y,z odd kx+b(y x)+c(z x)n <z b,c,x,y,z odd kx+by +cz n y <z b,c,x odd, y,z vn y z 3A 3 (y y x, z z x) 3 kx+(b+c)yn b,c,x odd, y vn yz (by Dfinition 4.).

10 3 Erin McAf, Knnth S. Williams 3A 3 <k x odd, y, vn 3A 3 R (, 0, 0), by Dfinition 3.. Nxt w considr th scond sum on th right hand sid of th xprssion for U. Thr ar six diffrnt cass in which xactly two of x, y, and z ar qual. Simpl transformations of th summation variabls show that th cass x y < z, x z < y, and y z < x, lad to th sam sum, and similarly for th cass x < y z, y < x z, and z < y x. As abov w obtain a+b+ck xactly of x,y,z qual 3 (kl (,, 0) R (,, 0) + R (, 0, 0)). Finally, w considr th third sum on th right hand sid of th xprssion for U. W hav a+b+ck xyz (a+b+c)xn a+b+ck a,b,c,x odd δ(n, k) a+b+ck a,b,c odd Putting ths thr sums togthr, w obtain ( k δ(n, k) ). U 3A 3 R (, 0, 0) + 3 (kl (,, 0) R (,, 0) + R (, 0, 0)) ( k ) + δ(n, k) as assrtd. (k ) δ(n, k) + 3 (kl (,, 0) R (,, 0)) + 3A, Lmma 5.. V (km (,, 0) + S (,, 0)) + B.

11 Proof. W hav V a+b ck An arithmtic formula of Liouvill 33 a+b ck x>y a+b ck + + (k b+c)x+by+czn b c<k b,c,x,y,z odd a+b ck a+b ck xy + kx+b(y x)+c(z+x)n b c<k b,c,x,y,z odd kx+by +c(z+x)n b c<k b,c,x,z odd, y vn kx+by+cz n b c<k, z >x b,c,x odd, y,z vn b c<k, z>x B + k + + +,x odd, y vn + + (a+b)x+czn a+b ck a,b,c,x,z odd a+b ck xy x+czn ck a+b a,b,c,x,z odd, vn x+czn c+k c,x,z odd, vn (c+k)x+czn c,x,z odd kx+cyn y>x c,x odd, y vn + c + k k + c,x odd, y vn B + (km (,, 0) + S (,, 0)), as rquird. Lmma 5.3. W S (, 0, 0) + C.

12 34 Erin McAf, Knnth S. Williams Proof. W hav W a b ck (k+b+c)x+by+czn b,c,x,y,z odd kx+b(x+y)+c(x+z)n b,c,x,y,z odd kx+by +cz n y >x, z >x b,c,x odd, y,z vn y>x, z>x, y<z <z C + C y>x b+c b,c,x odd,,y vn x odd,,y vn C + S (, 0, 0), y>x, z>x, y>z kx+(b+c)yn y>x b,c,x odd, y vn + (by Dfinition 4.) y>x, z>x,yz as claimd. Lmma 5.4. U 3V + 3W (k ) δ(n, k) 3 σ(o(n kx)). kx<n x odd

13 An arithmtic formula of Liouvill 35 Proof. Applying Lmmas 5., 5. and 5.3 w find U 3V + 3W (k ) δ(n, k) + 3 (kl (,, 0) R (,, 0)) + 3A ( ) ( ) 3 (km (,, 0) + S (,, 0)) + B + 3 S (, 0, 0) + C (k ) δ(n, k) + 3 ( k(l (,, 0) M (,, 0)) R (,, 0) ) S (,, 0) + S (, 0, 0) + 3(A B + C). Appaling to Lmmas.5 and 4.7, w obtain U 3V + 3W (k ) δ(n, k) + 3 ( ( ) (n/k) k(l (, 0, 0) L (,, 0) δ(n, k)) ) R (,, 0) S (,, 0) + S (, 0, 0) 3 ( ) R (, 0, 0) + S (, 0, 0) + S (, 0, 0) (k ) δ(n, k) + 3 ( (S (, 0, 0) + kl (,, 0)) (R (, 0, 0) ( ) n k ) kl (, 0, 0)) δ(n, k) R (,, 0) S (,, 0). Appaling to Lmmas 3.4 and 3.5 w obtain U 3V + 3W (k ) δ(n, k) + 3 ( R (,, 0) S (,, 0) R 3 (,, 0) ) R (,, 0) S (,, 0) (k ) δ(n, k) 3 ( R (,, 0) + R (,, 0) + R 3 (,, 0) ) + S (,, 0) + S (,, 0) + S 3 (,, 0) (k ) δ(n, k) 3 x,,y 0

14 36 Erin McAf, Knnth S. Williams as assrtd. (k ) δ(n, k) 3 (k ) δ(n, k) 3 kx<n yn kx x odd odd, y vn kx<n x odd n kx odd 6. Proof of Liouvill s formula W now us th rsults of Sctions -5 to prov Liouvill s formula. Thorm 6..Lt n b a positiv odd intgr, and lt F : Z C b an odd function. Thn (F (a + b + c) + F (a b c) F (a + b c) F (a b + c)) (d )F (d) 3 d n ax<n a,x odd σ(o(n ax))f (a). Proof. First w hav (F (a + b + c) + F (a b c) F (a + b c) F (a b + c)) F (a + b + c) + F (a + b c) a+b+ck F (a b c) Nxt w look at ach of ths sums individually. W hav F (a + b + c) F (k) k N k N k odd k odd and, as F is odd, F (a b c) k N k odd F (a + b c) k N k odd F (k) (W V ), F (k) (V W ), F (a b + c). F (k)u,

15 An arithmtic formula of Liouvill 37 F (a b + c) k N k odd F (k) (V W ). Thrfor by Lmma 5.4 (F (a + b + c) + F (a b c) F (a + b c) F (a b + c)) k N k odd F (k)(u 3V + 3W ) F (k) (k ) δ(n, k) 3 σ(o(n kx)) k N kx<n k odd x odd (d ) F (d) 3 σ(o(n ax))f (a), d n ax<n x,a odd as rquird. In th nxt sction w giv an application of Thorm Sums of twlv triangular numbrs Thr ar many applications of Thorm 6.. W just giv on application to triangular numbrs. Th triangular numbrs ar th nonngativ intgrs Lt T k k(k + ), k N {0}. {T k k 0,,, 3,...} {0,, 3, 6, 0, 5,...}, and for m N and n N {0}, lt δ m (n) card{(t,..., t m ) m n t + + t m }. Thn δ m (n) counts th numbr of rprsntations of n as th sum of m triangular numbrs. Th valuation of δ 4 (n) was known to Lgndr, namly (7.) δ 4 (n) σ(n + ), n N {0}. Ono, Robins and Wahl [6, p. 0] hav provd this rsult using modular forms. An lmntary proof has bn givn by Huard, Ou, Sparman and Williams [, p. 59]. Using Thorm 6. and th abov formula for δ 4 (n), w obtain a nw valuation of δ (n).

16 3 Erin McAf, Knnth S. Williams Thorm 7.. Lt n b a positiv intgr. Thn δ (n) 9 σ 5(n + 3) 9 σ 3(n + 3) Proof. By (7.) w hav δ (n) m +m +m 3 n m,m,m 3 0 m +m +m 3 n m,m,m 3 0 n +n +n 3 n+3 n,n,n 3 odd δ 4 (m )δ 4 (m )δ 4 (m 3 ) n + s n n+3 s, n,n odd σ(m + )σ(m + )σ(m 3 + ) σ(n )σ(n )σ(n 3 ). σ 3 (n )σ(n ). W apply Thorm 6. with F (x) x 3, so that F (a + b + c) + F (a b c) F (a + b c) F (a b + c) 4abc, and n rplacd by n + 3. Th lft-hand sid of Thorm 6. givs (F (a + b + c) + F (a b c) F (a + b c) F (a b + c)) ax+by+czn+3 ax+by+czn+3 4 4abc n +n +n 3 n+3 n,n,n 3 odd 4δ (n). Th right hand sid givs (d )d 3 3 d n+3 ax<n+3 a,x odd σ(n )σ(n )σ(n 3 ) σ(o(n + 3 ax))a 3 σ 5(n + 3) σ 3(n + 3) 3 n + s n n+3 s, n,n odd σ(n )σ 3 (n ). Equating th lft-hand sid and th right-hand sid, w obtain th rquird assrtion.

17 An arithmtic formula of Liouvill 39 Rfrncs [] L. E. Dickson, History of th Thory of Numbrs. Vol. (99), Vol. (90), Vol. 3 (93), Carngi Institut of Washington, rprintd Chlsa, NY, 95. [] J. G. Huard, Z. M. Ou, B. K. Sparman, K. S. Williams, Elmntary valuation of crtain convolution sums involving divisor functions. Numbr Thory for th Millnium II, M. A. Bnntt t al., ditors, A. K. Ptrs Ltd, Natick, Massachustts, 00. [3] J. Liouvill, Sur qulqus formul générals qui puvnt êtr utils dans la théori ds nombrs. (prmir articl) 3 (5), 43 5; (duxièm articl) 3 (5), 93 00; (troisièm articl) 3 (5), 0-0; (quatrièm articl) 3 (5), 4 50; (cinquièm articl) 3 (5), 73 ; (sixièm articl) 3 (5), ; (sptièm articl) 4 (59), ; (huitièm articl) 4 (59), 73 0; (nuvièm articl) 4 (59), 0; (dixièm articl) 4 (59), (onzièm articl) 4 (59), 304; (douzièm articl) 5 (60), ; (trizièm articl) 9 (64), 49 56; (quatorzièm articl) 9 (64), ; (quinzièm articl) 9 (64), 3 336; (sizièm articl) 9 (64), ; (dix-sptièm articl) 0 (65), 35 44; (dix-huitièm articl) 0 (65), [4] E. McAf, A thr trm arithmtic formula of Liouvill typ with application to sums of six squars. M. Sc. thsis, Carlton Univrsity, Ottawa, Canada, 004. [5] P. S. Nasimoff, Applications to th Thory of Elliptic Functions to th Thory of Numbrs. Moscow, 4. [6] K. Ono, S. Robins, P. T. Wahl, On th rprsntation of intgrs as sums of triangular numbrs. Aquations Math. 50 (995), Erin McAf and Knnth S. Williams School of Mathmatics and Statistics Carlton Univrsity Ottawa, Ontario, Canada KS 5B6 rinmcaf@rogrs.com, williams@math.carlton.ca