CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS III

rerit: October 1, 01 CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS III Zhi-Hog Su arxiv:101.v [math.nt] 5 Oct 01 School of Mathematical Scieces, Huaiyi Normal Uiversity, Huaia, Jiagsu 001, PR Chia Email: zhihogsu@yahoo.com Homeage: htt://www.hytc.edu.c/xsjl/szh Abstract. Let > be a rime, ad let R be the set of ratioal umbers whose deomiator is corime to. Let {P x} be the Legedre olyomials. I this aer we maily show that for m,,t R with m 0 mod, ]t 1 x x+t mod, 1 x +mx+ m [/] m +7 mod, 1 m where a is the Legedre symbol ad [x] is the greatest iteger fuctio. As a alicatio we solve some cojectures of Z.W. Su ad the author cocerig 1 /m mod, where m is a iteger ot divisible by. MSC: Primary 11A07, Secodary C5, 11E5, 11L10, 05A10 Keywords: Cogruece; Legedre olyomial; character sum; biary quadratic form; ellitic curve 1. Itroductio. Let {P x} be the Legedre olyomials give by P 0 x 1, P 1 x x, +1P +1 x +1xP x P 1 x 1. It is well ow that see [B,. 151], [G,.1-.1] 1.1 P x 1 From 1.1 we see that [/] 1 x 1! d dx x 1. 1. P x 1 P x, P m+1 0 0 ad P m 0 1m m We also have the followig formula due to Murhy [G,.15]: + x 1 + x 1. 1. P x 1 m. m

+ We remar that +. Let Z be the set of itegers, ad for a rime let R be the set of ratioal umbers whose deomiator is corime to. Let [x] be the greatest iteger ot exceedig x, ad let a be the Legedre symbol. I [S-S] the author showed that for ay rime > ad t R, 1. 1.5 1. P 1t ] t ] t 1 1 x t +x+tt 9 mod, x t+5 1 x+9t+7 mod, x +t 5x+t 1t+11 mod. I the aer, by usig elemetary ad straightforward argumets we rove that 1.7 ] t 1 x x+t mod. Moreover, for m, R with m 0 mod we have 1.8 ad 1 1 x +mx+ x +mx+ m m 1 m+1 m [/] m ] m ] m mod if 1 mod, m mod if mod m +7 1 m mod. It is well ow that see for examle [S,.1-] the umber of oits o the curve y x +mx+ over the field F with elemets is give by 1 x #E x +mx+ +mx+ +1+. For ositive itegers a,b ad, if ax +by for some itegers x ad y, we briefly say that ax + by. Recetly the author s brother Zhi-Wei Su[Su1,Su] ad the author[s] osed some cojectures for 1 /m modulo, where > is a rime ad m Z with m. For examle, Zhi-Wei Su cojectured that [Su, Cojecture.8] for ay rime >, 1.9 1 9 { 0 mod if 19 1, x mod if 19 1 ad so x +19y.

Usig 1.8 ad ow character sums we determie ] x mod for 11 values of x see Corollaries.1-.11, ad 1 /m mod for m 00, 580, 90, 9, 15,0,,55,5000, 188000. Thus we solve some cojectures i [Su1,Su] ad [S]. For examle, we cofirm 1.9 i the case 19 1 ad rove 1.9 whe 19 1 ad the modulus is relaced by. Let > be a rime. I the aer we also determie 1 /8 mod ad establish the geeral cogruece 1.10 1 x1 x 1 x mod, ad ose some cojectures o suercogrueces.. Cogrueces for ] x mod. Lemma.1. Let be a odd rime. The i ii iii 1 mod for 0,1,..., 1, mod for 0,1,..., [ ], [ ]+ 1 7 1 1 mod for ad 0,1,..., [ ]. Proof. For {0,1,..., 1 1 } we have holds. Now suose {0,1,...,[ 1 ]}. It is clear that 1 mod. Thus i 1 1 1!! +1+ 1!! +1+!!!! mod. Thus ii is true. iii was give by the author i [S, the roof of Lemma.]. The roof is ow comlete. Lemma.. Let > be a rime ad {0,1,...,[ 1 ]}. The [ ] [ ] [ ] 1 [ 1 ] +1 + [ ] [ ] + 1 mod.

Proof. Usig Lemma.1i we see that [ ] [ ] [ [ ] ]! [![ ]![ ]! ] [ [ ] ] 1 [ ] [ [ ] ] 1 1 [ ] [ ] [ ] If 1 mod, usig Lemma.1i we see that 1 1 1 + 1 1 + 1 + 1! [ mod. ]![+1 ]+! 1 + 1 1! 1 1 1! 1 mod.! Thus, from the above ad Lemma.1 we deduce that [ ] [ ] [ ] 1 1 1 + [ ] + 1! 1! 1! 1 +! 1 [ ]+ 1 [ ]+ / / 7 7 [ ] 1 [ ] [ ] mod. If mod, usig Lemma.1i we see that 1 +1 1 + +1 + 1 +1 5 +1 + +1 +1 1 +! +1 +1 5!! 1 1! +1 5! mod.! Thus, from the above ad Lemma.1 we deduce that [ ] [ ] [ ] 1 +1 + [ ] + 1!! +1 1! +! 1 [ ]+ [ ]+ / / 7 This comletes the roof. + 7 [ ] 1 [ ] +1 [ ] mod.

Theorem.1. Let > be a rime ad m, R with m 0 mod. The 1 x +mx+ m 1 m ] m mod if 1 mod, +1 m m m m mod if mod. ] Proof. For ay ositive iteger it is well ow that see [IR, Lemma,.5] 1 { 1 mod if 1, x 0 mod if 1. For,r Z with 0 r 1 we have 0 +r 1. Thus, ad therefore.1 1 { 1 mod if 1 r, x +r 0 mod if 1 r 1 x +mx+ 1 1 1 1/ r0 1 1 1/ 1/ 1/ r 1/ 1/ 1/ r0 1 1 r 1/ x +mx 1 r0 x r mx r 1 r m r 1 r 1/ 1 r 1 r r 1/ 1 r 1 r r 1 x +r m 1 r r 1 m 1 r r 1 mod. If 0 mod, from the above we deduce that { 1 x 1 +mx+ 1 1 x +mx+ 1 1 m mod if 1, 0 mod if. Thus alyig 1. ad Lemma. with [ 1 ] we get ] 0 { 1 [ [ 1 ] ] [ 1 [ 1 ] 1 ] 1 1 1 1 1 1 mod if 1, 0 if. 5

Hece the result is true for 0 mod. Now we assume 0 mod. From.1 we see that 1 x +mx+ 1 x +mx+ 1 1/ r O the other had, where ] m m [ [ 1 ] [ ] ] [ 1 ] 1 [ 1 r 1 ] [ ] 1 [ ] m [ m [ [ 1 ] [ ] ] [ 1 ] m [ ] m [ 1 [ 1 ] [ 1 ] [ ] ] [ ] m [ ] m δm, 1 [ 1 [ 1 ] 1 ] [ ] δm, Hece, by the above ad Lemma. we get δm, ] m m 1 [ 1 ] [ ] [ ] 1 r r 1 r m + [ ] + 1 m mod. [ ] 1 1 ] [ ] 7 [ 1 ] m 7 [ m 7 [ m m 1 if 1 mod, m +1 m if mod. ] 1 ] mod, Sice [ 1 ] 1 1 [ ] +1 [ ] [ ] 1 [ 1 ] [ ] 1 [ ] +1 1 [ ] 1 ]+[ [ 7 ] + + 1 [ ] [ ] [ ] [ ]+1 / 1 [ ] 1 ]+[ 7 [ ] mod. m 1 [ 1 ]+[ ] 1 1 [+1 ] 1 [ 1 ] 1 mod, 1 1

from the above we deduce that m δm, ] m This comletes the roof. [ 1 ] 1 1 + [ ] [ ] + 1 m x +mx+ mod. Remar.1 The cogruece.1 has bee give by the author i [S]. Corollary.1. Let,,11 be a rime. The { 1 1 ] a mod if 1, a +b ad a 1, 11 0 mod if mod.. Proof. By [S5, Corollary.1 ad.] we have 1 x 11x+1 { 1 x x 1 + a if 1, a +b ad a 1, 0 if mod. Thus, taig m 11 ad 1 i Theorem.1 we obtai the result. Corollary.. Let > 5 be a rime. The 7 10 ] 5 1 d 5 5 1 c mod if 8 1, c +d ad c 1, 5 5 d 10 mod if 8, c +d ad d 1, 0 mod if 5,7 mod 8. Proof. Usig [S, Lemma.] we see that 1 x 0x+5 1 x 0 x+5 1 0 if 5,7 mod 8. 1 x 0x 5 1 +7 8 c if 1 mod 8, c +d ad c 1, 1 8 c if mod 8, c +d ad c 1, By [S,.117] we have { c 1 1 [ ] 8 1 1 8 +d mod if c +d 1 mod 8, 1 c 1 d 8 c 1 d 8 c mod if c +d mod 8 with c d. Now taig m 0 ad 5 i Theorem.1 ad alyig the above we deduce the result. 7

Corollary.. Let > 5 be a rime. The 11 5 ] 5 5 1 5 A mod if 1 1, A +B ad A 1, 5 5 A 5 mod if 1 7, A +B ad A 1, 0 mod if mod. Proof. By [S, Lemma.] or [S5, Corollary.1 with t 5/ ad.] we have 1 x. 15x+ { A if 1, A +B ad A 1, 0 if mod. Thus, taig m 15 ad i Theorem.1 we obtai the result. Corollary.. Let > 5 be a rime. The 1010 1 L mod 5 10 ] 800 if 1 1, L +7M ad L 1, 10 10 L 10 mod if 1 7, L +7M ad L 1, 0 mod if mod. Proof. From [S, Corollary.] we ow that 1 x 10x+50 {. L if 1, L +7M ad L 1, 0 if mod. Thus taig m 10 ad 50 i Theorem.1 we deduce the result. Corollary.5. Let > 7 be a rime. The 15 15 1 C mod 105 ] 5 if 1,9,5 mod 8, C +7D ad C 1, 15 15 D 105 mod if 11,15, mod 8, C +7D ad D 1, 0 mod if,5, mod 7. Proof. Sice x 7 5 x 7+98 x +1x +11x, from [R1,R] we see that 1 x 5x+98 1 1 1 x +1x +11x.5 { 1 +1 C 7 C if 1,, mod 7 ad C +7D, 0 if,5, mod 7. Suose 1,, mod 7 ad so C +7D. By [S,.117] we have { C. 7 [ ] mod if 1,9,5 mod 8 ad C 1 mod, 7 C 7 D C mod if 11,15, mod 8 ad D 1 mod. Now taig m 5 ad 98 i Theorem.1 ad alyig all the above we deduce the result. 8

Corollary.. Let be a rime such that,,5,7,17. i If,5, mod 7, the ] 171 1785 85 0 mod. ii If 1,, mod 7 ad so C +7D for some C,D Z, the { 171 1785 55 55 1 C mod if 1 ad C 1, ] 85 5555 D 1785 mod if ad D 1. Proof. From [W,.9] we ow that 1 x +x+x +1x { C 7 C if 1,, mod 7 ad C +7D, if,5, mod 7. As x +x+x +1x x +/x+10/x +/x, we see that 1 x +x+x +1x 1 +/x+10/x +/x 1 +x+10x +x x1 x1 1 +8x+0x +x 1 x + 51 x +17x+ 1 x1 ad 1 x + 51 x +17x+ 1 x 17 1 + 51 x 595 1 x + 558 17 x +17x 17 + 1 1 x 595x+558 x 595 558 1 x+ Now combiig all the above we deduce.7 { 1 x 595x+558 1 +1 C C 7 if C +7D 1,, mod 7, 0 if,5, mod 7. Taig m 595 ad 558 i Theorem.1 ad the alyig.7 ad. we deduce the result. 9.

Corollary.7. Let,,11 be a rime. i If,,7,8,10 mod 11, the ] 7 0 mod. ii If 1,,,5,9 mod 11 ad hece u +11v for some u,v Z, the 1 u mod if 1 ad u 1, 7 ] 1 u mod if 1 ad 8 u, v mod if ad v 1, v mod if ad 8 v. Proof. It is ow that see [RP] ad [JM] 1 x 9 11x+11 11 {.8 u 11 u if 11 1 ad u +11v, 0 if 11 1. Thus alyig Theorem.1 we deduce ] 7 1 u 11 u mod if 11 1, 1 ad u +11v, u 11 u mod if 11 1, ad u +11v, 0 mod if 11 1. Now assume 11 1 ad so u +11v. If u v 1 mod, by [S, Theorem.] we have { u 11 [ ] mod if 1 mod, 11 u 11 v u mod if mod. If u v 0 mod, by [S, Corollary.] we have { u 11 [ ] 11 mod if 1 mod ad 8 u, u 11 v mod if mod ad 8 v. u Now combiig all the above we derive the result. From [RPR], [JM] ad [PV] we ow that for ay rime >,.9 1 x 8 19x+ 19 { u 19 u if 19 1 ad u +19v, 0 if 19 1, 1 x 80 x+ { u u if 1 ad u +v, 0 if 1, 1 x 0 7x+ 7 { u 7 u if 7 1 ad u +7v, 0 if 7 1, 1 x 80 9 1x+1 11 19 17 1 { u 1 u if 1 1 ad u +1v, 0 if 1 1. 10

Thus, usigthemethoditheroofofcorollary.7oecasimilarlydetermie ] 5 100, P [ ] 51 9800 110, P[ ] 5570 80 10815 mod. Lemma.. Let be a rime greater tha, ad let x be a variable. The ] x [/] 1 x mod. 8 Proof. Suose that r {1,5} is give by r mod. The clearly [ ]+ r + r + 1 r +1! + r+ r +r! 1 r r r rr+ +r! Hece.10 1! 9 15! 1!!!!!! mod.!! [ ] [ ]+ [ ]+ This together with 1. yields the result. mod. Theorem.. Let > be a rime ad m, R with m 0 mod. The ] 11, ] m [/] m 1 m m 1 x m x+ mod. Proof. Relacig m by m i Theorem.1 ad the alyig Lemma. we deduce the result. For ositive itegers a 1,a,a,a let q 1 q a1 1 q a 1 q a 1 q a 1 ca 1,a,a,a ;q q < 1. For a 1,a,a,a 1,1,11,11,,,10,10,1,,5,15,1,,7,1 ad,,8,8 it is ow that see [MO, Theorem 1] are weight ewforms. fz 1 ca 1,a,a,a ;q q e πiz 1 11

Corollary.8. Let be a odd rime. The 19 [/] c1,1,11,11; ] 1 1 8 5 mod. Proof. It is easy to see that the result holds for,11. Now assume,11. By the well ow wor of Eichler i 195, we have Sice we obtai {x,y F F : y +y x x } c1,1,11,11;. {x,y F F : y +y x x } { x,y F F : y + 1 x x + 1 { x,y F F : y x x + 1 } 1 + 1 + x x + 1 1 + x 1 19 x+ 108 + + 1 x 1x+8, } x+ 1 x+ 1 + 1 1 x 1 x + 19 108.11 c1,1,11,11; 1 x 1x+8. Usig Theorem. we see that c1,1,11,11; 1 x 1x+8 From 1. ad Lemma. we have 19 ] 1 [ ] P 8 [ ] 1 1 [/] 19 1 1 8 mod. 5 [/] 19 ] mod. 8 1+19/8 8 Thus the result follows. 1

Cojecture.1. Let > be a rime. The c,,10,10; 1 x 1x 11, c,,,1; 1 x 9x 70, c1,,5,15; 1 x x, c1,,7,1; 1 x 75x 50, c,,8,8; 1 x 99x 78. If > is a rime of the form +, from Cojecture.1 ad [S, Theorem.8] we deduce that +1 c,,10,10; 1 x 1x 11 +1+ #E x 1x 11 { N 1 δ if 7 mod 1, N + 7+ +δ if 11 mod 1, where N is the umber of a {0,1,..., 1} such that x x +x a mod is solvable, ad 0 if 7, mod 0, δ 1 if,7,1,9 mod 0, if 11,19 mod 0. Hece.1 c,,10,10; 5+1 N +δ for mod. Theorem.. Let > be a rime, ad let t be a variable. The.1 ] t 1 x x+t 1 mod. Proof. Taig m 1 ad t i Theorem. we see that.1 is true for t 0,1,..., 1. Sice both sides of.1 are olyomials i t with degree less tha 1/, alyig Lagrage s theorem we see that.1 holds whe t is a variable. 1

Theorem.. Let > be a rime ad let t be a variable. i If t + 0 mod, the t + P 1t ] t t 1 t +1 t + ii If t+5 0 mod, the t+10 1 t + mod if 1 mod, ] tt 9 ] tt 9 t + t + mod if mod. ] 9t+7 t+10 t+10 +1 t+10 ] 9t+7 t+10 t+5 mod if 1 mod, t+5 mod if mod. Proof. Sice both sides are olyomials of t with degree at most. It suffices to show that the cogrueces are true for t R. Now combiig 1.-1.5 with Theorem.1 we deduce the result. Corollary.9. Let > be a rime ad m R with m 0 mod. The ad P 1m m m m m ] m [/] m 5 m ] m 1/ m m 9m m+18 mod 8m m ] m [/] [/] m 19 m+1m mod. 8m Proof. Taig t m i Theorem.i ad the alyig [S,.] ad Lemma. we deduce the first cogruece. Taig t m 5/ i Theorem.ii ad the alyig [S5, Theorem.1ii] ad Lemma. we deduce the secod cogruece. Theorem.5. Let > be a rime ad let t be a variable. The ] t 5 t 1 ] t 1t+11 5 t 5 t mod if 1 mod, +1 5 t 5 t t ] 1t+11 5 t 5 t mod if mod. Proof. Sice both sides are olyomials i t with degree at most. It suffices to show that the cogruece is true for all t R with t 5 mod. Set m t 5 ad t 1t+11. The m m t 1t+11 5 t 5 t. 1

Thus, by 1. ad Theorem.1 we have ] t 1 x +mx+ 1 95t t ] 1t+11 5 t 5 t mod if 1, 95t +1 95 t ] t 1t+11 5 t 5 t mod if. For 1 mod we have 9 1 1 mod, For mod we have 9 +1 1 1 mod. Thus the result follows. Corollary.10. Let > be a rime ad m R with m 0 mod. The 5 m m m +18m 7 ] P [ ] 8m m [/] [/] m 1 1 m+1 m mod. 8 m Proof. Taig t 5 m i Theorem.5 ad the alyig [S, Lemma.] ad Lemma. we deduce the result. Corollary.11. Let > be a rime. The 7± { ] a ± 1 mod if 1 mod ad a +b with a 1 mod, 0 mod if mod. Proof. Set t 7± /. The t 1t+11 0. Thus, from Theorem.5 ad the cogruece for ] 0 i the roof of Theorem.1 we deduce { 7± 1 9 ] ± 1 1 1 mod if 1 mod, 0 mod if mod. It is well ow that 1 1 a mod for 1 mod see [BEW,.9]. Thus the corollary is roved. Theorem.. Let > be a rime ad m, R with m 0 mod. The 1 x +mx+ { 1 m m +1 m [/1] [ 1 ] [/1] [ [ 5 1 ] m +7 m mod if 1, 1 ] [ 5 1 ] m +7 mod if. m 15

Proof. Let P α,β x be the Jacobi olyomial defied by +α P α,β x 1 It is ow that see [AAR,.15] +β x+1 x 1..1 P x P 0, 1 x 1 ad P +1 x xp 0,1 x 1. From [B,.170] we ow that Thus, P α,β x.15 P 0,β x +α +α+β +1 1 x α+1! +α α β 1 x 1. 1 α Hece, if 1 mod, the [ ] [ 1 ] ad so m ] P 0, 1 m [ 1 ] 7 1 m [ 1 ] [ 1 ] 1 β 1 1 x. [ 1 ] [ 1 ] [ 1 ] m +7 m 1 [ 1 ] [ 1 ] [ 5 1 ] m +7 mod ; m if mod, the [ ] [ ]+1 ad so 1 m ] m P 0,1 m m m m m m m m [ 1 ]+1 [ 1 ] [ 1 ] [ 1 ] [ 1 ] [ 1 ] 7 1 m [ 1 ] 1 7 m [ 1 ] m +7 m 1 7 m [ 1 ] [ 1 ] [ 5 1 ] m +7 mod. m Now combiig the above with Theorem.1 we deduce the result. 1

. A geeral cogruece modulo. Lemma.1. For ay oegative iteger we have Proof. Let m be a oegative iteger. For {0,1,...,m} set F 1 m, m, m m m F m,. m m For {0,1,...,m+1} set G 1 m, 18 m+m+1 +1 G m, 1 m m +19m +11 m + m +1 m +1. m +1 m +1 For i 1, ad {0,1,...,m}, it is easy to chec that.1. +1 m, m+ m+ F i m+, m+18m +5m+1F i m+1, +07m+1m+1m+5F i m, G i m,+1 G i m,. Set S i F i, for 0,1,,... The m+ S i m+ F i m+,m+ F i m+,m+1 m+18m +5m+1S i m+1 F i m+1,m+1 +07m+1m+1m+5S i m m m m+ F i m+, m+18m +5m+1 F i m+1, +07m+1m+1m+5 m F i m, m G i m, +1 G i m, G i m,m+1 G i m,0 G i m,m+1. Thus, for i 1, ad m 0,1,,...,. m+ S i m+ m+18m +5m+1S i m+1 +07m+1m+1m+5S i m G i m,m+1+m+ F i m+,m++f i m+,m+1 m+18m +5m+1F i m+1,m+1 0. Sice S 1 0 1 S 0 ad S 1 1 10 S 1, from. we deduce S 1 S for all 0,1,,... This comletes the roof. For give rime ad iteger, if α but α+1, we say that α. 17

Lemma.. Let be a odd rime ad,r {0,1,..., 1} with +r. The r r r r 0 mod. Proof. If > 5, the 5!,!,! ad so!!!! 0 mod. If < 5, the <, < 5,!,!,! ad so!!!! 0 mod. If < <, the < <, < <,!,!,! ad so!!!! 0 mod. If <, the < <,!,!,! ad! so!!! 0 mod. If < <, the < < ad so!!!! 0 mod. From the above we see that for >. Therefore, if > ad r >, the r r r r 0 mod. If r < 5, the r > ad so by the above. If <, the r > 5 ad so r r r r by the above. Now uttig all the above together we rove the lemma. Theorem.1. Let be a odd rime ad let x be a variable. The 1 x1 x 1 x mod. Proof. It is clear that 1 1 1 m0 x1 x mi{m, 1} x m x r0 x r r m. m Suose m ad 0 1. If, the <,,!,! ad so!!! 0 mod. If < <, the <, >,! ad! ad so! 0 mod. If!! <, the m > ad so m 0. Thus, from the above ad Lemma.1 18

we deduce that 1 1 m0 1 m0 1 1 1 1 x m m x m m x1 x x 1 m m m m m m m m m m m 1 x r0 x 1 r0 r r r r 1 x By Lemma. we have Thus 1 r r r r r r r x 1 r x r x r 1 r 1 x x m r r r r r r r x r r r x r mod. r for 0 1 ad r 1. r r r r Now combiig all the above we obtai the result. x r 0 mod. Corollary.1. Let be a rime greater tha ad m R with m 0 mod. The 1 m 1 1 1 178/m mod. 8 Proof. Taig x 1 1 178/m 8 i Theorem.1 we deduce the result. Lemma.. Let be a rime of the form + 1 ad a + b a,b Z with a 1 mod. The 1 a if 1 mod 1 ad a, ] 0 [ 1 ] a if 1 mod 1 ad a, b if 5 mod 1 ad a b. Proof. By Lemma.1i ad the roof of Theorem.1 we have 1 [ ] 0 ] 1 1 [ 1 ] [ 1 ] [ 1 ] 1 1 19 mod.

By Gauss cogruece [BEW,.9], 1 a mod. By [S1, Theorem.], 1 1 mod if 1 mod 1 ad a, 1 1 mod if 1 mod 1 ad a, b a a b mod if 5 mod 1 ad b a mod. Thus the result follows. Let > be a rime. By the wor of Morteso[M] ad Zhi-Wei Su[Su],. 1 { a mod if a +b 1 mod ad a, 178 0 mod if mod. I [Su1] Zhi-Wei Su cojectured that 1 0 mod if 7,11 mod 1, 1 [a 8 ] a a mod if 1 1, a +b ad a 1, ab b b mod if 1 5, a +b ad a 1. I [Su], Zhi-Wei Su cofirmed the cojecture i the case mod. Now we rove the above cojecture for rimes 1 mod. Theorem.. Let be a rime of the form +1 ad so a +b with a,b Z ad a 1. The 1 a a mod if 1 mod 1 ad a, a+ 8 a mod if 1 mod 1 ad a, b b mod if 5 mod 1 ad a b. Proof. From Lemma. we have ] 0 r mod, where a if 1 mod 1 ad a, r a if 1 mod 1 ad a, b if 5 mod 1 ad a b. By the roof of Lemma. we have for > >. Thus, alyig Lemma. ad the above we get Set 1 1 1 8 [/] 8 ] 0 r mod. r +q. Usig Corollary.1 we see that 8 178 1 r +q 8 r +rq mod. Thus, alyig. we obtai a r + rq mod. Hece q 1 r mod ad the roof is comlete. 0

. Cogrueces for 1 /m. Theorem.1. Let > be a rime, m R, m 0 mod ad t 1 178/m. The 1 1 P m [ ] t x x+t 1 mod. Moreover, if ] t 0 mod or 1 x x+t 1 0 mod, the.1 1 0 mod. m Proof. Sice 1 t 1 t 8 1 8 1 m, by Theorem.1 we have 1 t 1 m 1 8 mod. From the roof of Lemma. we ow that for [ ] < <. Thus, usig Lemma. ad Theorem. we see that 1 1 t P[ 8 ] t 1 x x+t 1 mod. This together with.1 yields the result. Theorem.. Let > be a rime ad m, R with m 0 mod. The x +mx+ m 1 Moreover, if 1 +mx+ x 1 m 1 [/] 0, the m +7 Proof. By the roof of Lemma. we have m +7 0 mod. As x +mx+ x m 1 x +mx+ 1 x m x+ 1 t t0 m +7 1 m m +7 1 m 0 mod. m m m 1 1 m mod. for < <. We first assume x+ m mod we see that x m 1 m mod. x m

Sice m 0 mod we have 0 mod ad so 1 +mx+ m 0. x Thus the result holds i this case. Now we assume m +7 0 mod. Set t m m ad m 1 178 m m +7. The t 1 178 m 1. From Theorems.1 ad.1 we have 1 x +mx+ 1 m P[ ] t If 1 +mx+ x so 1 m 1 m 1 mod. 0, usig Theorems.1 ad.1 we see that ] t 0 mod ad 0 mod. This comletes the roof. m 1 Theorem. [Su, Cojecture.7]. Let,11 be a rime. The 1 { 0 mod if 11 1, x mod if 11 1 ad so x +11y. Proof. Taig m 9 11 ad 11 11 i Theorem. ad the alyig.8 we deduce the result. Theorem. [Su, Cojecture.8]. Let,,19 be a rime. The 1 9 { 0 mod if 19 1, x mod if 19 1 ad so x +19y. Proof. Taig m 8 19 ad 19 i Theorem. ad the alyig.9 we deduce the result. Theorem.5 [Su, Cojecture.9]. Let,,5, be a rime. The 1 90 { 0 mod if 1, 15 x mod if 1 ad so x +y. Proof. Taig m 80 ad i Theorem. ad the alyig.9 we deduce the result. Theorem. [Su, Cojecture.9]. Let be a rime such that,,5,11,7. The { 1 0 mod if 7 1, 580 0 x mod if 7 1 ad so x +7y. Proof. Taig m 0 7 ad 7 i Theorem. ad the alyig.9 we deduce the result.

Theorem.7 [Su, Cojecture.10]. Let be a rime with,,5,,9,1. The 1 { 0 mod 00 if 1 1, 10005 x mod if 1 1 ad so x +1y. Proof. Taig m 80 9 1 ad 1 11 19 17 1 i Theorem. ad the alyig.9 we deduce the result. Theorem.8 [S, Cojecture.8]. Let > 7 be a rime. The 1 15 { 0 mod if,5, mod 7, 15 C mod if C +7D 1,, mod 7. Proof. Taig m 5 ad 98 i Theorem. ad the alyig.5 we deduce the result. Theorem.9 [S, Cojecture.9]. Let > 7 be a rime ad 17. The 1 { 0 mod if,5, mod 7, 55 55 C mod if C +7D 1,, mod 7. Proof. Taig m 595 ad 558 i Theorem. ad the alyig.7 we deduce the result. Theorem.10 [S, Cojecture.]. Let be a rime such that,,11. The 1 { 0 mod if mod, a mod if a +b 1 mod ad a. Proof. Taig m 11 ad 1 i Theorem. ad the alyig. we deduce the result. Theorem.11 [S, Cojecture.5]. Let > 5 be a rime. The 1 { 0 mod if 5,7 mod 8, 0 5 c mod if c +d 1, mod 8. Proof. Taig m 0 ad 5 i Theorem. ad the alyig the result i the roof of Corollary. we deduce the result. Theorem.1 [S, Cojecture.]. Let > 5 be a rime. The 1 5000 { 0 mod if mod, 5 A mod if A +B 1 mod. Proof. Taig m 15 ad i Theorem. ad the alyig. we deduce the result.

Theorem.1 [S, Cojecture.7]. Let > 5 be a rime. The 1 188000 { 0 mod if, 10 L mod if 1 ad so L +7M. Proof. Taig m 10 ad 50 i Theorem. ad the alyig. we deduce the result. Remar.1 From [O] we ow that the oly j-ivariats of ellitic curves over ratioal field Q with comlex multilicatio are give by 0,1, 15,0,, 0,, 9, 10,55, 90, 580, 00, coicidig with the values of m i. ad Theorems.-.1. 5. Some cojectures o suercogrueces. Cojecture 5.1. Let > 5 be a rime. The 1 1 1 1 1 1 +8 15 1 +8 55 8 + 0 +5 11 +1 5000 50+1 188000 15 8 mod, 55 8 mod for 17, 5 mod, 5 mod for 11, 15 mod, 0 1 mod. Cojecture 5.1 is similar to some cojectures i [Su1].

Cojecture 5.. Let > be a rime. The 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9+ 5 5+ 1 9+ 50 5+1 9 +1 0 90+1 89 10+11 1000 mod 5 for > 5, mod, 1 mod for 5, mod, mod for 5, 15 1 mod 7 for 7, 11 mod for 5,1. 9 Cojecture 5.. Let > be a rime. The 1 0 1 0 1 0 1 0 1 0 1 0 +1 1 15+ 9 9+ 11 99+17 00 855+109 70 585+58 0 1 mod, mod for 7, mod 7 for 7, 1 17 mod, 1 109 mod for 1, 1 58 mod for 7,1. For a iteger m ad odd rime with m let 5

1 Z m 0 m. The we have the followig cojectures cocerig Z m modulo. Cojecture 5.. Let be a odd rime. The x mod if x +y 1 mod 1 with y, x mod if x +y 1 mod 1 with x, Z 1 xy xy mod if x +y 5 mod 1, 0 mod if mod. Cojecture 5.5. Let be a odd rime. The Z 9 { x mod if x +y 1, mod 8, 0 mod if 5,7 mod 8. Cojecture 5.. Let > 5 be a rime. The { x mod if x +y 1 mod, Z Z 50 0 mod if mod. Cojecture 5.7. Let > 5 be a rime. The x mod if 1,9 mod 0 ad so x +5y, Z 1 x mod if,7 mod 0 ad so x +5y, 0 mod if 11,1,17,19 mod 0. Cojecture 5.8. Let > be a rime. The x mod if 1,7 mod ad so x +y, Z 8x mod if 5,11 mod ad so x +y, 0 mod if 1,17,19, mod. Cojecture 5.9. Let > 7 be a rime. The x mod if x +15y 1,19 mod 0, Z 5 Z 9 1x mod if x +5y 17, mod 0, 0 mod if 7,11,1,9 mod 0.

Cojecture 5.10. Let b {7,11,19,1,59} ad let fb 11, 00, 70, 0, 1100 accordig as b 7,11,19,1,59. If is a rime with,,b ad fb, the x mod if x +by, 1x mod if x +by, Z fb x mod if x +by, x mod if x +by, 0 mod if b 1. Cojecture 5.11. Let b {5,7,1,17} ad fb 0,89,1000,900 accordig as b 5,7,1,17. If is a rime with,,b ad fb, the x mod if x +by, 8x mod if x +by, Z fb 1x mod if x +by, x mod if x +by, 0 mod if b 1. Refereces [AAR] G. Adrews, R. Asey, R. Roy, Secial Fuctios,, Ecycloedia Math. Al., vol. 71, Cambridge Uiv. Press, Cambridge, 1999. [B] H. Batema, Higher Trascedetal Fuctios Vol. I, McGraw-Hill Boo Com. Ic., US, 195. [BEW] B.C. Berdt, R.J. Evas ad K.S. Williams, Gauss ad Jacobi Sums, Joh Wiley & Sos, New Yor, 1998. [G] H.W. Gould, Combiatorial Idetities, A Stadardized Set of Tables Listig 500 Biomial Coefficiet Summatios, Morgatow, W. Va., 197. [IR] K. Irelad ad M. Rose, A Classical Itroductio to Moder Number Theory d editio, Grad. Texts i Math. 8, Sriger, New Yor, 1990. [JM] A. Joux et F. Morai, Sur les sommes de caractères liées aux courbes ellitiques à multilicatio comlexe, J. Number Theory 55 1995, 108-18. [MO] Y. Marti ad K. Oo, Eta-quotiets ad ellitic curves, Proc. Amer. Math. Soc. 15 1997, 19-17. [M] E. Morteso, Suercogrueces for trucated +1 F hyergeometric series with alicatios to certai weight three ewforms, Proc. Amer. Math. Soc. 1 005, 1-0. [PV] R. Padma ad S. Veatarama, Ellitic curves with comlex multilicatio ad a character sum, J. Number Theory 1 199, 7-8. [RP] A.R. Rajwade ad J.C. Parami, A ew cubic character sum, Acta Arith. 0 198, 7-5. [R1] A.R. Rajwade, The Diohatie equatio y xx + 1Dx + 11D ad the cojectures of Birch ad Swierto-Dyer, J. Austral. Math. Soc. Ser. A 1977, 8-95. [R] A.R. Rajwade, O a cojecture of Williams, Bull. Soc. Math. Belg. Ser. B 198, 1-. [RPR] D.B. Rishi, J.C. Parami ad A.R. Rajwade, Evaluatio of a cubic character sum usig the 19 divisio oits of the curve y x 19x + 19, J. Number Theory 19 198, 18-19. [S1] Z.H. Su, Sulemets to the theory of quartic residues, Acta Arith. 97 001, 1-77. [S] Z.H. Su, O the umber of icogruet residues of x +ax +bx modulo, J. Number Theory 119 00, 10-1. [S] Z.H. Su, O the quadratic character of quadratic uits, J. Number Theory 18 008, 195-15. 7

[S] Z.H. Su, Cogrueces cocerig Legedre olyomials, Proc. Amer. Math. Soc. 19 011, 1915-199. [S5] Z.H. Su, Cogrueces cocerig Legedre olyomials II, arxiv:101.898v. htt://arxiv.org/ abs/101.898. [S] Z.H. Su, Cogrueces ivolvig, J. Number Theory, to aear, htt://arxiv.org/ abs/110.789. [Su1] Z.W. Su, Oe cojectures o cogrueces, arxiv:0911.55v59. htt://arxiv.org/abs/0911.55. [Su] Z.W. Su, Suer cogruecesad elliticcurvesover F, rerit, arxiv:1011.7. htt://arxiv. org/abs/1011.7. [Su] Z.W. Su, O sums ivolvig roducts of three biomial coefficiets, rerit, arxiv:101.11. htt://arxiv.org/abs/101.11. [Su] Z.W. Su, Cojectures ad results o x mod with x + dy, Proceedigs of Number Theory ad Related Fields, to aear, arxiv:110.5v7. htt://arxiv. org/abs/110.5. [W] K.S. Williams, Evaluatio of character sums coected with ellitic curves, Proc. Amer. Math. Soc. 7 1979, 91-99. 8