plays an important role in many fields of mathematics. This sequence has nice number-theoretic properties; for example, E.

Transcription

1 Tiwnese J. Mth , no., FIBONACCI NUMBERS MODULO CUBES OF PRIMES Zhi-Wei Sun Dertment of Mthemtics, Nnjing University Nnjing 10093, Peole s Reublic of Chin zwsun@nju.edu.cn htt://mth.nju.edu.cn/ zwsun Abstrct. Let be n odd rime. It is well nown tht F 0 mod, where {F n } n 0 is the Fiboncci sequence nd is the Jcobi symbol. In this er we show tht if then we my determine F mod 3 in the following wy: 1/ F mod 3. We lso use Lucs quotients to determine 1/ 0 /m modulo for ny integer m 0 mod ; in rticulr, we obtin 1/ mod. In ddition, we ose three conjectures for further reserch. 1. Introduction The well nown Fiboncci sequence {F n } n 0, defined by F 0 0, F 1 1, nd F n+1 F n + F n 1 n 1,, 3,..., lys n imortnt role in mny fields of mthemtics. This sequence hs nice number-theoretic roerties; for exmle, E. Lucs showed tht F m, F n F m,n for ny m, n N {0, 1,... }, where m, n denotes the gretest common divisor of m nd n. 010 Mthemtics Subject Clssifiction. Primry 11B39, 11B6; Secondry 0A10, 11A07. Keywords. Fiboncci numbers, centrl binomil coefficients, congruences, Lucs sequences. Suorted by the Ntionl Nturl Science Foundtion grnt of Chin nd the PAPD of Jingsu Higher Eduction Institutions. 1

2 ZHI-WEI SUN Let, be rime. It is nown tht F 0 mod, where denotes the Jcobi symbol. In 199 Z. H. Sun nd Z. W. Sun [SS] roved tht if F then the Fermt eqution x + y z hs no integrl solutions with xyz. When F 0 mod, is clled Wll-Sun- Sun rime cf. [CDP] nd [CP,. 3]. It is conjectured tht there should be infinitely mny but rre Wll-Sun-Sun rimes though none of them hs been found. There re some congruences for the Fiboncci quotient F / modulo cf. [W], [SS] nd [ST]; for exmle, in 198 H. C. Willims [W] roved tht F mod. Quite recently H. Pn nd Z. W. Sun [PS] roved tht for ny Z + {1,, 3,... } we hve 1 1 F 1 0 mod 3, which ws conjecture in [ST]. Now we give the first theorem of this er. Theorem 1.1. Let be n odd rime nd let be ositive integer. If, then If 3, then 1/ 0 1/ F mod mod Let be n odd rime nd let Z +. For 0, 1,..., 1/, clerly 1/ 1 1 mod j + 1 nd hence 1/ 0 j< 1/ 1/ 4 mod. 1.3

3 FIBONACCI NUMBERS MODULO CUBES OF PRIMES 3 Thus, for ny integer m 0 mod we hve since nd 1 0 m 1/ 0 1/ 0 1/ m mm 4 4 m 1 4 1/ m mod, mod + for ech + 1/,..., 1 by Lucs theorem cf. [St,. 44]. Recently the uthor [Su10] determined 1 0 /m mod in terms of Lucs sequences. See lso [SSZ], [GZ] nd [Su11] for relted results on -dic vlutions. Let A, B Z. The Lucs sequences u n u n A, B n N nd v n v n A, B n N re defined by nd u 0 0, u 1 1, nd u n+1 Au n Bu n 1 n 1,, 3,... v 0, v 1 A, nd v n+1 Av n Bv n 1 n 1,, 3,.... The sequence {v n } n 0 is clled the comnion of {u n } n 0. Note tht F n u n 1, 1, nd those L n v n 1, 1 re clled Lucs numbers. It is nown tht for ny rime not dividing B we hve u mod nd u 0 mod where A 4B see, e.g., [Su10, Lemm.3]; the integer u / is clled Lucs quotient. The reder my consult [Su06] for connections between Lucs quotients nd qudrtic fields. Our second theorem is s follows. Theorem 1.. Let be n odd rime nd let Z +. integer not divisible by. Then 1/ 0 m mm 4 m mm Let m be ny mu 4 m 4, m mod, 1.

4 4 ZHI-WEI SUN where We lso hve 1/ 0 1 if m 4 mod, m if 4 m 1, /m if 4 m 1. C m 4 m 1/ 0 m + m m δ,1 mod, where C denotes the Ctln number +1 Kronecer symbol δ s,t tes 1 or 0 ccording s s t or not. +1, nd the Remr 1.1. For ny m Z \ {0} nd n N, the sum n 0 /m is closely relted to n 0 /m vi the identity n 0 1 m 4 which cn be esily roved by induction. n + 1 m Now we resent two consequences of Theorem n n Corollry 1.1. Let be n odd rime nd let Z +. Then 1/ 8 mod 1.7 nd 1/ Corollry 1.. Let > 3 be rime. Then tht is, 1/ 0 1/ m n mod. 1.8 mod, mod if 1 mod 1, / mod if mod 1, 1 mod if 7 mod 1, 1/ mod if 11 mod We will show Theorems 1.1 nd 1. in Sections nd 3 resectively. Section 4 is devoted to the roofs of Corollries To conclude this section we ose three conjectures.

5 FIBONACCI NUMBERS MODULO CUBES OF PRIMES Conjecture 1.1. For ny n N we hve 1 n { n mod 9 if 3 n, n 16 4 mod 9 if 3 n. n 0 Also, 3 1 1/ mod 7 0 for every 1,, 3,.... Let > 3 be rime. In 007 A. Admchu [A] conjectured tht if 1 mod 3 then 3 0 mod. 1 Motivted by this nd Theorems 1.1 nd 1., we ose the following conjecture bsed on the uthor s comuttion vi the softwre Mthemtic. Conjecture 1.. Let be n odd rime nd let Z +. i If 1 mod 3 or > 1, then mod. ii Suose. If 1, mod or mod or >, then 4 1 mod. 0 If 1, 3 mod or 3 mod or >, then mod. iii If 1, mod or mod or >, then mod. If 1, 3 mod or 3 mod or >, then mod.

6 6 ZHI-WEI SUN Conjecture 1.3. Let, be rime nd set q : F /. Then 1 1 F 3 q + 4 q mod nd 1 1 L q 1 4 q mod. Remr 1.. identities It is interesting to comre Conjecture 1.3 with the two 1 F 4π nd 1 L π obtined by utting x ± 1/ in the nown formul rcsin x 1 x x <.. Proof of Theorem 1.1 Lemm.1. Let be n odd rime nd let {0,..., 1/} with Z +. Then 1/ / 0<j j 1 mod 3..1 Proof. Clerly.1 holds for 0. Below we ssume 1 1/. Note tht 1/ + j1 j 1 4! j1 j 1 4! j j1 1 j 1 j 1 mod 4

7 FIBONACCI NUMBERS MODULO CUBES OF PRIMES 7 which ws observed by Z. H. Sun [S11, Lemm.] in the cse 1. Thus, in view of 1.3 we hve 16 1/ 4 1/ 4 1/ 1 4 1/ 4 1/ mod. 1/ So.1 holds. Lemm.. Let be n odd rime nd let Z +. If 3, then / mod 3.. When, we hve L 1 F F mod 4..3 Proof. Note tht 1/ mod nd 1 1 1/ 1/ 1/ + mod. Thus / / mod 3.

8 8 ZHI-WEI SUN It follows tht 1/ 1/ / / / mod 3, 1 which is equivlent to. times 3 1/. So. is vlid if > 3. Now ssume tht. As L n Fn + 1 n L n 1 n for ll n N, by [SS, Corollry 1] we hve L mod. Thus, in view of [Su10, Lemm.3], L 1 / L mod. Write L + Q with Q Z. Then F L 4 1 Note tht nd L Q 4 Q mod 4. L F + F 1 F +1 F F + F 1 + L 1 F +1 L +1 F + Therefore L 1 F + 1 F L. F 4 L 1 L 1 F 4 F Q 1 F mod 4.

9 This roves.3. FIBONACCI NUMBERS MODULO CUBES OF PRIMES 9 The following Lemm ws osed s [Su1, Conjecture 1.1]. Lemm.3. Let be n odd rime nd let H 0<j 1/j for 0, 1,,.... If, then H q δ,3 mod,.4 where q denotes the Fiboncci quotient F /. If > 3, then 1 0 H 3 q mod,. where q denotes the Fermt quotient 1 1/. Proof. It is esy to verify.4 for 3. Below we ssume > 3. The desired congruences essentilly follow from [MT, 37]. Here we rovide the detils. Putting t 1, 1/ in [MT, 37] we get nd H H u 3, 1 mod.6 u /, 1 mod..7 Note tht u 3, 1 F nd u /, 1 /3 for ll 0, 1,,.... Since 1 1 q mod nd q mod by [Gr] nd [S08, Theorem 4.1iv] resectively,.7 imlies tht 1 0 H q mod. Now we wor with >. Recll tht for ny n N we hve F n αn β n α β nd L n α n + β n,

10 10 ZHI-WEI SUN where α nd β re the two roots of the eqution x x 1 0. By [PS, 3. nd 3.7], 1 β 1 α 1 1 α L 1 α 1 L 1 1 mod. Since αβ 1 nd α α + 1 α + 1 mod, we hve 1 1 Hence α α + 1α F 1 F α β 1 L 1 α + L 1 mod. 1 α β α β β α L 1 1/ L 1 1 α β α β+1 4 q mod L 1 since L 1 F mod by [ST,. 139]. Combining this with.6 we obtin H The roof of Lemm.3 is now comlete. 4 q q mod. Proof of Theorem 1.1. Let us first recll the following two identities: F n+1 Thus we hve n/ 0 n nd n 0 n+ n n. F 1/ / j0 1/ + j j

11 nd FIBONACCI NUMBERS MODULO CUBES OF PRIMES 11 1/ 0 Therefore, with the hel of.1, nd 1/ 0 1/ 0 1/ 0 1/ 0 1/ 0 16 F 1/ / / 1/ 1 3 0<j 1/ / 1 1/ For 0,..., 1/, clerly 1/ j1 1/ j 1 j+1 0<j 1/ 0<j j 1 j 1 mod 3 j 1 mod 3. 1/ j / 4j j1 0<j 4j 1/ 4 1 i i1 0<i / i mod 3. Since 1/ 1/ 1 i 1 i i i δ,3 mod i1 i1 i1 with the hel of Wolstenholme s congruence cf. [Wo] nd [Z], by the bove we hve 4 1/ F 1 H / 1 δ,3 mod.8

12 1 ZHI-WEI SUN nd 4 1/ / H / 1 mod..9 Our next ts is to simlify the right-hnd sides of congruences.8 nd.9. Let u {1, }. Then u H / r0 u H r 1 u 1+r H + r 1 1 r0 1 + r 1 u r + r mod. For {0,..., 1} nd r {0,..., 1 1}, by the Chu-Vndermonde identity cf. [GKP,. 169] we hve 1 + r 1 + r If 1 j, then 1 j 1 +r j0 1 j 1 j r 1 + r j mod. j 1 Thus 1 + r 1 r 1 r 1 + r 1 j 1 j + r 1 r j0 r mod by Lucs theorem. r Therefore u H / 1 u H r0 r u r r mod...10

13 FIBONACCI NUMBERS MODULO CUBES OF PRIMES 13 In view of 1.4, 1 1 r0 r 1 r r 1 mod, nd lso 1 1 r0 r r r 1/ 1/ 4 1 mod 1 rovided 3. Combining this with.8 nd.10, we obtin 4 1/ F 1 H 1 H 1 1 r0 1 r r 1 1 r δ,3 δ,3 0 mod, 1 nd hence 4 1/ F 1 H + δ,3 mod..11 Similrly, when 3 we hve 4 1/ H / mod..1 Now ssume tht 3. By. nd.1, 1/ / 6 q mod 3.

14 14 ZHI-WEI SUN Since mod ϕ, we hve mod nd hence 1/ / 6 mod 3. Combining this with. we immeditely obtin 1.. Below we suose tht. By.4 nd.11, 1/ 0 16 F 8 F mod 3. In view of [Su10, Lemm.3], F 1 / 1 F F mod nd thus 1/ 0 16 F 8 F mod 3. Combining this with.3 we get 1/ 0 16 F L F + 4 mod 3. Therefore 1/ 0 16 L 4 F L F F mod 3. This roves 1.1. So fr we hve comleted the roof of Theorem 1.1.

15 FIBONACCI NUMBERS MODULO CUBES OF PRIMES 1 3. Proof of Theorem 1. We need some reliminry results bout Lucs sequences. Let A, B Z nd A 4B. The eqution x Ax + B 0 hs two roots α A + nd β A which re lgebric integers. It is well nown tht for ny n N we hve u n A, B If is rime then 0 <n α β n 1 nd v n A, B α n + β n. v A, B α + β α + β A A mod. Lemm 3.1. Let A, B Z nd n N. Then u n+1 A, B n/ 0 n A n B. 3.1 Remr is well-nown formul due to Lgrnge, see, e.g., H. Gould [G, 1.60]. Lemm 3.. Let A, B Z nd let be n odd rime not dividing B where A 4B. Then u A, B A B 1/ u A, B+ B mod. 3. Proof. For convenience we let u n u n A, B nd v n v n A, B for ll n N. Let α nd β be the two roots of the eqution x Ax + B 0. Then v n u n α n + β n α n β n 4αβ n 4B n for ny n N. As u see, e.g., [Su10, Lemm.3], divides v 4B B v B B / v + B /.

16 16 ZHI-WEI SUN On the other hnd, by [Su10, Lemm.3] we hve v B1 B / B / mod. Therefore v B By induction, for ε ±1 we hve B / mod. for ll n Z +. Thus Au n + εv n B 1 ε/ u n+ε B 1 / u Au + nd hence So 3. is vlid. Au + v B B / mod u AB 1/ u B B B 1/ + B mod. Lemm 3.3. Let be n odd rime nd let Z +. Let m be n integer not divisible by. Then 1/ +1 m m 1/ m m m + δ,1 mod. 0 Proof. Observe tht 1/ 0 1/ /m 1/ + 1/ δ,1 m m m +1/ m 1/ m 1/ + m 1/ 0 3/ 0 1/ m m m m mod 3.3

17 FIBONACCI NUMBERS MODULO CUBES OF PRIMES 17 nd hence 3.3 follows. Proof of Theorem 1.. Set n 1/. By Lemm 3.3, n 0 C m 1 m n m + m δ,1 0 m mod. This roves 1.6. It remins to show 1.. By Lemms 3.1 nd.1, u 4, m n n 4 n m n n 16 n m n 0 0 n 0 n n m n m n n 0 16 m n m mod. Note tht m n m 1/ 1 s0 s m 1 s0 s m mod nd hence Thus m m n m m n m n m m m m n + m m mod. m m 1 m 1 1/ mod nd hence n m m u 4, m m u 4, m m 1 m 1 1 mod

18 18 ZHI-WEI SUN since m 1 m 1 mod by Euler s theorem. By [Su10, Lemm.3], 4 u 4, 4m m 1 4 4m u 4, m mod u 1 4, m 4 m mod. Therefore n m m m m u 4, m u 4, 1 1 m 0 4 m m m4 m m u 4, m m mm 4 mm 4 m u 4, m mod. In view of Lemm 3., 4 m m 1 1 u 4, m 4 m mu 4 m 4, m+ mod. So, by the bove, n 0 /m is congruent to mm 4 m 4 m 1 mu 4 m 4, m + mm 4 m mm mu 4 m 4, m modulo. This roves 1.. We re done. 4. Proofs of Corollries Proof of Corollry 1.1. Note tht n mod 4. The eqution x 4x hs two roots ± i where i 1. Thus u n 4, 8 + in i n 4i nd hence by Theorem 1. we hve i in i n 4i 0 1/ mod.

19 FIBONACCI NUMBERS MODULO CUBES OF PRIMES 19 Clerly q is divisible by 3 nd the two roots of the eqution x 4x re + 3 4ω nd 3 4ω, where ω 1 + 3/ is rimitive cubic root of unity. Thus u q 4, 16 4ω q 4ω q since 3 q. Alying 1. with m 16 we get 1/ The roof of Corollry 1.1 is now comlete. 3 Proof of Corollry 1.. Set n 1/. Then mod. n n n mod with the hel of [Su11b, 1.4]. Observe tht n n n n 1 C j 16 j j0 n C 16 C n 8 4 n. Also, C n 4 n 1 1/ / 1 1/ 1 mod in view of Morley s congruence [Mo] 1 1 1/ 4 1 mod 3. 1/

20 0 ZHI-WEI SUN By 1.6 nd 1.8, n 0 C mod. Therefore n mod nd hence n n which yields 1.9 nd its equivlent form We re done. mod, Acnowledgment. The uthor would lie to thn the referee for helful comments. References [A] [CDP] [CP] [G] [Gr] [GKP] [GZ] [MT] [Mo] [PS] [St] [SSZ] A. Admchu, Comments on OEIS A in 007, On-Line Encycloedi of Integer Sequences, htt:// njs/sequences/a R. Crndll, K. Dilcher nd C. Pomernce, A serch for Wieferich nd Wilson rimes, Mth. Com , R. Crndll nd C. Pomernce, Prime Numbers: A Comuttionl Persective, Second Edition, Sringer, New Yor, 00. H. W. Gould, Combintoril Identities, Morgntown Printing nd Binding Co., 197. A. Grnville, The squre of the Fermt quotient, Integers 4 004, #A, 3 electronic. R. L. Grhm, D. E. Knuth nd O. Ptshni, Concrete Mthemtics, nd ed., Addison-Wesley, New Yor, V. J. W. Guo nd J. Zeng, Some congruences involving centrl q-binomil coefficients, Adv. in Al. Mth , S. Mttrei nd R. Turso, Congruences for centrl binomil sums nd finite olylogrithms, J. Number Theory , F. Morley, Note on the congruence 4n 1 n n!/n!, where n + 1 is rime, Ann. Mth , H. Pn nd Z. W. Sun, Proof of three conjectures on congruences, rerint, rxiv: htt://rxiv.org/bs/ R. P. Stnley, Enumertive Combintorics, Vol. 1, Cmbridge Univ. Press, Cmbridge, N. Struss, J. Shllit nd D. Zgier, Some strnge 3-dic identities, Amer. Mth. Monthly ,

21 FIBONACCI NUMBERS MODULO CUBES OF PRIMES 1 [S08] Z. H. Sun, Congruences involving Bernoulli nd Euler numbers, J. Number Theory , [S11] Z. H. Sun, Congruences concerning Legendre olynomils, Proc. Amer. Mth. Soc , [SS] Z. H. Sun nd Z. W. Sun, Fiboncci numbers nd Fermt s lst theorem, Act Arith , [Su06] Z. W. Sun, Binomil coefficients nd qudrtic fields, Proc. Amer. Mth. Soc , 13. [Su10] Z. W. Sun, Binomil coefficients, Ctln numbers nd Lucs quotients, Sci. Chin Mth , htt://rxiv.org/bs/ [Su11] Z. W. Sun, -dic vlutions of some sums of multinomil coefficients, Act Arith , [Su11b] Z. W. Sun, On congruences relted to centrl binomil coefficients, J. Number Theory , [Su1] Z. W. Sun, On hrmonic numbers nd Lucs sequences, Publ. Mth. Debrecen 80 01, 41. [ST] Z. W. Sun nd R. Turso, New congruences for centrl binomil coefficients, Adv. in Al. Mth , [W] H. C. Willims, A note on the Fiboncci quotient F ε /, Cnd. Mth. Bull. 198, [Wo] J. Wolstenholme, On certin roerties of rime numbers, Qurt. J. Al. Mth. 186, [Z] J. Zho, Wolstenholme tye theorem for multile hrmonic sums, Int. J. Number Theory 4008,