AN EXTENSION OF LUCAS THEOREM. Hong Hu and Zhi-Wei Sun. (Communicated by David E. Rohrlich)

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1 Proc. Amer. Mah. Soc. 19(001, o. 1, AN EXTENSION OF LUCAS THEOREM Hog Hu ad Zhi-Wei Su (Commuicaed by David E. Rohrlich Absrac. Le p be a prime. A famous heorem of Lucas saes ha p+s p+ ( s (mod p if m,, s, are oegaive iegers wih s, < p. I his paper we aim o prove a similar resul for geeralized biomial coefficies defied i erms of secod order recurre sequeces wih iiial values 0 ad Iroducio Le N {0, 1,, }, Z + {1,, 3, } ad Z Z \ {0}. Fix A, B Z. The Lucas sequece {u } N is defied as follows: (1 u 0 0, u 1 1 ad u +1 Au Bu 1 for 1,, 3,. Is compaio sequece {v } N is give by ( v 0, v 1 A ad v +1 Av Bv 1 for 1,, 3,. By iducio, for 0, 1,, we have u α β 1 ad v α + β where I follows ha α A +, β A ad A 4B. v u +1 Au, u u v ad v v B for N. For a, b Z le (a, b deoe he greaes commo divisor of a ad b, a ice resul of E. Lucas assers ha if (A, B 1 he (u m, u u (m, for m, N (cf. L. E. Dicso [ Mahemaics Subjec Classificaio. Primary 11B39; Secodary 11A07, 11B65. The secod auhor is resposible for all he commuicaios, ad suppored by he Teachig ad Research Award Fud for Ousadig Youg Teachers i Higher Educaio Isiuios of MOE, ad he Naioal Naural Sciece Foudaio of P. R. Chia. 1

2 HONG HU AND ZHI-WEI SUN I he case A B 1, by iducio o N we fid ha u 0 if 3, ad { 1 if A 1 & 3 1, or A 1 & 1, (mod 6; u 1 if A 1 & 3 + 1, or A 1 & 1, (mod 6. We se [ 0< u for N, ad regard a empy produc as value 1. For, N wih [ 0, we defie he Lucas u-omial coefficie [ as follows: [ { [ [[ if, (3 0 oherwise. I he case A ad B 1, [ ( is exacly he biomial coefficie ; whe A q +1 ad B q where q Z ad q > 1, [ coicides wih Gaussia q-omial coefficie ( because u q j (q j 1/(q 1 for j 0, 1,,. For geeralized biomial coefficies formed from a arbirary sequece of posiive iegers, he reader is referred o he elega paper of D. E. Kuh ad H. S. Wilf [5. Le d > 1 ad q > 0 be iegers wih d u q. If (A, B 1 ad d u for 1,, q 1, he for ay N we have d u d divides (u, u q u (,q q (, q q, his propery is usually called he regular divisibiliy of {u } N. If (d, u 1 for all 0 < < q, he we wrie q d ad call d a primiive divisor of u q while q is called he ra of appariio of d. Whe (A, B 1, q d, N ad q, we have (d, u ((d, u q, u (d, (u, u q (d, u (,q 1. Whe p is a odd prime o dividig B, p exiss because p u p ( p as is wellow where ( deoes he Legedre symbol. O he oher had, drawig upo some ideas of A. Schizel [6, i 1977 C. L. Sewar [7 proved ha if A is prime o B ad α/β is o a roo of uiy, he u has a primiive prime divisor for each > e ; i 1995 P. M. Vouier [9 cojecured ha he lower boud e ca be replaced by 30. For m Z we use Z m o deoe he rig of raioals i he form a/b wih a Z, b Z + ad (b, m 1. Whe r Z m, by x r (mod m we mea ha x ca be wrie as r + my wih y Z m. For coveiece we se R(q {x Z : 0 x < q} for q Z +. Our mai resul is as follows. Theorem. Suppose ha (A, B 1, ad A ±1 or B 1. The u 0 for every 1,, 3,. Le q Z +, m, N ad s, R(q. The (4 [ mq + s q + [ s u (q+(m +(s q+1 (mod w q

3 AN EXTENSION OF LUCAS THEOREM 3 where w q is he larges divisor of u q prime o u 1,, u q 1. If q or m(++(s+1 is eve, he (5 [ mq + s q + [ s ( 1 (m s(q 1 B q ((q+(m +(s (mod w q. Remar 1. Providig (A, B 1 ad q Z +, (u q, 0<<q u 1 if ad oly if u d ±1 for all proper divisors d of q (his is because (u q, u u (q,, herefore u q is prime o u 1,, u q 1 if q is a prime. Whe A ad B 1, we have u for all N, hece he Theorem yields Lucas heorem which assers ha ( mp + s p + ( s (mod p where p is a prime ad m,, s, are oegaive iegers wih s, < p. I he case A a + 1 ad B a where a Z ad a > 1, as u q+1 (a q+1 1/(a 1 au q (mod u q for q Z +, our Theorem implies Theorem 3.11 of R. D. Fray [. Theorem 3 of B. Wilso [10 follows from our Theorem i he special case A 1, B 1 ad s. Wilso used a resul of Kummer cocerig he highes power of a prime dividig a biomial coefficie, see Kuh ad Wilf [5 for various geeralizaios of Kummer s heorem. Our proof of he Theorem is more direc, we do use Kummer s heorem i ay form. Example. (i Se A 4 ad B 1. The u 0 0, u 1 1, u 4, u 3 15, u 4 56, u 5 09, u Clearly p 13 is he larges primiive divisor of u By he Theorem, [ 71 5 [ ( [ 11 5 ( u 5 ( (mod 13. (ii Tae A 1 ad B 7. The w 3 u 3 8 ad u By he Theorem, [ [ ( 11 3 [ (11 3+3( 1 3 (mod 8.

4 4 HONG HU AND ZHI-WEI SUN. Several Lemmas Lemma 1. Le ad be posiive iegers wih > ad [ 0. The [ [ [ 1 1 (6 u +1 Bu 1. 1 If A ad B, he [ ( (mod. Proof. Clearly he righ had side of (6 coicides wih [ 1 u +1 [[ 1 Bu [ 1 1 [ 1[ [ [ 1 [[ (u +1u Bu u 1 where i he las sep we use he ideiy u +1 u l Bu u l 1 u +l which ca be easily proved by iducio o l Z +. Now suppose ha (A 1B. The u 1, u 3, u 5, are odd ad u, u 4, u 6, are eve. If [ ( 1 1 (mod ad [ 1 1 ( 1 (mod, 1 he (6 yields ha [ ( ( 1 1 ( + 1 ( 1 1 ( ( ( 1 ( + 1 (mod. 1 So [ ( (mod by iducio. Remar. I he ligh of Lemma 1, by iducio, if N ad [ 0 he [ Z for all N. This was also realized by W. A. Kimball ad W. A. Webb [4. I 1989 Kuh ad Wilf [5 proved ha geeralized biomial coefficies, formed from a regularly divisible sequece of posiive iegers, are always iegral. Lemma. Le q be a posiive ieger. The u q+1 B q (mod u q. If q, he u q+1 B q/ (mod d for ay primiive divisor d of u q. Proof. As ( uq u q 1 u q+1 u q ( ( uq 1 u q A 1 u q u q 1 B 0 ( ( q 1 u1 u 0 A 1, u u 1 B 0

5 AN EXTENSION OF LUCAS THEOREM 5 we have u q u q 1 u q+1 B q 1 ad hece u q+1 Bu q 1 u q+1 B q (mod u q. Now assume ha q where Z +. Le d be a primiive divisor of u q. Sice u v u q 0 (mod d ad (d, u 1, we have d v ad hece u q+1 Au q + v q This eds he proof. Au v + v B Lemma 3. Le, q Z +. The u +1 v B B (mod d. (7 u q+l u q+1u l (mod u q for l 0, 1,,. If u q 0, he (8 u q u 1 q+1 u + ( 1Au q q (mod u q. Proof. Le l N. By Lemma of Z.-W. Su [8, u q+l r0 ( c r u r r qu l+r. where c Bu q 1 u q+1 Au q. Clearly u q+l u q+1u l (mod u q. I he case u q 0, u q ( 1 ( 1 u r 1 c r u r 1 q q u r c r u r. u q r r 1 r r1 For ay prime p ad ieger r > 3 we have r1 p r (1 + 1 r 1 + (r + 1 r, herefore u r q /r Z uq for r 3, 4,. If A ad B, he u q u q 1 u 1 1 (mod. Whe u q, we have A(B 1, c u q+1 B q (mod u q ad hece c B (mod. Thus (8 holds providig u q 0. Lemma 4. Assume ha (A, B 1, q Z + ad u 0 for all Z +. The for ay m, N ad s, R(q we have [ [ [ mq + s mq s (9 u (m +(s q+1 (mod w q q + q where w q is he larges divisor of u q prime o u 1,, u q 1. Proof. Le m, N ad s, R(q. If m <, he mq + s < (m + 1q q + ad hece [ [ mq+s q+ 0 mq [ q. If m ad s <, he mq+s [ q+ 0 s. Below we assume ha m ad mq + s q +.

6 6 HONG HU AND ZHI-WEI SUN Observe ha w q is prime o u q+1 0<r<q u r, ad [ mq + s (m q<j mq u j q + 0<j q u j { 0<r s u mq+r 0<r u q+r 0<r s u 1 (m q+r if s, 0 r< s u (m q r if s <. By Lemma 3, u q+r u q+1u r (mod w q for ay, r N. So [ mq + s q + [ mq q 0<r s (um q+1u r 0<r (u q+1 u r { 0<r s (u m q+1 u 1 r (mod w q if s, 0 (mod w q oherwise, [ mq [s q [ [ mq s q q+1 [ ums { u ( m(s q+1 /[s (mod w q if s, 0 (mod w q if s <. u (m +(s q+1 (mod w q. This cocludes he proof. 3. Proof of he Theorem Le us firs show ha u 1, u, u 3, are all ozero. If 0, he α β A/ ad hece u 0 r< α r β 1 r ( 1 A 0 for 1,, 3,. Suppose ha u 0 for some Z +. The 0, α β ad α β. Sice he field Q( coais he roo α/β ±1 of uiy, by Proposiios ad of K. Irelad ad M. Rose [3 here exiss a posiive ieger D such ha D ad α/β {±i}, or 3D ad α/β {±ω, ±ω } where ω ( 1 + 3/. I he former case, (A + Di/(A Di {±i}, hece A D ad B (A / D, his is impossible sice A or B is odd. Thus he laer case happes. Now ha A + D 3 A D 3 A 3D { + AD 3 1 ± 3 A + 3D, 1 ± } 3, we have A 3D ±AD ad hece A {D, 9D }. If A D, he B (A /4 D, hece (A, B > 1 or A B 1; if A 9D, he B (A /4 3D ad hece 3 (A, B. This leads a coradicio.

7 AN EXTENSION OF LUCAS THEOREM 7 Nex we come o show (4. Le u 0 0, u 1 1 ad u j+1 v qu j Bq u j 1 for j 1,, 3,. Noe ha α q + β q v q ad α q β q B q. Fix Z +. If A 4B 0, he u q (αq β q /(α β u q (α q β q /(α β (αq (β q α q β q u ; if 0, he α β A/, u q q(a/ q 1, u q q(a/ q 1 ad u 0 r< (α q r (β q 1 r So we always have u q /u q u. By (8, ( q( 1 A u q. u q u q u q r (mod u q where r u 1 q+1 + { ( 1Auq / if u q, 0 oherwise. Noice ha (r, u q 1 if u q, ad (r, u q / 1 if u q. Suppose m > > 0. We asser ha ( u (m q m (10 u (m q+1 (mod u q. u ( q If u q or 4 u q, he (r, u q 1 for all 1,, 3,, hece ( m ( m u (m q u ( q u (m q+1 + m u m 1 q+1 + (m 1Au q / u 1 q+1 + ( 1Au q / ( ( u m q+1 + (m Au q m ( m(m 1 m Au q 1 I he case u q (mod 4, by he above mehod ( u (m q m u (m q+1 u ( q u m u u (m q /((m u q u ( q /(( u q ( u (m q+1 + (m A u q ( m u (m q+1 (mod u q. ( mod u q as v q u q+1 Au q 0 (mod ad B 1 (mod (oherwise A, u 1, u, u 3, are all odd we also have u (m q ( ( m m u (m q+1 (mod u ( q,

8 8 HONG HU AND ZHI-WEI SUN by Lemma 1. This proves (10. Now we claim ha (11 [ mq q ( m u (m q q+1 (mod w q. This is obvious if m or 0. I he case m > > 0, if 0 < j < q ad q j he (u q j, w q 1 ad u mq j u q j u (m q+q j u q j u m q+1 (mod w q by Lemma 3, hus [ mq u mq j u (m q q u q j u 0 j<q ( q I view of (9 ad (11, u (m q+1 u (m (q q+1 [ mq + s q + ( m u (m q q+1 [ s 0<j<q q j u mq j u q j u (m q q+1 (mod w q. [ s u (m +(s q+1 u (q+(m +(s q+1 (mod w q. Fially we say somehig abou (5. If q, he (q + (m + (s (m + (s m s (mod, ad u q+1 B q/ (mod w q by Lemma. Whe q is odd ad l m(++(s+1 is eve, (q + (m + (s ( + (m + (s l 0 (mod ad u q+1 B q (mod w q by Lemma. Thus (5 follows from (4 if ql. We are doe. Acowledgeme. The auhors are idebed o he referee for his (or her helpful suggesios.

9 AN EXTENSION OF LUCAS THEOREM 9 Refereces 1. L. E. Dicso, Hisory of he Theory of Numbers, vol. I, Chelsea, New Yor, 195, p R. D. Fray, Cogruece properies of ordiary ad q-biomial coefficies, Due Mah. J. 34 (1967, K. Irelad ad M. Rose, A Classical Iroducio o Moder Number Theory (Graduae exs i mahemaics; 84, d ed., Spriger-Verlag, New Yor, 1990, pp W. A. Kimball ad W. A. Webb, Some cogrueces for geeralized biomial coefficies, Rocy Mouai J. Mah. 5 (1995, D. E. Kuh ad H. S. Wilf, The power of a prime ha divides a geeralized biomial coefficie, J. Reie Agew. Mah. 396 (1989, A. Schizel, Primiive divisors of he expressio A B i algebraic umber fields, J. Reie Agew. Mah. 68/69 (1974, C. L. Sewar, Primiive divisors of Lucas ad Lehmer sequeces, i: Trascedece Theory: Advaces ad Applicaios (A. Baer ad D.W. Masser, eds., Academic press, New Yor, 1977, pp Zhi-Wei Su, Reducio of uows i Diophaie represeaios, Scieia Siica (Ser. A 35 (199, o. 3, P. M. Vouier, Primiive divisors of Lucas ad Lehmer sequeces, Mah. Comp. 64 (1995, B. Wilso, Fiboacci riagles modulo p, Fiboacci Quar. 36 (1998, Deparme of Mahemaics, Huaiyi Normal College, Huaiyi 3001, Jiagsu Provice, P. R. Chia Deparme of Mahemaics, Najig Uiversiy, Najig 10093, P. R. Chia. zwsu@ju.edu.c