(m) (-I)P-I(ap b. rn+s n. kpn kpn-1 AND LINEAR SECOND ORDER RECURRENCES SOME CONGRUENCE PROPERTIES OF BINOMIAL COEFFICIENTS NEVILLE ROBBINS (1.

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1 Iterat. J. Math. & Math. Sci. VOL. II NO. 4 (1988) SOME CONGRUENCE ROERTIES OF BINOMIAL COEFFICIENTS AND LINEAR SECOND ORDER RECURRENCES NEVILLE ROBBINS Departmet of athematlcs Sa Fracisco State Uiversity.Sa Fracisco, CA 9413 (Received May 1, 1987) ABSTRACT. Usig elemetary methods, the followig results are obtaled:(1) If p is -m prime, 0 m If r,s are the roots of x Ax-B, where (A,B) ad D A_4B > 0, if -, 0 < b < ap -m, ad p ab, the (m) (-I)-I(ap b (rood p). u v r+s, ad > O, the (II) v =- v (rood p). r-s p p-1 (III) If p is odd ad p D, the u (_D) u (rood p); p p -I (IV) u (_1)Bu ) -I (rood KEYWORDS AND HRASES. Biomial coefficiet, liear secod order recurrece AMS SUBJECT CLASSIFICATION CODE: loal0 IOA35. I. INTRODUCTION. Followig Lucas [I], let A, B be itegers such that (A,B) ad D-- A-4B > O. Let the roots of x Ax-B be: r 1/( ), s--i/ (A- ). Let 0. Let the sequeces u v be defied by: u r -s r-s (1.1) v r+s (1.) the u 0 0 u u Au Bu for -l- - v, v A, v Av -Bv for o -I - (1.4) r+s=a

2 744 N. ROBBINS rs B (1.6) U V UV _B v (1.8) (I.9) Um+ UmV + UVm (I.I0) Let p be prime. Let O (t) j if J t p 3+ p t. If 0 < < p, the (1.11) If 0, the (- () (1.1) -I a p a p (rood p) (1.13) If x +/- a (rood ), the x a (rood +l (1.14) e If 0 < < pe ad p, the Op(()) e (I.5) REMARK. (I.I) through (I.I0) appear i Lucas [I]. (I.II) through (1.14) are elemetary. (1.15) is theorem i Robbls [].. MAI N RE SULTS. Lemma ( I), If 0 m, 0 < b < ap -m ad p a b, -m b(pm-l) (mod p). the (m) (a b (-I) bp m -I ROOF. (pm) j--0 ap-j bpm-j I where 1 is the product of all such factors where pmlj, ad is the product of all such factors where Now s-m 1 bpm-i j jip b-1 a m b-i -m_i -m I _ap ap i--o bpm-ip m i=o b-i b a_ (_l)bpm_b (rood J pro_ am) (a b( I) (bp (-I) (rood p). Therefore ) while

3 SOME CONGRUENCE ROERTIES OF BINOMIAL COEFFICIENTS 745 -m THEOREM.1. If 0 m, 0 < b < ap ad p ab, -m bp (rood p). the (am) (-I)-I (a b ROOF. If p is odd, the b(pm-l) -- p-! 0 (rood ); if p, the by hypothesis, b is odd, so b(m-l) -1 (rood ). I either case, the coclusio ow follows from Lemma.1. LEMMA.. V+! A+l (+l B ROOF. By (1.) ad the biomial theorem, we have +I +l- +l +l +l s V+l r + s (r+s) [ r = so (1.5) implies V+ A+l-{ +l +l- +I +l-l.. )r s + (+l_)r s Now (1.1) implies +l) V+l A +l l (rs) (r+l-+s+i- ), so (1.) ad (1.6) imply V+l A+l (+l)bv LEMMA.3. v -- v (rood p) +l- ROOF. Lemma. ad (I.II) imply v p-- A p ); too, (1.4) implies v v (rood p). LEMMA.4. If i ) J, the vi+j viv j ROOF. By (1.) ad (1.6), BJv (rood p); (I.13) implles A p -- A (rood isj+r j i J viv j vi+j (ri+si)(rj+sj)-(ri+j+si+j) r s (rs) (rl-j+s LEMMA.5. If 0 m, y z (rood pro), w x (mod p), ad x 0 (rood p-m), the wy xz (rood p). --- ROOF. Hypothesis implies Y z+ip m, w x+jp so wy E xz+ipmx (rood p ). Hypothesis also implies p-mlx, so wy xz (rood p). LEMMA.6. If v (rood pm m m-i ), the v ---v (mod pm p m p m-1 ROOF. (Iductio o ). Lemma 6 holds trivially for =O, ad by hypothesis for. O, m ) ad v ---

4 746 N. ROBBINS m v =v =v v -B v =v v B v m m t m m m (+!)p m pm+p m p m p p -p p p (-l)p m by Lemma.4. Now iductio hypothesis ad (1.13) imply that v =v v -B p v (+ )pro p m- p (-I )p v v (mod pm) (+ )pro (+ )pro- m-1 (rood pro). Now Lemma.4 implies m-1 m-1 v (rood p ) -I ROOF. (Iductio o ) Lemma.7 holds for =1, by Lemma.3. Suppose Lemma LEMMA 7. If p is odd ad I, the v holds for all m <, where. The Lemma. implies -I I/ (p-l-l) -I v A p p )Biv -I i - p i=o p I-i I/ (p- I) v Ap p j=o ()BJVp_ ] If p j, the (I.15) implies 0 (rood p J also Therefore I/ (p-1-1) v Ap p j=o > (pp)biv p - ip Let i p hpm, where p h ad m <. Now Bi Bh Bh B i (rood pm), by (1.13); also v p _ ip m m-i V V V p_hpm (p-m_h)pm (p-m_h)pm- =- v -I p -i (rood pro) by iductio hypothesis ad Lemma.6. Therefore B ip v B i v (rood pm -1 p-ip p -i Also p -I -I -m _= (p) i hpm) -= (hp m-l) p ) i h implies ( 0 (rood p -m) Therefore Lemma 5 h implies (rood p by Theorem I, ad (I.15) -I (p)biv ( )Biv (mod -ip p ) Now (I 13) i implles p p -l_ i

5 SOME. CONGRUENCE ROERTIES OF BINOMIAL COEFFICIENTS / (p-l-l) v A p (p - )B i v (rood p ) that is i -I p i=o p -i v E v (mod p). -I LEMMA.8. If 1A, the v (rood +l) for 0. ROOF. (Iductio o ) v0 -I -I =- 4 (mod +l) Hypothesis implies B v A (rood by (1.4) ad hypothesis. Now iductio hypothesis implies v (rood ), so (1.14) implies v implies v v -1 -I -B 4-(I) (rood +l). -I B Now (1.9) so (rood ) LEMMA.9. If AB, the v -I (rood +l) for 0. ROOF. (Iductio o ) v0 v A =- -I (rood ) by (1.4) ad hypothesis. Now iductio hypothesis implies v =-I (rood ), so (1.14) implies -I v (rood +l).- Agai, B odd implies -I _B-1 v v =- I-(I) -I (rood +lj. -I LEMMA.10. If B, the v (rood +l)- for O. -I B -= (rood ), so (1.9) implies ROOF. (Iductio o ) Hypothesis implies A is odd, so (1.4) implies v 0 v A (rood ). By hypothesis, B 0 (rood ), so -1 B -= 0 (rood -1 Sice -I -I for we have B 0 (rood ) By (rood ), so (1.14) implies v (mod +l). iductio hypothesis, V_1 _l -I Now (1.9) implies v v -B E 1-(0) =- (rood +l)." -I LEMMA.11. V v (mod ) -I ROOF. Lemmas.8,.9,.10 imply v t (rood -I ), V where t or +/-I. Therefore V t V_l (rood ). E t (rood +l),

6 748 N. ROBBINS THEOREM.. I I, the v =- v (rood )- ROOF. Follows from Lemmas.7 ad.11. THEOREM.3. If ad 0, the v v (rood p) p p -I ROOF. Fol-los from Theorem. ad Lemma.6. LEMMA.1. U+l D + (-B)U+l_ + + ROOF. (i.i) ad (1.7) imply U+l r s +l (_l)(+l)r+l-. s so (1.7) implies DI/ Um+l D + I/-{ (-l)(+l)r+l- s. Settig j +l- i the secod sum, we obtai I/9 D U+ +I/_{ (-I) (+1)r s (-i) = D + I/-. D/- U+l D. (-l)(rs)( I) (r (+l +l + (-l)j( +1)r J=+ J +l- -( +l- +l- s by (1.1) (-B) U+l_ by (1.6) ad (I.7), so (-B) u + l- LEMMA.13. If p is odd, the u =- () (rood p) +l-j sj ROOF. Follows from Lemma.1, (I.II), ad Euler s criterio. +l +l- +l- r s LEMMA.14. If p is odd, pd, ad I, the DI/ (p)-- ( D) (mod p). ROOF. (Iductio o ) Lemma.14 holds for =l by Euler s criterio. Let () t 11. Now iductio hypothesis implies DI/(p)E t (rood p), I/(p). + so D t+ip. D I/( )= (D i/ (p ) ) (t+ip) p t p + pt p-i (ip) + (.) t-j(ip)j. Now t p t 3 j--

7 SOME CONGRUENCE ROERTIES OF BINOMIAL COEFFICIENTS 749 ad p+llpj for j ), so Dl/(p+l )-- (rood p+l). LEMMA.15. If ) 0, m > I, p is odd, (), ad u tu (rood pm ), the u tu (mod pm). m m-i ROOF. (Iductio o ) Lemma.15 is trivially true for =O, ad is true by hypothesis for. Now (I.I0) implies U(+ m u u v + u v pm m m m m l)p pm+ p p p p By iductio hypothesis, ), ad Upm --tupm_l pm pm- u E tu (rood pm m pm- Theorem. implies v v (rood pm m pm- m tu u( +1)p (rood pro); for O. Therefore m-i v + tu v 9tu (rood pro). Sice m-i m-i m p p p pm-i p p is odd, we have U(+l)p m -= tu (mod p m LEMMA.16. If p is odd, pd, ad ) I, the u -= (--D)u (mod p -I ROOF. (Iductio o ) Lemma.16 is true for by Lemma.13. -I DI/ (p-l_l) I/ (p -I) -I i Lemma.1 implies u -I (pff (-B) u p i p _l_ i - (p) 0 p). j (J) (-B)jup- j. If pj, the (1.15) implies (rood p). Therefore u DI/ (p-i) I/ (p-i-i) )-. (pp) (-B) iu - (rood p i= p ip Let ip hp m where ph ad m <. Let t (). m )hpm- Now (-B) ip s (-B) hp (-B (-B) i (rood pro) by (1.13). By iductio p). hypothesis ad Lemma.15, we have u E u E u =- tu m-i E tu -I m-i p-ip p-hpm (p-m-h) pm (p-m-h) p p -hp

8 750 N. ROBBINS tu _l_ (rood pro) Therefore (-B)iu t(-b)iu -l_ p i p-ip p i (mod pro). As i the proof of Lemma.7, we have -! (-B)iu t( p (-B)i () ip p-ip p -i u -I (rood p). Therefore u E DI/ (p-l) -t(d 1/ (p-l-l) -u (rood p) that is p ; -I u tu + DI/ (pr-l_l) (DI/ (p_p-l)_t) (rood p) -I Sice (p) -I - Lemma.14 implies u )u (rood p) -I LEMMA.17. If ad D is odd, the u =- (-l)bu_l (rood ). ROOF. By hypothesis ad by the defiitios of A, B, ad D, A must be odd. If B is odd, the Lemma.9 implies implies v -= (rood ). -1 Now (1.8) implies u (-l)bu_l (D) if p is odd B (-I) if p v -I -= -I (rood ); if B is eve the Lemma.10 Therefore, i either case, v (-I)B (rood ). -I (mod ). THEOREM.4. If ad p D, the u tu (rood p), where -l ROOF. Follows from Lemmas.16 ad.17. THEOREM.5. If ) 0, > I, ad pd, the u p where is defied as i Theorem.4. -= tu (mod p ) -I p ROOF. Follows from Lemma.15 ad Theorem.4. Cocludig Remars. Let be defied as i Theorem.4 as a result of Theorems.3 ad 5, the sequeces v t u determie p-adic itegers for each 0. p p REFERENCE S I. LUCAS, E.. ROBBI NS, N. Theorie des foctios umeriques simplemet periodiques, Amer. J. Math. (187 7) ; O the umber of biomial coefficiets which are divisible by their row umber, Caad. Math. Bull. 5 (3) 198,