Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)
GRYTCZUK, A., Remark on Ankeny, Artin and Chowla conjecture .
Remark on Ankeny, Artin and Chowla conjecture 25 Then, for k = m + 1 in virtue of Q s-(m+i) — Qs-(m+i)Qs-m-2 + Qs — m—ó and 9 s_( m+1 ) = q m +i we get Q a-( m+i) = qm+iQs-m-2 +Qs-m3- By (17) and the last equality it follows that Q s—\ = Q m+iQs-m-2 + Q mV s — m — ó and inductive proof of (16) is finished. Putting k = ^y^ and ovbserving that s-k- 1 = s-k- 2 = ^ - 1, we ob at in Q s_i = Q 2 3l + q2_ i . i i i —j— In similar way we obtain that P s-\ = P rQ r + Pr-iQr-i a^d the proof of Lemma 3 is complete. Proof of Theorems Proof of Theorem 1. Suppose that p\yo. Then by (13) of Lemma 2 we have (18) c = (-l) r+ 1(P 2 r -pQl). Prom Lemma 2 we also obtain (19) b = (-1 Y + l(pQrQr-l - PrPr-l). Let L = cQ r + 6Q r_i. Then by (18) and (19) it follows that (20) L = (-l) r+ 1 (P r(PrQr ~ Pr-lQr-l) ~ pQr(Ql ~ Q\1 )) • On the other hand from Lemma 2 we have (21) PrQr - Pr — lQr — l = KQl + Qr-lb Substituting (21) to (20) we obtain (22) L = (-1 (bP r(Q 2 + Ql_,) - pQ r(Ql - Q 2^)) . By Lemma 3 it follows that y 0 = Q s-i = Q 2 r + Q 2 r_ l and therefore from (22) we get p \ L. Prom (10) and (11) of Lemma 2 we have (23) P 2 + P 2_, = (bQ r + cQr-i) 2 + (cQr - bQ r_x) 2. On the other hand it is well-known the following indentity: (24) {bQr+cQr^) 2 +{cQ r-bQ r_ 1) 2 = {cQr+bQr^y+ibQr-cQr^) 2. Prom (23) and (24) we obtain (25) P 2 + P 2_, = (cQr + 6g r_0 2 + (bQ r - cQ r-i) 2
