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 .
24 Aleksander Grytczuk (3) if s = 2r + 1 then minimal number k, for which c^+i — c^ is k = (4) dQ n-1 = b nP n-i + c nP n_ 2, (6) -Pn-i = b nQ n-1 + C nQ n—2 j (7) - dgU where P njQ n is the n-th convergent of \fd. This Lemma is a collection of well-known results of the theory of continued fractions. Prom Lemma 1 we can deduce for the case d = p = 1 (mod 4) and r = —" the following: Lemma 2» Let p = 1 (mod 4) be a prime and let yjp — [<?o ; qi , • • •, Qs], where s = 2r + 1 tiien (8) p = b 2 r+ l + c 2 = b 2 + c 2; b r+ 1 = b, c r = c (9) = + cP r_! (10) P r - bQ r + cQ r_ 1 (11) P r_i - cQr (12) PrQr-l - QrPr-l =(-1 ) r+1 (is) p 2-pg 2 = (-ir + 1c (14) P^-pQ^ =(-l) r C (15) Lemma 3. Let \[d — [go ; qi , • • •, qs} and s = 2r + 1, then Q s-\ — 2 1 2 Ps — 1 — PrQr ~f~ 1 V — 1 Qr—1 • Proof. First we prove that for k = 1,2,..., we have (16) 1 = QfcQs-(fc+l) + Qk-lQs-(k+2). Really, since q s_ 1 = q x , Q 1 = q x , Q 0 = 1 then we obtain Q s_i = q s_i Q s_ 2 + Qs-3 = QiQs-2 + QoQs-3 and (16) is true for k = 1. Suppose that (16) is true for k = m, i.e. (17) Qs1 = g m^i s —(m + l)
