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 Chowlá conjecture ALEKSANDER GRYTCZUK Abstract. In this paper we give two new criteria connected with well-known and still open conjecture of Ankeny, Artin and Chowla. Introduction In the paper [2] Ankeny, Artin and Chowla conjectured that, if p = 1 (mod 4) is a prime and £ — 1/2(T + Uy/p) > 1 is the fundamental unit of the quadratic number field K = Q(^/p) then p\U. It was shown by Mordell [5] in the case p = 5 (mod 8) and by Ankeny and Chowla [3] for the remaining primes p = 1 (mod 4) that p \ U if and only if p\Bp^±> where is 2n-th Bernoulli number. Another criterion has been given by T. Agoh in [1]. Beach, Williams and Zarnke [4] verified the conjecture of Ankeny, Artin and Chowla for all primes p < 6270713. Sheingorn [6], [7] gave interesting connections between the fundamental solution (xo,yo) of the non-Pellian equation (1) x

2 — py

2 = —1, p = 1 (mod 4), p is a prime and the manner of the reflection lines on the modular surface and also of the yjp Riemann surface. We prove the following two theorems: Theorem 1. Let p = 1 (mod 4) be a prime and p = b

2 +c

2

. Moreover, let yjp — [fjo

; i/i

, i/2

,..., r/

s] be the representation of ^Jp as a simple continued fraction and let (Xo,yo

) be the fundamental solution of (1). Then p \ yo if and only if p \ cQ

r + bQ

r_i and p | Q

r — cQ

r-\, where r — and P

nIQn is n-th convergent of yjp. Theorem 2. Assume that the assumptions of the Theorem 1 are satisfied. Then p \ yo if and only if p \ 4bQ

rQ

r-i — (-l)

r+

1, where r = and Pn/Qn is n-th convergent of ^/p. Basic Lemmas Lemma 1. Let \/d = [go

; Qi

, • • •, Q

s] b

e the representation of \/d as a simple continued fraction. Then (2) q

n = <7o + b

7 ? b

n + frn+i — c

ng

n, d — b

2

n+

l + c

nc

n_f_i