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 .

28 Aleksander Grytczuk On the other hand it is well-known (see, [8]; Thm. 4, p. 323) that all natural numbers d

, for which the representation of \fd as a simple continued fraction has the period s — 3 are given by the formula: 2 (40) d((ql + l)* + fj +2<?ifc + l, where qi is an even natural number and k — 1, 2,3,... Suppose that d | y

0, then we have d < y

0. By (39) and (40) it follows that d > yo and we get a contradiction. From this observation follows that A-A-C conjecture is true for all primes p = 1 (mod 4), having the representation in the form (40). References [1] T. AGOH, A note on unit and class number of real quadratic fields Acta Math. Sinica 5 (1989), 281-288. [2] N. C. ANKENY, E. ARTIN and S. CHOWLA, The class number of real quadratic number fields Annals of Math. 51 (1952), 479-483. [3] N. C. ANKENY and S. CHOWLA, A note on the class number of real quadratic fields, Acta Arith. VI. (1960), 145-147. [4] B. D. BEACH, H. C. WlLLIMSand C. R. ZARNKE, Some computer results on units in quadratic and cubic fields, Proc. 25 Summer Mitting Canad. Math. Congr. (1971), 609-649. [5] L. J. MORDELL, On a Pellian equation conjecture, Acta Arith. VI. (1960), 137-144. [6] M. SHEINGORN, Hyperbolic reflections on Pell's equation, Theory 33. (1989), 267-285. [7] M. SHEINGORN, The y/p Riemann surface, Acta. Arith. LXIII. 3. (1993), 255-266. [8] W. SLERPINSKI, Elementary Thory of Numbers, PWN-Warszawa, (1987) ALEKSANDER GRYTCZUK INTITUTE OF MATHEMATICS DEPARTMENT OF ALGEBRA AND NUMBER THEORY T. KOTARBINSKI PEDAGOGICAL UNIVERSITY PL. SLOWIANSKI 9, 65-069 ZIELONA GORA POLAND