Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1993. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 21)
A. Grytczuk and J. Kacierzynski: On Factorization in real quadratic number fields
2° d- 2 app v..p k, p = l(mod4); a = Oorl, p and p i are odd distinct primes. 3° d-2 p xp 2...p ky k> 2, p t = 3 (mod 4) for /' = 1,2,...,*.. 4° d-p xp 1...p k y k >3, p i = 3 (mod4) for /' = 1,2,..., A:. Consider the case 1°. Let r denote the quadratic nonresidue ( r\ for prime p = 1 (mod 4). Hence — = -1. Suppose that m 0 is a KPJ positive integer such that (2.4) pm 0 + r = 5 (mod 8). We note that m 0 satisfying (2.4.) exist since the number pj+r for j = 1,2,...,8 gives distinct residues modulo 8. Let (2.5) r - /?(8w + /w 0) +r = 8/?w + (pm 0 +r), m = 1,2,.... From (2.4) it follows that (8p,pm 0 +r) = 1. Therefore by Dirichlefs theorem we obtain from (2.5) that for some m (2.6) q* = r 0 where q* is a prime number. On the other hand by (2.4) it follows that pm 0 + r = &k + 5 thus by (2.5) and (2.6) we obtain (2.7) q* = 8/ +5. 83
