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
Let m 0 denote the positive integer such that (2.12) pp x ..p km 0+u = (mod 8). It is easy to see that such m 0 exist, because the number PPv Pj + 1 1 gives distinct residues (mod 8). Let (2.13) r m - pp x... p k (8m + m 0 ) + u = 8pp }... p km + (pp ]...p km 0+u) m= 1,2.... Since (&pp ]...p k,pp ]...p km 0+u) = \ then by Dirichlefs Theorem we have for some m (2.14) q=r„ where q is a prime number. It is easy to see that by (2.12) it follows that q = 1 (mod 8) Therefore similarly as in the case 1° we obtain (2.15) and (2.16) £ KP J ( kPÍ \P) ( \ \Pij r} PJ V| Pij ( -l, VPi ( \ VPJ -1 <PIJ = -1 = 1 for / = 2,3,...,*. From (2.15), (2.16) and reciprocity law of Gauss we get (JL) M * U ) W) W) and therefore we have WJ =i = 1 for I =2,3,..., A: = -1 and ( * \ q ^ f * \ -q ^ PJ V P = -i. 85
