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

has a solution in positive integers x,y for every prime p such that yP) = +1, where a = 0 \f d = 2,3 (mod 4) 1 \í d = \ (mod 4) Proof. From the assumption that rd} <Pj = 41 it follows that there exists a positive integer x such that r

2 = J(mod p). From this follows that (2.19) p\x

2-d = (x-y[d)(x + Jd) Suppose that the number p is an irreducible element of R

k. Since the ring R

k is the ring with uniqueness factorization we get that the number p is also prime number in R

k. The by (2.19) it follows that (2.20) p\x-yfd or p\x + 4d x — -Jd it is impossible, because the elements and are no P elements of R

k. Therefore we obtain x

]+y

]*Jd^\ fx

2+y

2yfd 2

a 2

a (2.21) p = where (2.22) a = , [1 d = 1 (mod 4) and the elements ^ and *

2 are noninvertible. Í0 d = 2,3 (mod 4) 2" 2

a 88