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

that w = j;. (mod/?,.), for j = l,2,...,£. Let m

0 denote the number such that P\P

2 - Pk

mo +ti = 3 (mod 4) and r

m=4p

l:.p

km + (j>i...p

km

0+u), m = 1,2,... then we have (4p

v.p

k,p

l...p

km

0+u) = \ and for some m, r = q = 3 (mod 4). Since p

i = 3 (mod 4) for i = 1,2,..., k thus we have M m Jl for i = 1,2, {Pi) <pí> [pi) for i = 3,4, By Gauss theorem we get ( P

i\ [-1 for / = 1,2, Therefore * U ) q J = i, 1 fori = 3,4,.. • \ Pi J = -1 , f * \ v P\ j = -1 and proof the case 4 and our Lemma is finished. Lemma 2. Let R

k denote the ring of all integers of K = QÍ*Jd)> d

>0. If R

k is the ring with uniqueness of factorization then the Diophantine equation (2.18) x

2 ~dy

2 — ±4"

p 87