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

Since r

m-q - p(8m + m

0) + r thus by well-known property of Legendre's symbol we have (2.8) (<n ÍkPJ yPj = -1 By reciprocity law of Gauss in our case <7* = 5 (mod 8), p = 4k +1 we get (2.9) = -1 and Thus by (2.9) we have W ) = -1 (2.10) d VÍ ) \

cl J Í ^ \ W ) / \ u

; = +1. By (2.8) and property of Legendre's symbol we have {-JL V P p J V p1 =(-1) ( " \ \p = -1 and the case 1° is proved. For the proof of case 2° suppose that r,s are the residues for p and p

x and r

2,r

2,...,r

k are non residues for modulo p

2,pi,...,p

k. Since (p,p,) = (p

i,p

J) = l for thus by Chinese remainder theorem we obtain, that there exists positive integer u such that (2.11) u = r (mod p), z/ = s(mod /?,), w = r

{ (mod /?,); / = 2 ,...,k

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