Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

GRYTCZUK, J., On Perron's proof of Fermât's two square theorem

96 Jaroslaw Grytczuk Analysing the last équation we conclude that = 2 or = 1. However, the second possibility occurs only if k is a multiple of s [1, p.171]. Hence q

k and qk-i are even a

k is odd and because of (3) so is mFrom (1) ra^-i is odd. too. Actually, the parity of qi and rrii remains unchanged for further indices i = k — 2, k — 3,..., 1,0. Indeed, puting (1) to (2) we obtain (5) qi = qi2 + ai_i(mi_i - m*) and now looking by turns on (5) and (1) we get the announced effect. But this is contrary to the initial conditions rao = 0, ço = 1 and the proof is complété. References [1] I. NIVEN and H. ZUCKERMAN, An Introduction to The Theory of Numbers, Third Edition, John Wiley and Sons, (1972). [2] 0. PERRON, Die Lehre von den Kettenbrüchen, Teubner, Stuttgart, (1954).