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

GRYTCZUK, A., On a theorem of G. Baron and A. Schinzel

On a theorem of G. Baron and A. Schinzel ALEKSANDER GRYTCZUK Abstract. G. Baron and A. Schinzel [l] generalized the wellknown Wilson's theorem. In this paper

—under Theorem B—an extension of their theorem can be found. 1. Introduction In 1979 an extension of Wilson's theorem was given by G. Baron and A. Schinzel [1]. Namely they proved the following: Theorem A. For any prime p and any residues X{ mod p we have Xa(l) {

xa( 1) + Xa(2)) • • • (

xa( 1) H + xa(p-l)) = (1) crÇSp— 1 = H \-Xp-iY'

1 (mod p) where summation is taken over ail permutation a of {1, 2, • • • ,p — 1}. In the present Note we prove the following extension of Theorem A: Theorem B. For any prime p and any residues X{ mod p and for fixed natural number k such that p — l\k we have Y^ + ®ï(2)) " " " (®£(1) + * * * + Xt(p-l)) = (2) ^eVi - (xí + + (mod p) and if Xi ^ 0 are residues mod p, p is an odd prime such that p — 1 | k then Xa(l) (®í(l) + Xt(2)) • • • (

Xa(l) + • • • + (3) vés,,-! +®Í(P-I)) = 1 (

MO

D P) where summation is taken over ail permutation a of {1,2, • • • ,p — 1}. We note that Ch. Snyder [3] gave interesting applications of (1) to differentials in rings of characteristic p.