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

128 Aleksander Grytczuk 2. PROOF of the Theorem B. Let (4) S

lf<

7 = Y^ (Xcr(l) +Z<r(2)) (^(1) + ' " ' + Z<r(p-1) ) crES

p-i and (5) Sk,a - ^^(í) + xí(2)) ' " ' + 1" ^(p-i)) First we note that if p — l\k and k > p — 1 then k — (p—l)t + r, 1 < r < p—l and by Fermât 's theorem we obtain S

k,a = Sr,a (mod p). Thus it suffices to prove (2) in the case k < p — 1. It is easy to see that for such k we have (6) x- = x

a(i) (mod p) for somé a and i = 1, 2, • • •, p — 1. From (6) we obtain Sk,a= i

Xcr(a{l)) +X

a(a(2))) • • • {

xa(a(l)) + (7) <r<=S

p-i + "• + ^-1))) (mod p) By (7) and (1) it follows that (8) S

k}(

X = (x

ail

) +'•• + x

a{

p_

1

)y~

1 (mod p). Now by (8) and (6) we obtain — (arf + + (mod p) and (2) is pro ved. For the proof (3) we remark that k = (p — 1 )t and by Fermât 's theorem we obtain (9) S

K<

J = Y, l-2---(p-l) = (p-l)!(p-l)! (mod p) CR£S

P-1 From (9) and Wilson's theorem we obtain S

k,a = 1 (mod p) and the proof is finished.