Let p be a prime. In 1878 \'{E}. Lucas proved that the congruence (kpβ1β)β‘(β1)k(modp) holds for any nonnegative integer
kβ{0,1,β¦,pβ1}. The converse statement was given in Problem 1494 of
{\it Mathematics Magazine} proposed in 1997 by E. Deutsch and I.M. Gessel. In
this note we generalize this converse assertion by the following result: If
n>1 and q>1 are integers such that (knβ1β)β‘(β1)k(modq) for every integer kβ{0,1,β¦,nβ1}, then q is a prime and
n is a power of q.Comment: 6 page