On a class of prime-detecting congruences

Abstract

Gould (The Fibonacci Quarterly 2 (1964) 241–260) proved the general inversion theorem: for any ordered sequence pair ( f( n, k), g( n, k)) ƒ(n,k)= ∑ j=k n g(n,j)R(j,k) if and only if g(n,k)= ∑ j=k n ƒ(n,j)A(j,k) where R( n, k) is the number of compositions of n ⩾ 1 into k relatively prime parts and A(n,k)= ∑ j=k n (-1) n−j n j [ j k ] is its inverse. In this paper, we obtain a variety of such ordered inversion pairs. Further, we give necessary and sufficient conditions for the congruence f( n, k)  g( n, k) (mod k) to hold, in particular criteria for k ⩾ 2 to be a prime when the congruence holds for all n ⩾ 1.