Polynomial generators of recursively enumerable languages

For each language L, let <(F)overcap >(boolean AND)(L) be the smallest intersection-closed full AFL generated by the language L. Furthermore, for each natural number k >= 2 let P-k = {a(nk) vertical bar n is an element of N}. By applying certain classical and recent results on Diophantine equations we show that L-RE = (F) over cap (boolean AND)(P-k), i.e., the family of all recursively enumerable languages coincides with the smallest intersection-closed full AFL generated by the polynomial language Pk for all k >= 2. This allows us to answer to an open problem of S. Ginsburg and J. Goldstine in [2].