Numbers of Solutions of Congruences and Rationality of Generating Functions

LetRbe a UFD andpa prime ofR. Letl1a1,…,lrarbe prime top. In this paper a formula is derived for the number of solutions of the diagonal congruencea1xl11+•••+arxlrr≡b(pn). The corresponding generating function is also constructed and a conjecture of Wan is proved in this case.