On the number of solutions of some congruences modulo a product of odd prime powers
Let \(P_1,P_2,\ldots,P_r\) be distinct odd prime powers and suppose that \(g\) is a primitive root \(\pmod{P_i}\) for each \(i\). Put \(P= \prod_{i=1}^r P_i\) and for each \(i\) put \(f_i = \varphi(P_i)\) and \(f = \text{lcm}_i (f_i)\). Let \(N(g)\) be the number of solutions of the congruence \(g^s + 1 \equiv ^t\pmod P\) with \(0\le s,t< f\). Explicit formulas for \(N(g)\) are obtained in terms of Jacobi sums, thus generalizing work of Carlitz, White, and Whiteman.
congruences modulo a product of odd prime powers
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