The Number of Solutions of a System of Congruences

where A1 = {x : 0 ≤ x < p, p f1(x)f2(x)} . In order to do this, Burgess found estimates for the inner summation over various subsets of A1 and then counted the number of m for which these subsets were non-empty. The counting process was carried out using different methods, one of which concerned the estimation of the cardinality of S = {m : 0 < mi ≤ h, f1(X) ≡ f2(X) (mod pμ)} . The estimation of such character sums in the case r = 2 is contained in [2]. The case r = 4 has yet to be proved. This paper estimates #S when r = 4, as a step in the direction of a proof. The result obtained is given by the following theorem.