This paper considers the cyclic system of n ≥ 2 simultaneous congruences [Formula: see text] for fixed nonzero integers (r, s) with r > 0 and (r, s) = 1. It shows there are only finitely many solutions in positive integers qi ≥ 2, with gcd (q1q2 ⋯ qn, s) = 1 and obtains sharp bounds on the maximal size of solutions for almost all (r, s). The extremal solutions for r = s = 1 are related to Sylvester's sequence 2, 3, 7, 43, 1807,…. If the positivity condition on the integers qi is dropped, then for r = 1 these systems of congruences, taken ( mod |qi|), have infinitely many solutions, while for r ≥ 2 they have finitely many solutions. The problem is reduced to studying integer solutions of the family of Diophantine equations [Formula: see text] depending on three parameters (r, s, m).
A Note on Simultaneous Congruences
