Cyclic Systems of Simultaneous Congruences

for fixed nonzero integers (r, s) with r > 0 and (r, s) = 1. It shows there are finitely many solutions in positive integers qi ≥ 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 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 r (