Cyclic systems of simultaneous congruences

TL;DR

This paper investigates the solutions of a cyclic system of simultaneous congruences, establishing bounds on solutions, conditions for infinitude, and reducing the problem to a family of Diophantine equations.

Contribution

It provides a finite solution bound for positive solutions and characterizes when infinitely many solutions exist, linking the problem to Diophantine equations.

Findings

Finite solutions have a doubly-exponential size bound in n.

Infinite solutions occur when r=1 or positivity constraints are relaxed.

The problem reduces to a family of Diophantine equations with three parameters.

Abstract

This paper considers solutions (x_1, x_2, ..., x_n) to the cyclic system of n simultaneous congruences r (x_1x_2 ...x_n)/x_i = s (mod |x_i|), for fixed nonzero integers r,s with r>0 and gcd(r,s)=1. It shows this system has a finite number of solutions in positive integers x_i >1 having gcd(x_1x_2...x_n, s)=1, obtaining a sharp upper bound on the maximal size of the solutions in many cases. This bound grows doubly-exponentially in n. It shows there are infinitely many such solutions when the positivity restriction is dropped, when r=1, and not otherwise. The problem is reduced to the study of integer solutions of a three parameter family of Diophantine equations r(1/x_1 + 1/x_2 + ...+ 1/x_n)- s/(x_1x_2...x_n) = m, with parameters (r,s,m).