Abstract
In this paper some extensions of the Farey fraction method as presented in [Methods and Applications of Error-Free Computation, Springer-Verlag, Berlin, New York, 1984] are discussed for determining solutions of systems of the form $x \equiv ay\bmod m$ in unknown integers $x,y$ where $x,y$ are restricted by $0\leqq x\leqq N,1\leqq y\leqq N$, $\gcd (y,m) = 1$. Since $x \equiv ay\bmod m$ can be used as definition of $x/y \equiv a$ such a system is spoken of as a “rational congruence” with bounded variables $x,y$. Interval conditions are analyzed for $x/y$ to enlarge the set of rational congruences that can be handled by procedures of Euclidean type (gcd computations). Several approaches are proposed for solving rational congruences with one or more moduli.
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