Article ID Journal Published Year Pages File Type
8896933 Journal of Number Theory 2018 11 Pages PDF
Abstract
Consider the linear congruence equationa1sx1+…+aksxk≡b(mod ns) where ai,b∈Z,s∈N. Denote by (a,b)s the largest ls∈N which divides a and b simultaneously. Given ti|n, we seek solutions 〈x1,…,xk〉∈Zk for this linear congruence with the restrictions (xi,ns)s=tis. Bibak et al. [2] considered the above linear congruence with s=1 and gave a formula for the number of solutions in terms of the Ramanujan sums. In this paper, we derive a formula for the number of solutions of the above congruence for arbitrary s∈N which involves the generalized Ramanujan sums defined by E. Cohen [5].
Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
Authors
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