In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence. As a consequence, we derive necessary and sufficient conditions under which the this restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in a special case. Since then, this problem has been studied, in several other special cases, in many papers. The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions.
Bibak, K., Kapron, B., Srinivasan, V., Toth, L., Tauraso, R. (2017). Restricted linear congruences. JOURNAL OF NUMBER THEORY, 171, 128-144 [10.1016/j.jnt.2016.07.018].
Restricted linear congruences
Tauraso R
2017-01-01
Abstract
In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence. As a consequence, we derive necessary and sufficient conditions under which the this restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in a special case. Since then, this problem has been studied, in several other special cases, in many papers. The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
