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ON CONGRUENCES OF THE FORM σ(n) ≡ a (mod n)

ARIA ANAVI

Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA

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PAUL POLLACK

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

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, and

CARL POMERANCE

Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA

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https://doi.org/10.1142/S1793042112501266Cited by:8 (Source: Crossref)

Abstract

We study the distribution of solutions n to the congruence σ(n) ≡ a (mod n). After excluding obvious families of solutions, we show that the number of these n ≤ x is at most x^½+o(1), as x → ∞, uniformly for integers a with ∣a∣ ≤ x^¼. As a concrete example, the number of composite solutions n ≤ x to the congruence σ(n) ≡ 1 (mod n) is at most x^½+o(1). These results are analogues of theorems established for the Euler ϕ-function by the third-named author.

Keywords:

AMSC: 11A25, 11N25, 11A07

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