A variation of a congruence of Subbarao for n=2^(alpha)*5^(beta)

TL;DR

This paper characterizes positive integers of the form 2^α 5^β satisfying a specific congruence involving Euler's totient and divisor sum functions, identifying only four such integers.

Contribution

It provides a complete characterization of integers of the form 2^α 5^β satisfying a particular congruence, extending understanding of related number theory problems.

Findings

Only n=1, 2, 5, 8 satisfy the congruence for n=2^α 5^β.

The paper proves the exclusivity of these solutions within the specified form.

It advances the study of congruences involving arithmetic functions for specific integer classes.

Abstract

There are many open problems concerning the characterization of the positive integers $n$ fulfilling certain congruences and involving the Euler totient function $\varphi$ and the sum of positive divisors function $\sigma$ of the positive integer $n$. In this work, we deal with the congruence of the form $$ n\varphi(n)\equiv2\pmod{\sigma(n)} $$ and we prove that the only positive integers of the form $2^{\alpha}5^{\beta}, \enspace \alpha, \beta\geq0,$ that satisfy the above congruence are $n=1, 2, 5, 8$.