# On the simultaneous equations $\sigma(2^a)=p^{f_1}q^{g_1}, \sigma(3^b)=p^{f_2}q^{g_2}, \sigma(5^c)=p^{f_3}q^{g_3}$

## TL;DR

This paper investigates solutions to a set of simultaneous equations involving the sum-of-divisors function applied to powers of 2, 3, and 5, expressed as products of two distinct primes, aiming to classify all such solutions.

## Contribution

The paper provides a complete solution to the system of equations involving the sum-of-divisors function and two primes, extending understanding of divisor sum equations with prime factorizations.

## Key findings

- Classified all solutions to the equations involving σ(2^a), σ(3^b), σ(5^c).
- Established conditions under which the equations hold for distinct primes p and q.
- Enhanced the understanding of divisor sum equations with prime power arguments.

## Abstract

We shall solve the simultaneous equations $\sigma(2^a)=p^{f_1}q^{g_1}, \sigma(3^b)=p^{f_2}q^{g_2}, \sigma(5^c)=p^{f_3}q^{g_3}$ with $p, q$ distinct primes.