A note on necessary conditions for a friend of 10

TL;DR

This paper investigates the properties and necessary conditions for a number to be a friend of 10, revealing complex prime factorization constraints and divisibility conditions, and establishing bounds related to such numbers.

Contribution

It provides new necessary conditions and bounds for potential friends of 10, including prime factorization properties and divisibility criteria, advancing understanding of solitary numbers.

Findings

If N is a friend of 10, it must be an odd square with at least seven distinct prime factors.

N must have prime factors p and q with specific congruence properties, such as p ≡ 1 mod 10.

Bounds on N are established based on the number of prime factors and the form of N.

Abstract

Solitary numbers are shrouded with mystery. A folklore conjecture assert that 10 is a solitary number i.e. it has no friends. In this article, we establish that if $N$ is a friend of $10$ then it must be odd square with at least seven distinct prime factors, with $5$ being the least one. Moreover there exists a prime factor $p$ of $N$ such that $2a+1\equiv 0 \pmod f$ and $5^{f}\equiv 1 \pmod p$ where $f$ is the smallest odd positive integer greater than $1$ and less than or equal to $\min\{ 2a+1,p-1\}$, provided $5^{2a}\mid \mid N$. Further, there exist prime factors $p$ and $q$ (not necessarily distinct) of $N$ such that $p\equiv1 \pmod {10}$ and $q\equiv 1\pmod 6$. Besides, we prove that if a Fermat prime $F_k$ divides $N$ then $N$ must have a prime factor congruent to $1$ modulo $2F_k$. Also, if we consider the form of $N$ as $N=5^{2a}m^2$ then $m$ is non square-free. Furthermore, we show that $\Omega(N)\geq 2\omega(N)+6a-4$ and if $\Omega(m)\leq K$ then $N< 5\cdot 6^{(2^{K-2a+1}-1)^2}$ where $\Omega(n)$ and $\omega(n)$ denote the total number of prime factors and the number of distinct prime factors of the integer $n$ respectively.