On Characterizing Potential Friends of 20

TL;DR

This paper investigates the properties and constraints of potential 'friends' of the number 20, providing characterizations, bounds, and conditions that such numbers must satisfy, thus addressing a folklore conjecture about their existence.

Contribution

The paper offers a detailed characterization of potential friends of 20, including explicit forms, prime divisor bounds, and modular conditions, advancing understanding of this folklore problem.

Findings

Potential friends of 20 have a specific algebraic form involving powers of 5 and 2.

There are lower bounds on the number of prime divisors of such friends.

Prime divisors of friends of 20 are bounded above by a function of their total number of divisors.

Abstract

Does $20$ have a friend? Or is it a solitary number? A folklore conjecture asserts that $20$ has no friends i.e. it is a solitary number. In this article, we prove that, a friend $N$ of $20$ is of the form $N=2\cdot5^{2a}\cdot m^2$, with $(3,m)=(7,m)=1$ and it has at least six distinct prime divisors. Furthermore, we show that $\Omega(N)\geq 2\omega(N)+6a-5$ and if $\Omega(m)\leq K$ then $N< 10\cdot 6^{(2^{K-2a+3}-1)^2}$, where $\Omega(n)$ and $\omega(n)$ denote the total number of prime divisors and the number of distinct prime divisors of the integer $n$ respectively. In addition, we deduce that, not all exponents of odd prime divisors of friend $N$ of $20$ are congruent to $-1$ modulo $f$, where $f$ is the order of $5$ in $(\mathbb{Z}/p\mathbb{Z})^\times$ such that $3\mid f$ and $p$ is a prime congruent to $1$ modulo $6$. Also, we prove necessary upper bounds for all prime divisors of friends of 20 in terms of the number of divisors of the friend. In addition, we prove that, if $P$ is the largest prime divisor of $N$ then $P<N^{\frac{1}{4}}$.