Upper bounds for the prime divisors of friends of 10

TL;DR

Let $I(n)=\frac{\sigma(n)}{n}$ be the abundancy index; the paper investigates the (unknown) friend of $10$ by deriving explicit upper bounds on the second, third, and fourth smallest prime divisors $q_2,q_3,q_4$ of any such friend $n$ in terms of the number of distinct prime divisors $\omega(n)$. The method combines elementary properties of $I(n)$ (weak multiplicativity and a prime-factor formula) with monotonicity arguments on carefully chosen rational functions and standard prime-bound estimates to show $q_2<p_{\lceil 7\omega(n)/3\rceil}$, $q_3<p_{\lceil 180\omega(n)/41\rceil}$, and $q_4<p_{\lceil 390\omega(n)/47\rceil}$. Each bound is established by contradiction: if the corresponding $q_k$ were at least the stated prime, the resulting upper bound on $I(n)$ would fall below a feasible threshold (e.g., $\frac{9}{5}$), contradicting the existence of a friend of $10$. These results significantly constrain the possible prime structure of any friend of $10$, although extending the method to higher primes remains challenging. Computational notes indicate that any potential friend would have to be extraordinarily large. $I(n)$ and the derived inequalities hinge on manipulations of products over primes and the behavior of the function $\psi(x)=\frac{x}{x-1}$ and related monotone rational functions.

Abstract

In this paper we propose necessary upper bounds for the second, third and fourth smallest prime divisors of friends of 10 based on the number of distinct prime divisors of it.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Remark 3.3
• Theorem 3.4
• proof