Exploring the Relationships Between the Divisors of Friends of $10$

TL;DR

The paper explores the structure of friends of $10$ via the abundancy index $I(F)=\frac{\sigma(F)}{F}$, showing that every friend can be written as $F=5^{2a}\cdot Q^{2}$ with $Q$ odd and coprime to $15$, and deriving strong congruence constraints on $\sigma(5^{2a})$ and $\sigma(Q^{2})$ depending on the parity of $a$. It proves that not all exponents $a_i$ in the prime factorization $F=5^{2a_1}\cdot\prod_{i\ge 2} p_i^{2a_i}$ can satisfy certain congruences (e.g., $a_i\equiv 1 \pmod{3}$ or $a_i\equiv 13 \pmod{27}$ for all $i$), using results on prime-divisor divisibility of divisor sums. The work also establishes explicit lower bounds for $F$ in terms of the exponents, namely $F>\frac{25}{81}\cdot\prod_{i=1}^{\omega(F)}(2a_i+1)^2$, which implies $F>625\cdot 9^{\omega(F)-3}$, and discusses implications for the solitary number problem. These results advance understanding of the arithmetic constraints governing friends of $10$ and contribute to narrowing the search for possible friends. Overall, the findings provide concrete divisibility and congruence criteria that any friend of $10$ must satisfy, with potential implications for larger unsolved questions about solitary numbers.

Abstract

A solitary number is a positive integer that shares its abundancy index only with itself. $10$ is the smallest positive integer suspected to be solitary, but no proof has been established so far. In this paper, we prove that not all half of the exponents of the prime divisors of a friend of 10 are congruent to $1$ modulo $3$. Furthermore, we prove that if $F=5^{2a}\cdot Q^2$ ($Q$ is an odd positive integer coprime to $15$) is a friend of $10$, then $σ(5^{2a})+σ(Q^2)$ is congruent to $6$ modulo $8$ if and only if $a$ is even, and $σ(5^{2a}) + σ(Q^2)$ is congruent to $2$ modulo $8$ if and only if $a$ is odd. In addition, if we set $Q={\displaystyle \prod_{i=2}^{ω(F)}}p_{i}^{a_i}$ and $a=a_1$, where $p_i$ are prime numbers, then we establish that $$F>\frac{25}{81}\cdot\prod_{i=1}^{ω(F)}(2a_i + 1)^2,$$ in particular $F> 625\cdot 9^{ω(F)-3}.$

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• proof
• Lemma 2.1: rlpaw
• Theorem 2.2: SS2024, Theorem 1.3
• Proposition 2.3