Year: 2025
Type: preprint
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 $\sigma(5^{2a})+\sigma(Q^2)$ is congruent to $6$ modulo $8$ if and only if $a$ is even, and $\sigma(5^{2a})+\sigma(Q^2)$ is congruent to $2$ modulo $8$ if and only if $a$ is odd. In addition, if we set $Q={\displaystyle \prod_{i=2}^{\omega(F)}}p_{i}^{a_i}$ and $a=a_1$, where $p_i$ are prime numbers, then we establish that $$F>\frac{25}{81}\cdot\prod_{i=1}^{\omega(F)}(2a_i + 1)^2,$$in particular $F> 625\cdot 9^{\omega(F)-3}.$
Source: arXiv (Cornell University)
Author: Mandal, Sagar
Institution: Indian Institute of Technology Kanpur
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6949315583
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