Prime Divisors of 10's Friends: A Generalization of Prior Bounds
Sagar Mandal
TL;DR
This work addresses the problem of bounding the prime divisors of a friend of $10$, i.e., numbers $n$ with $I(n)=I(10)=9/5$ where $I(n)=\sigma(n)/n$. It introduces a general framework that yields an upper bound of the form $q_r < L\{\log L + 2\log\log L\}$ for the $r$-th smallest prime divisor $q_r$, with $L=\lceil \mathcal{A}\omega(n)/\mathcal{B}\rceil$ under a diophantine constraint on coprime integers $\mathcal{A},\mathcal{B}$ and $r$, and then proves that this bound forces a contradiction unless $q_r$ stays below $p_L$, the $L$-th prime. The paper further provides corollaries that yield sharper bounds for $q_3$ and $q_4$, supplies an alternative, parameter-based proof of existing bounds, and discusses computational implications, suggesting that any such friend must be enormous (beyond $10^{30}$). Overall, the results offer a systematic, parameterized method to constrain all prime divisors of a friend of $10$, guiding both theoretical analysis and computational searches.
Abstract
10 is the smallest positive integer which is whether solitary or friendly is still an open question in mathematics. In this paper, we provide upper bounds for each of the prime divisors of a friend of 10. This paper is precisely a generalization of a recent paper [4] in which necessary upper bounds for the 2nd, 3rd, and 4th smallest prime divisors of a friend of 10 have been proved. Further, we establish better upper bounds for the 3rd, and 4th smallest prime divisors of a friend of 10 than the bounds given in [4].