Prime Divisors of 10's Friends: A Generalization of Prior Bounds

TL;DR

This paper generalizes previous work by providing improved upper bounds for the prime divisors of a 'friend' of 10, an open problem in number theory, extending bounds for the 2nd, 3rd, and 4th smallest prime divisors.

Contribution

It introduces a broader framework for bounding prime divisors of a friend of 10, improving upon prior bounds for specific prime positions.

Findings

Established upper bounds for prime divisors of a friend of 10.

Improved bounds for the 3rd and 4th smallest prime divisors.

Generalized previous results to all prime divisors of a friend of 10.

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].