On the largest prime factor of $x^2-1$

TL;DR

This paper characterizes integers whose $x^2 - 1$ factors only into primes less than 100, leading to applications in finding large integers with consecutive numbers having prime factors below 100.

Contribution

It provides a complete classification of integers with $x^2 - 1$ having prime factors under 100, and derives related numerical corollaries.

Findings

Identified all integers with $x^2 - 1$ prime factors less than 100.

Established existence of large integers with consecutive numbers all having prime factors below 100.

Derived numerical corollaries related to prime factor bounds.

Abstract

In this paper, we find all integers $x$ such that $x^{2}-1$ has only prime factors smaller than 100. This gives some interesting numerical corollaries. For example, for any positive integer $n$ we can find the largest positive integer $x$ such that all prime factors of each of $x, x+1,..., x+n$ are less than 100.