Maximum GCD Among Pairs of Random Integers

TL;DR

This paper investigates the asymptotic behavior of the maximum GCD among pairs of randomly chosen integers from a large set, revealing that it typically scales around $N^2/ ext{log} N$, with implications for prime factor sharing.

Contribution

The paper provides new asymptotic estimates for the maximum GCD among pairs of random integers, extending the analysis to general arithmetical semigroups under prime number theorem conditions.

Findings

Maximum GCD scales around $N^{2}/ ext{log} N$ with high probability.

Pairs of integers are likely to share a prime factor of order $N^2/ ext{log} N$.

Results generalize to arithmetical semigroups satisfying a prime number theorem.

Abstract

Fix $\alpha >0$, and sample $N$ integers uniformly at random from $\{1,2,\ldots ,\lfloor e^{\alpha N}\rfloor \}$. Given $\eta >0$, the probability that the maximum of the pairwise GCDs lies between $N^{2-\eta }$ and $N^{2+\eta}$ converges to 1 as $N\to \infty $. More precise estimates are obtained. This is a Birthday Problem: two of the random integers are likely to share some prime factor of order $N^2/\log [N]$. The proof generalizes to any arithmetical semigroup where a suitable form of the Prime Number Theorem is valid.