The sharp threshold for making squares

TL;DR

This paper establishes the exact sharp threshold for the appearance of a perfect square product subsequence in a random integer sequence, confirming previous conjectures and employing advanced probabilistic and combinatorial techniques.

Contribution

It proves the conjectured sharp threshold for making squares in random sequences, resolving a problem posed by Pomerance and extending prior approximate results.

Findings

Confirmed the sharp threshold for subsequences with perfect square products.

Provided a new proof for the upper bound in previous work by Croot et al.

Used the method of self-correcting martingales to analyze the prime factor hypergraph.

Abstract

Consider a random sequence of $N$ integers, each chosen uniformly and independently from the set $\{1,\dots,x\}$. Motivated by applications to factorisation algorithms such as Dixon's algorithm, the quadratic sieve, and the number field sieve, Pomerance in 1994 posed the following problem: how large should $N$ be so that, with high probability, this sequence contains a subsequence, the product of whose elements is a perfect square? Pomerance determined asymptotically the logarithm of the threshold for this event, and conjectured that it in fact exhibits a sharp threshold in $N$. More recently, Croot, Granville, Pemantle and Tetali determined the threshold up to a factor of $4/\pi + o(1)$ as $x \to \infty$, and made a conjecture regarding the location of the sharp threshold. In this paper we prove both of these conjectures, by determining the sharp threshold for making squares. Our proof combines techniques from combinatorics, probability and analytic number theory; in particular, we use the so-called method of self-correcting martingales in order to control the size of the 2-core of the random hypergraph that encodes the prime factors of our random numbers. Our method also gives a new (and completely different) proof of the upper bound in the main theorem of Croot, Granville, Pemantle and Tetali.