The sharp threshold for making squares
Paul Balister, B\'ela Bollob\'as, Robert Morris

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 integers, each chosen uniformly and independently from the set . 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 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 . More recently, Croot, Granville, Pemantle and Tetali determined the threshold up to a factor of as , 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…
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