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July 2018 The sharp threshold for making squares
Paul Balister, Béla Bollobás, Robert Morris
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Ann. of Math. (2) 188(1): 49-143 (July 2018). DOI: 10.4007/annals.2018.188.1.2


Consider a random sequence of $N$ integers, each chosen uniformly and independently from the set $\{1,\ldots,x\}$. Motivated by applications to factorization 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.


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Paul Balister. Béla Bollobás. Robert Morris. "The sharp threshold for making squares." Ann. of Math. (2) 188 (1) 49 - 143, July 2018.


Published: July 2018
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2018.188.1.2

Primary: 11Y05
Secondary: 60C05

Keywords: integer factorization , perfect square , random graph process

Rights: Copyright © 2018 Department of Mathematics, Princeton University


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Vol.188 • No. 1 • July 2018
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