  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.
