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{\bf Andrew Granville and J.L. Selfridge}
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{\bf Product of Integers in an Interval, Modulo Squares}
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We prove a conjecture of Irving Kaplansky which asserts that between any
pair of
consecutive positive squares there is a set of distinct integers whose product
is twice a square. Along similar lines, our main theorem asserts that if
prime $p$
divides some integer in $[z,z+3\sqrt{z/2}+1)$ (with $z\geq 11$) then there
is a set
of integers in the interval whose product is $p$ times a square. This is
probably best possible, because it seems likely that there are arbitrarily
large
counterexamples if we shorten the interval to $[z,z+3\sqrt{z/2})$.


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