Geodesic
Product of integers in an interval, modulo squares
The electronic journal of combinatorics, Tome 8 (2001) no. 1
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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 $[z,z+3\sqrt{z/2})$.
DOI : 10.37236/1549
Classification : 11N64, 11B75
Mots-clés : distribution of integers with specified multiplicative constraints 
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     author = {Andrew Granville and J. L. Selfridge},
     title = {Product of integers in an interval, modulo squares},
     journal = {The electronic journal of combinatorics},
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     number = {1},
     doi = {10.37236/1549},
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Andrew Granville; J. L. Selfridge. Product of integers in an interval, modulo squares. The electronic journal of combinatorics, Tome 8 (2001) no. 1. doi: 10.37236/1549

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