Reducing Every Set of 36 Consecutive Integers to Zero by Differences of Squares

Abstract

For a finite multiset of integers, repeatedly choose two entries $a,b$ and replace them with $|a^{2}-b^{2}|$. Hickerson and Kleber asked whether, for every integer $n$, the set ${n,n+1,\ldots,n+35}$ can be reduced to zero. We give an explicit reduction. Together with their results for lengths 12 and 24, this gives, for every positive integer $L$, [ {n,n+1,\ldots,n+L-1}\text{ reduces to zero for every }n\in\mathbb Z \Longleftrightarrow 12\mid L\text{ and }L\ge 24. ]