Sieving for Twin Smooth Integers with Solutions to the Prouhet-Tarry-Escott Problem

We give a sieving algorithm for finding pairs of consecutive smooth numbers that utilizes solutions to the Prouhet-Tarry-Escott (PTE) problem. Any such solution induces two degree-n polynomials, a(x) and b(x), that differ by a constant integer C and completely split into linear factors in \(\mathbb {Z}[x]\). It follows that for any \(\ell \in \mathbb {Z}\) such that \(a(\ell ) \equiv b(\ell ) \equiv 0 \bmod {C}\), the two integers \(a(\ell )/C\) and \(b(\ell )/C\) differ by 1 and necessarily contain n factors of roughly the same size. For a fixed smoothness bound B, restricting the search to pairs of integers that are parameterized in this way increases the probability that they are B-smooth. Our algorithm combines a simple sieve with parametrizations given by a collection of solutions to the PTE problem.