Diophantine quadruples

A Diophantine quadruple is a set \{a,b,c,d\} of four positive integers such that ab+1, ac+1, ad+1, bc+1, bd+1, and cd+1 are all perfect squares. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by x. In doing so, we describe two existing tools in analytic number theory and how we extended them. The "Erdős-Turán inequality" bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also adapt an argument of Hooley on the equidistribution of solutions of polynomial congruences to handle reducible quadratic polynomials.

Greg Martin