D(n)-quintuples with square elements

TL;DR

This paper proves the existence of infinitely many D(n)-quintuples with square elements by constructing genus one curves on specific algebraic surfaces, advancing understanding of special integer sets related to perfect squares.

Contribution

It demonstrates the infinite existence of D(n)-quintuples with squares through algebraic geometric constructions, a novel approach in this area.

Findings

Infinitely many essentially different D(n)-quintuples with square elements exist.

Construction of genus one curves on double covers of affine planes.

New method linking algebraic geometry to Diophantine tuples.

Abstract

For an integer n, a set of m distinct nonzero integers {a_1,a_2,...,a_m} such that a_i a_j+n is a perfect square for all 0<i<j<m+1, is called a D(n)-m-tuple. In this paper, we show that there are infinitely many essentially different D(n)-quintuples with square elements. We obtained this result by constructing genus one curves on a certain double cover of A^2 branched along four curves.