Original scientific paper

https://doi.org/10.3336/gm.54.1.05

Diophantine m-tuples with the property D(n)

Riley Becker ; Department of Mathematics, Queen's University, Kingston, Ontario, K7L 3N6, Canada
M. Ram Murty ; Department of Mathematics, Queen's University, Kingston, Ontario, K7L 3N6, Canada

Abstract

Let n be a non-zero integer. A set of m positive integers { a1,a2,⋯ ,am} such that aiaj+n is a perfect square for all 1≤ i < j≤ m is called a Diophantine m-tuple with the property D(n). In a series of papers, Dujella studied the quantity Mn= sup {|????|: ???? has the property D(n)} and showed for |n|≥ 400 that Mn ≤ 15.476 log |n| and if |n| >10100, then Mn < 9.078 log |n|. We refine his argument to show that Cn≤ 2log |n|+ O(log |n|/(log log |n|)2), where the implied constant is effectively computable and Cn = sup {|???? ∩ [1,n2]|:???? has the property D(n)}. Together with earlier work of Dujella, this implies Mn≤ 2.6071 log |n|+ O(log |n|/ (log log |n|)2), where the implied constant is effectively computable.

Keywords

Diophantine m-tuples; Gallagher's sieve; Vinogradov's inequality

Hrčak ID:

220843

URI

https://hrcak.srce.hr/220843

Publication date:

7.6.2019.

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