Complete n-sets of integer triples, and improved bounds for n₁(k)

Abstract

For a nonzero integer n, a D(n)-set is a set of distinct nonzero integers whose pairwise products, increased by n, are all perfect squares. Adžaga, Dujella, Kreso and Tadić asked, for each k, how small in absolute value a nonzero integer n₁(k) can be when some triple of nonzero integers is a D(n)-set for k distinct values of n, one of them n₁(k), and gave a table of upper bounds for 5 ≤ k ≤ 20. Their bounds come from points on the elliptic curve induced by a triple, so each row exhibits some of the admissible n for its triple. We instead determine, for a fixed triple, all of them: the set of admissible n is the image of an explicit finite divisor enumeration, and sweeping that enumeration proves completeness rather than mere existence. This improves eight of the sixteen published rows, among them |n₁(5)| ≤ 4 and |n₁(6)| ≤ 4 in place of 36 and 215, |n₁(7)| ≤ 144, |n₁(8)| ≤ 2304, |n₁(10)| ≤ 103684 and |n₁(18)| ≤ 493214400. All sixteen rows are reproduced; the other eight are matched but not improved. Two of the improvements need no new triple: the triples printed on the rows k=16 and k=18 are D(n)-sets for 17 and for 19 values of n respectively. We also show that each of the seven published Diophantine triples that are D(n)-sets for four values of n carries exactly four and no more, and we add two further such triples, {20,1980,637602} and {140,204,77913732}. All statements are formally verified in Lean 4.

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