Two-parameter families of uniquely extendable Diophantine triples

Abstract

Abstract. Let A, K be positive integers and ε ∈ {−2, −1, 1, 2}. The main contribution of the paper is a proof that each of the D(ε 2 )-triples {K, A 2 K + 2εA, (A + 1) 2 K + 2ε(A + 1)} has unique extension to a D(ε 2 )-quadruple. This is used to slightly strengthen the conditions required for the existence of a D(1)-quintuple whose smallest three elements form a regular triple.

The regularity of Diophantine quadruples

“…(ii) {1, 3} ( [17]); (iii) {k − 1, k + 1} ( [5,21]); (iv) {k, A 2 k +2A, (A+1) 2 k +2(A+1)} with positive integers k and A satisfying A ≤ 10 or A ≥ 52330 ( [25][26][27]); (v) {a, b} with b < a + 4 √ a ( [20]). Recently, Cipu, Mignotte, and the first author showed that any Diophantine quadruple containing the regular Diophantine triple {k, A 2 k + 2A, (A + 1) 2 k + 2(A + 1)} with any positive integers k and A, which appears in (iv), is always regular ( [9]).…”