Any Diophantine quintuple contains a regular Diophantine quadruple

Abstract

A set {a 1 , . . . ,a m } of m distinct positive integers is called a Diophantine m-tuple if a i a j + 1 is a perfect square for all i, j withIn this paper, we show that if {a, b, c, d, e} is a Diophantine quintuple with a < b < c < d < e, then d = d + .

An irregular D(4)-quadruple cannot be extended to a quintuple

“…Let us mention that recently Fujita [13] has proved the analogous result for D(1)-m-tuples. The main difference in our proof is that we consider the binary recurrence sequences more carefully, so we obtain significantly improved gap principles.…”

The number of Diophantine quintuples

“…It suffices to show that u 4 = v 4 and u 6 > v 4 . The latter immediately follows from Lemma 2.10 (1) in [12] and its proof, in view of…”

“…It is known that there does not exist a Diophantine sextuple and that there does not exist only finitely many Diophantine quintuples ( [8]). Recently, we ( [12]) showed that if {a, b, c, d, e} is a Diophantine quintuple with a < b < c < d < e, then d = d + (note that all the quadruples contained in the quintuple, other than {a, b, c, d}, are irregular; see Section 1 in [12]). …”

“…We first consider the number of Diophantine quintuples {a, b, c, d, e} with a < b < c < d < e for a fixed triple {a, b, c}. Since the above-mentioned result ( [12]) implies that d is unique, it suffices to bound the number of the e's. The essential tool to do this is an exponential gap principle between the solutions due to Okazaki (cf.…”

On a family of two-parametric D(4)-triples

“…All we have to do is to find the real numbers satisfying the assumptions in Lemma 3.2. Using formulas (24) and (25) in [12], we have…”