Finding all S-Diophantine quadruples for a fixed set of primes S

Given a finite set of primes S and an m-tuple $$(a_1,\ldots ,a_m)$$ ( a 1 , … , a m ) of positive, distinct integers we call the m-tuple S-Diophantine, if for each $$1\le i < j\le m$$ 1 ≤ i < j ≤ m the quantity $$a_ia_j+1$$ a i a j + 1 has prime divisors coming only from the set S. For a given set S we give a practical algorithm to find all S-Diophantine quadruples, provided that $$|S|=3$$ | S | = 3 .

There are no Diophantine quadruples of Pell numbers

In this paper, we prove that there are no positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] such that [Formula: see text] is a Diophantine quadruple, where for a positive integer [Formula: see text], [Formula: see text] is the [Formula: see text]th Pell number.

The extensibility of the Diophantine triple {2, b, c}

The aim of this paper is to consider the extensibility of the Diophantine triple {2, b, c}, where 2 < b < c, and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of c’s (depending on b). As corollary, for example, we prove that for b/2 − 1 prime, all Diophantine quadruples {2, b, c, d} with 2 < b < c < d are regular.

CONSTRUCTION OF IRRATIONAL GAUSSIAN DIOPHANTINE QUADRUPLES

Given any two non-zero distinct irrational Gaussian integers such that their product added with either 1 or 4 is a perfect square, an irrational Gaussian Diophantine quadruple ( , ) a0 a1, a2, a3 such that the product of any two members of the set added with either 1 or 4 is a perfect square by employing the non-zero distinct integer solutions of the system of double Diophantine equations. The repetition of the above process leads to the generation of sequences of irrational Gaussian Diophantine quadruples with the given property.