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Diophantine tuples are sets of positive integers with the property that the product of any two elements in the set increased by the unity is a square. In the main theorem of this paper it is shown that any Diophantine triple, the second largest element of which is between the square and four times the square of the smallest one, is uniquely extended to a Diophantine quadruple by joining an element exceeding the largest element in the triple. A similar result is obtained under the hypothesis that the two smallest elements have the form T 2 + 2 T , 4 T 4 + 8 T 3 − 4 T for some positive integer T , which we encounter as an exceptional case. The main theorem implies that the same is valid for triples with smallest elements K A 2 , 4 K A 4 ± 4 A for some positive integers A and K ∈ { 1 , 2 , 3 , 4 }. [ABSTRACT FROM AUTHOR]

Let k , m and n be integers. In this paper, for a fixed integer μ ≠ 0 , we show that the family of Thue equation x 4 − k m n x 3 y + (k m 2 − k n 2 + 2) x 2 y 2 + k m n x y 3 + y 4 = μ , is reducible by Tzanakis's method into a system of pellian equations k V 2 − (k m 2 + 4) U 2 = − 4 μ , k Z 2 − (k n 2 − 4) U 2 = 4 μ , with any triple of integers (k , m , n) such that k > 0 , | n | ≥ 2 , | m | ≥ 2. We consider this system for any even integer k ≠ 2 □ , μ = 1 and we prove that for all integers | n | ≥ 2 and | m | ≥ 2 that are sufficiently large and have sufficiently large common divisor this system has only the trivial solutions (U , V , Z ,) = (± 1 , ± m , ± n). We also show that if k ≠ 2 □ is even, then the system has in general at most 8 solutions in positive integers. [ABSTRACT FROM AUTHOR]