On some particular regular Diophantine 3-tuples

Abstract

Diophantine n-tuple where n=3 is called as a Diophantine triple. It means that Diophantine triple is a set of three positive integers satisfying special condition. For example, {a, b, c} is called a D(k)-Diophantine triple if multiplying of any two different of them plus k is a perfect square integer where k is an integer. In this work, we take in consideration some kind of regular D(±3 3)-Diophantine triples. We demonstrate that such sets can not be extendible to D(±3 3)-Diophantine quadruple by using algebraic methods such as classical Pell equations solutions, solutions of ux 2 + vy 2 = w Diophantine equations where u, v, w ∈ Z, factorization in the set of integers, and so on. Besides, we obtain some notable characteristic properties for such sets.

A Certain Type of Regular Diophantine Triples and Their Non-Extendability

“…The author Özer [5,6,7,8,9] has prepared her papers on various types of Diophantine triple with different algebraic methods. Some of the basic informations (Definitions, Examples, Theorems, Lemmas etc…) such as quadratic residues, quadratic reciprocity law, legendre & Jacobi symbols are given for algebraic and elementary number theory in [10], [11] and [12].…”