Some Notes on the Extendibility of an Especial Family of Diophantine P_2 Pairs

Year 2023, Volume: 6 Issue: 2, 1 - 7, 17.05.2023

Özen Özer

Abstract

Although it is known that there are an infinite number of Diophantine P_1 triples, there is no complete characterization for these triples.
This paper is a continuation and a generalization of one of the recent papers (see [ ref. 35 ]) in which several numerical results are demonstrated and some properties are given for special Diophantine P_2 pairs and triples. Here, the expansion of the single-element set {2} into a Diophantine P_2 binary special family as {2, s} (with s values expressed as a recurrence/iteration of natural numbers) is obtained firstly. Then, binary special family {2, s} is extended as {2, s, a_s} Diophantine P_2 triples ( a_s is determined in the terms of s ) using solutions of Diophantine equations. Lastly, it is proved that {2, s, a_s} can not be extended Diophantine P_2 quadruples using elementary and algebraic methods different from other works in the literaure.

Keywords

Diophantine P_2 sets, System of Equations, Integral Solutions, Non- extendable Diophantine P_2 triples, Elementary Number Theory, Natural Numbers.

Supporting Institution

Kırklareli Üniversitesi BAPKO

Project Number

KLUBAP-233

Thanks

The study is supported by Scientific Research Project with number KLUBAP-233 of Kırklareli University.

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