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

Authors : Özen ÖZER

Pages : 1-7

View : 32 | Download : 38

Publication Date : 2023-05-17

Article Type : Research Paper

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 insert ignore into journalissuearticles values(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} insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values( 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