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DETERMINATION OF EXTENDIBLE OR NON-EXTENDIBLE PARTICULAR REGULAR DIOPHANTINE 3-TUPLES
 
Let s≠0, s∈Z and R={W_1,W_2,W_3 } be a set of three distinct positive integers. If the product of any distinct two elements of R by adding s, i.e. W_i W_j + s , (i≠j,i,j=1,2,3) is a perfect square in the set of integers, then R is called Diophantine triple with property P_s such that s∈Z. The extendibility of the sets is an old problem constructed by Diophantus (He is known as 'father of algebra' due to his significant book Arithmetica, a study on the solution of algebraic equations and the theory of numbers). Diophantus discovered some types of Diophantine triples and found the first Diophantine quadruple {1,33,68,105} with property P_256 but there are still remained unsolved problems on it. The aim of this work is to determine some extendible or non-extendible Diophantine triples for positive fixed integer s=+33. Congruence types are specified for such sets. Then, it is shown that all of the sets are regular. Besides, question “what kind of elements are not in the P_s?” is replied when s=+33. To prepare our paper, some algebraic and elementary methods and notions such as Legendre Symbol, Quadratic Reciprocity Law so on… are used.

Anahtar Kelimeler: Property P_s, Simultaneous Pellian Equations, Quadratic Reciprocity, Congruence Types, Legendre and Jacobi Symbols.



 


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