A {f_1,f_2,f_3,…,f_k } Diophantine k tuple with the property P_r over a commutative ring is a set of k distinct nonzero elements of the ring with the property that the product of any two different elements of the ktuple added by r (such that r is an element of the ring) is a squared element in the ring
f_i f_j+r={Y_(ij)}^2 , (i≠j,i,j=1,…,k)
Although a lot of popular mathematicians such as Diophantus, Fermat, Euler,etc…studied the problem and constructed significant results, many problems are remained open in this area.
In this presentation, we consider set of integers as ring and Diophantine 3tuple with property P_r. We look for nonzero different positive integers h_1, h_2,h_3 satisfying a condition as follows:
if negative integer r=35 is added to different product of them (positive integers h_1, h_2,h_3), the results are all squares in the set of integers. It is known that some of Diophantine triple can be extended to a Diophantine quadruple. We demonstrate that some especial different sets of h_1, h_2,h_3 are regular Diophantine triple sets and nonextendible to Diophantine quadruple. In this paper, we use basic concepts from algebraic and elementary number theory. (Readers also can see basic concepts in the keywords part).
Anahtar Kelimeler: P_rTriples, Modular Arithmetic, Quadratic Reciprocity and Residue Theorems, Solutions of Diophantine Equations, Legendre Symbol, Factorization in the Set of Integers.
