The literature includes a big number number of papers on Diophantine sets. Despite these manuscripts, there are unsolved problems in the field of Diophantine sets. For replies to these problems, a lot of mathematicians have been worked by using different methods from number theory.
There is relation between Diophantine sets and quadratic equations with two unknowns. Though these types of equations are ancient, there is not any general method exists to decide upon a given such equation has how many solutions.
We know that B={b_1,b_2,b_3 } is a Diophantine sets with size three for the property P_k if the product of any different two elements of B by supplementation integer k≠0 is a perfect square in the set of integers (integer ring).
In this brief paper, we consider the analogous problem for Diophantine triples. We investigate several Diophantine sets with size three for the property P_k where k≠0 is a fixed integer. We examine regularity and extendibility of the such sets. While proving our results, we obtain quadratic equations with two unknowns. Using factorization method of integers, we solve quadratic equations with two unknowns and complete proofs. By use of the results of the paper, we will prepare our next generalized paper.
Anahtar Kelimeler: Diophantine Triples, Solutions of Quadratic Equations With Two Variables, Simultaneous Equations
