An old topic which is called Diophantine sets was started to work by Diophantus. The topic is work over some kind of rings such as set of integers, set of rational numbers, so on… Still,some significant questions have been waited to reply about Diophantine rtuples over the set of integers.
Let G={g_1,g_2,g_3,…,g_r } be a set of positive integer elements. G is called Diophantine r tuple with the property P_t over a integer ring if the product of any two distinct elements of the rtuple increased by integer t is a perfect square element in the set of integers (i.e. integer ring) g_i g_j+t=Z_ij^2 where i,j=1,…,r and i,j are different integers.
In this brief work, we consider some Diophantine sets with property P_t over the set of integer for some t integers such that t≡3 (mod 4). We determine some properties of the elements which have not been in these types of Diophantine sets. To prove them, we use Modular Arithmetic, Quadratic Reciprocity Law, Legendre Symbol, Jacobi Symbol, Quadratic Residue Theorems etc… All of them are basic and crucial notations in the number theory for mathematics. For our next paper, our aim will be to generalize and extend these classifications and properties.
Anahtar Kelimeler: Diophantine rTuple With The Property P_t, Modular Arithmetic, Quadratic Reciprocity Law, Legendre Symbol, Jacobi Symbol, Quadratic Residue Theorems.
