Let k be an integer, A is a set {x 1 , x 2 , x 3 , ...x n } with n positive different integers. A set is called P k , if i, j ∈ N and i = j, x i x j + k is a square of an integer. In our research we studied on set P k .We proved that sets P 7 = {1, 2, 9}, P −7 = {1, 8, 11, 16} and P −7 = {2, 4, 8} can not be extended. In addition, we showed that there is no P −7 set that contains any multiple of 3 or 5. Furthermore it is proven that sets P k , {k = 4t + 2, k = 4t + 3, t ∈ Z} can contain at most 1 even number as an element. Mathematics Subject Classifications: 10B05, 10A10, 10A35