A Note on the Particular Set with Size Three

For a fixed integer k , a P k – set is defined as a set of n positive integers { x 1 , x 2 , x 3 ,..., x n } with the property that x i , x j + k is a perfect square, whenever i ≠ j . In this paper, we prove P 11 = {1,14,25}, P −11 = {1,15,36}, and P −11 {4,9,23} sets cannot be extendable. It is also proved that P 11 sets don’t contain any multiple of 3 and P − 11 sets don’t include any multiple of 7. Moreover, it is demonstrated that all of the elements of the sets P − 11 of size three cannot be odd positive integer. Mathematics Subject Classifications: 11A07, 11D45, 11A15.