Primes differing by a fixed integer

TL;DR: In this article, it was shown that the problem is solvable by n PI P2 where Pi, P2 are primes differing by the integer m. This is called the "Standard" solution of (*) and an m for which this is the only solution is called a "-number".

Abstract: Abtract. It is shown that the equation (*) (n 1)2 u(n)+(n) = m2 is always solvable by n PI P2 where Pi, P2 are primes differing by the integer m. This is called the "Standard" solution of (*) and an m for which this is the only solution is called a "-number". While there are an infinite number of non *-numbers there are many (almost certainly infinitely many) *-numbers, including m = 2 (the twin prime case). A procedure for calculating all non *-numbers less than a given bound L is devised and a table is given for L = 1000.