Primes differing by a fixed integer

Source abstract

It is shown that the equation ( ∗ ) ( n − 1 ) 2 − σ ( n ) ϕ ( n ) = m 2 ( \ast )\;{(n - 1)^2} - \sigma (n)\phi (n) = {m^2} is always solvable by n = p 1 p 2 n = {p_1}{p_2} where p 1 , p 2 {p_1},{p_2} are primes differing by the integer m . This is called the "Standard" solution of ( ∗ ) ( \ast ) and an m for which this is the only solution is called a " ∗ ^\ast -number". While there are an infinite number of non ∗ ^\ast -numbers there are many (almost certainly infinitely many) ∗ ^\ast -numbers, including m = 2 m = 2 (the twin prime case). A procedure for calculating all non ∗ ^\ast -numbers less than a given bound L is devised and a table is given for L = 1000 L = 1000 .