twin composite numbers
this comes from my friend Marty: Three reminders: (1) x^3 - 1 = (x - 1)(x^2 + x + 1) (2) x^3 + 1 = (x + 1)(x^2 - x + 1) (3) A number, other than one, which is not prime is called 'composite.' --------------- Interesting fact: There exist infinitely many positive integers, n, such that 6n - 1 and 6n + 1 are composite. --------------- Interesting proof: For any positive integer, k, let n = 6^(3k-1). Then 6n = 6^(3k) = (6^k)^3. Then 6n - 1 = (6^k)^3 - 1, and 6n + 1 = (6^k)^3 + 1, both of which can be factored using equations (1) and (2) above.