Let P = {p_{1}, ... , p_{n}} be the set of the first n prime numbers. Let S be an arbitrary (possibly empty) subset of P. Let A be the product of the elements of S, and B the product of the elements of S', the complement of S. (An empty product is assigned the value of 1.)

Prove that each of A + B and |A − B| is prime, provided that it is less than p_{n+1}^{2} and greater than 1.

Firstly, by the Fundamental Theorem of Arithmetic, each of p_{1}, ... , p_{n} divides one of A, B, but not the other.

Hence none of p_{1}, ... , p_{n} divides A + B or |A − B|.

Therefore, if |A − B| > 1 (we always have A + B > 1), any prime divisor of A + B or |A − B| must be greater than or equal to p_{n+1}.

Further, there can only be one such prime divisor, as we have stipulated that A + B and |A − B| are less than p_{n+1}^{2}.

Therefore, each of A + B and |A − B| is prime, provided that it is less than p_{n+1}^{2} and greater than 1.

Source: More Mathematical Morsels (Morsel 20), by Ross Honsberger