Pairs of primes separated by a single number are called prime pairs. Examples are 17 and 19. Prove that the number between a prime pair is always divisible by 6 (assuming both numbers in the pair are greater than 6).

(From http://www.ocf.berkeley.edu/~wwu/riddles/intro.shtml )

Any number greater than 6 can be expressed as n = 6a + b, where 0 ≤ b ≥ 5. When b = 0, 2, or 4, n is divisible by 2; when b = 0 or 3, n is divisble by 3.

A prime number, must therefore be of the forms 6a + 1 or 6a + 5, or more compactly, 6a ± 1. Therefore prime pairs must be 6a' - 1 and 6a' + 1 for some a', and the number between them is 6a'.

Posted by TomM on 2002-09-02 12:00:23