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).
Let the prime pairs be denoted by (a, b), with a< b, and a> 6, are:
Being primes, both a and b must be odd.
Then, the possible values for (a,b) are:
(i) (a, b) (mod 6) = (1,3)
(ii) (a, b) (mod 6) = (3,5)
(iii) (a, b) (mod 6)= (5,7) (mod 6) = (-1,1)
But in (i), b (mod 6) = 3, is always divisible by 3, for b>6 and in (ii), a (mod 6) = 3, is always divisible by 3, for a>6. This is a contradiction.
Thus, (a, b) (mod 6) = (-1,1), so that: (a+1) mod 6 = (b-1) mod 6 = 0
Hence, the proof.
Note: For a < 6, the pair a=5, b=7 also satisfy the given conditions.