Twin Prime Sandwich

Building up on the lesson from a prior post, "Prime Numbers - a property rediscovered," it appears that all numbers between every pair of twin primes are multiples of 6. 

This isn't too surprising given that all primes are in the form 6n+/-1, but that fact alone doesn't guarantee that the number between a pair of twin primes is of the form 6n.

Here are the steps of the proof.

1. Let a, b, c, d, e represent 5 consecutive integers for which b and d are twin primes.

2. Consider the prime, b. Given the 6n+/-1 theorem, then either a is a multiple of 6 or c is a multiple of 6.

If a is a multiple of 6, then d cannot be prime because d=a+3 making d a multiple of 3. Therefore c must be a multiple of 6.

3. Likewise, consider d. Either d is prime because c is a multiple of 6 or e is a multiple of 6. However, if e is a multiple of 6 then b=e-3 so b would be a multiple of 3 and not prime. Therefore c must be a multiple of 6 if b and d are twin primes.