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Getting Primed With Subtraction And Addition (Posted on 2007-12-26) Difficulty: 2 of 5
Determine all possible primes p such that each of p-8, p-4, p+8 and p+12 are also primes.

See The Solution Submitted by K Sengupta    
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solution/spoiler | Comment 2 of 5 |
(In reply to computer exploration (spoiler) by Charlie)

By inspection, p=2 and p=3 aren't solutions.

All other primes can be expressed in one of 2 forms, either 6k+1 or 6k-1.

For p=6k+1, p-4=6k-3, which can only be prime if k=1.

But then p-8=6k-7= -1 which is impossible.

So p=6k-1 and p-8=6k-9 which is prime only for k=2 and p=11, whence p-4=7, p+8=19, and p+12=23.

  Posted by xdog on 2007-12-26 12:28:44
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