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        riddles >> easy >> Prime triplet
        
(Message started by: NickH on Apr 15^th, 2003, 2:25pm)

Title: Prime triplet
Post by NickH on Apr 15^th, 2003, 2:25pm

Show that, if 0 < p and p² + 2 are primes, then p³ + 2 is also prime.

Title: Re: Prime triplet
Post by SWF on Apr 16^th, 2003, 7:33pm

This seems like sort of a trick question. My solution (hidden):
[hide]The only p with p²+2 also prime is p=3, and for that p³+2=29, prime, so the statement is true.

For all other primes, p, p²+2 is a multiple of 3. Obvious, for p=2, and the rest of the primes are either a multiple of 6 plus 1 or a multiple of 6 plus 5. Plug 6*n+1 or 6*n+5 into p²+2, and you will see it is a multiple of 3.[/hide]

Title: Re: Prime triplet
Post by NickH on Apr 16^th, 2003, 8:21pm

Spot on! So, in fact, p⁴ + 2 is also prime!

Title: Re: Prime triplet
Post by LZJ on Apr 17^th, 2003, 1:50am

As for proof as to why 3 is the only value of p possible, p² (mod 3) = 1 for all odd positive integers that are not multiples of 3, so all other odd primes can be ruled out.