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(p+5)/8 and (p+1)/4 = Prime Numbers (Posted on 20100412) 

Determine all possible value(s) of a prime number p such that each of (p+5)/8 and (p+1)/4 is a prime number.
Analytical solution

 Comment 4 of 5 

The smallest of the three primes is (p + 5)/8.
If this is 2 then p = 11 and we have three primes (2, 3, 11) as required.
Otherwise, the smallest prime is odd, so let (p + 5)/8 = 2n + 1 (n = 1, 2, ..), which gives p = 16n + 3 so that the three potential primes are:
q = 2n + 1, r = 4n + 1, p = 16n + 3 (n = 1, 2, ... )
when n = 1, p = 19 and we have three primes (3, 5, 19) as required.
qrp = (2n + 1)(4n + 1)(16n + 3) = (2n + 1)(n + 1)n (mod 3) = (2n  2)(n + 1)n (mod 3) = 2n(n  1)(n + 1) (mod 3)
Since n  1, n and n + 1 are successive integers, one of them must be 0 (mod 3). Thus one of the numbers p, q, r must have the factor 3.
For n > 1: p, q, r > 3, so whichever has the factor 3 must be a multiple of 3, thus ruling out the possibility of it being a prime number.
So we have only two possible values of p: 11 and 19.

Posted by Harry
on 20100413 07:49:46 


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