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 (p+5)/8 and (p+1)/4 = Prime Numbers (Posted on 2010-04-12)
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.

 See The Solution Submitted by K Sengupta Rating: 3.0000 (1 votes)

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 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 2010-04-13 07:49:46

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