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| (p+5)/8 and (p+1)/4 = Prime Numbers (Posted on 2010-04-12) |
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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
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 | Comment 4 of 5 | 
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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.
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Posted by Harry
on 2010-04-13 07:49:46 |
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