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Old 2007-01-31, 19:22   #19
alpertron
 
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Aug 2002
Buenos Aires, Argentina

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p*2p+1 cannot be prime if p>3 is a prime twin.

First suppose p=1 (mod 3) and let p=2q+1 (because p is odd). Operating modulo 3 we find:

p*2p+1 = 1*22q+1+1 = 22q*2+1 = (22)q*2+1 = 1q*2+1 = 2+1 = 0 (mod 3)

The other possibility is p=2 (mod 3) and p+2 also a prime. Operating modulo p+2 we find:

p*2p+1 = (-2)*2p+1 = -2p+1+1 = -1+1 = 0 (mod p+2)

So in both cases p*2p+1 is composite.
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