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Time for some Prime (Posted on 2013-05-05) Difficulty: 3 of 5
Prove that if m=5^n+3^n+1 is a prime, then 12 divides n.

No Solution Yet Submitted by Danish Ahmed Khan    
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The expression is divisible by 3 if n is odd, and divisible by 5 if n is of the form 4n+2. The net effect is that n must be a multiple of 4. But even then, the expression is divisible by 7, unless the said multiple is also divisible by 3.  Hence, if the expression is prime, then 12 divides n.

n=0, n=12, n=36, n=48, are prime solutions.

 n=24 is 1034576133120646607.

 n=60 is 31448533847314239680874654589105287080263229

  Posted by broll on 2013-05-06 01:29:17
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