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An exponent and prime problem (Posted on 2007-01-06) Difficulty: 2 of 5
Given that n is a positive prime number. Determine analytically all possible n such that 2n + n2 is a prime number.

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (2 votes)

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unique solution? | Comment 5 of 6 |
Since n=2 can be ruled out and n=3 --> 2^3 + 3^2 = 17 works, all other n values equal to 0, 2, 3, and 4 mod 6 can be ruled out since n is prime. Also since 2^(6k+1) + (6k+1)^2 and 2^(6k+5) + (6k+5)^2 both equal 0 mod3 (using Fermat's Thm.), any n value equal to 1 or 5 mod6 forces 2^n + n^2 to be divisible by 3 and so not prime. 
  Posted by Dennis on 2007-01-06 15:58:03
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