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Sum of two powers (Posted on 2004-08-12) Difficulty: 4 of 5
If n is an integer, show that n4 + 4n is never a prime for n>1.

  Submitted by Federico Kereki    
Rating: 4.2500 (4 votes)
Solution: (Hide)
If n is even, n^4+4^n is even, and greater than 2, so it's not a prime, so we need consider only odd numbers.

We write n^4+4^n= (n²)²+(2n)²= (n²+2n)²-n²2^(n+1)= (n²+2n+n2^(n+1)/2)(n²+2n-n2^(n+1)/2).

As n is odd, (n+1)/2 is an integer, both factors are integers. Also, it's easy to prove that for n>1, both factors are also greater than 1, so n^4+4^n is composite.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionComplete solutionDanish Ahmed Khan2012-10-24 14:42:01
SolutionSolutionK Sengupta2007-05-16 11:13:54
re: Complete answerSilverKnight2004-08-12 16:25:19
SolutionComplete answerBon2004-08-12 15:31:37
re: a startBon2004-08-12 14:51:04
Some ThoughtsIncomplete answerBon2004-08-12 14:46:53
Some Thoughtsa startCharlie2004-08-12 13:52:16
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