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Primes (Posted on 2003-09-09) Difficulty: 3 of 5
If x and (x² + 8) are both primes, then prove that (x³ +16) is also a prime.

  Submitted by Ravi Raja    
Rating: 4.2500 (12 votes)
Solution: (Hide)
First, we need to realize that for any whole number n, one of n+1, n, or n-1 is divisible by three.

Also, if n is prime, then n is not divisible by three (unless n=3), so either n+1 or n-1 is divisible by three. In that case,
(n+1)(n-1)=n²-1
and
n²-1+9=n²+8
will also be divisible by three.

Therefore, the only time that n and n²+8 will both be prime is if n is equal to three. In this situation, x³+16=43 is indeed prime, so we have proven the original supposition.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSolutionK Sengupta2021-12-14 01:11:42
SolutionSolutionPraneeth2007-09-11 07:49:21
re: Solution (spoiler)iamkobe2006-05-16 14:45:46
SolutionSolution (spoiler)SilverKnight2003-09-10 01:40:51
Solutionre(3): Second thoughtSilverKnight2003-09-10 00:44:23
re(2): Second thoughtSilverKnight2003-09-10 00:19:16
re: Second thoughtSilverKnight2003-09-10 00:04:37
Some ThoughtsSecond thoughtDJ2003-09-09 22:39:20
Hints/TipsFirst thoughts...SilverKnight2003-09-09 20:53:17
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