If x and (x² + 8) are both primes, then prove that (x³ +16) is also a prime.

Primes are of form: 6k+1, 6k-1 for k≥1 (excluding 2,3)
Case(1): x=6k+1
x²+8 is prime => (6k+1)²+8 is prime => 3(12k²+4k+3) is not prime
which is contradiction

Case(2): x=6k-1
Similarly 3(12k²-4k+3) is not prime, which is contraction

If x=2, x²+8 is not prime.
If x=3,x²+8=11 is prime => x³+16=43 is also a prime. Hence Proved

Edited on September 11, 2007, 7:49 am

Posted by Praneeth on 2007-09-11 07:49:21