# prime number conjecture

• Sep 2nd 2008, 02:26 PM
rmpatel5
prime number conjecture
disprove the conjecture: There are infinitely many prime numbers expressible in the form n^3 +1 where n is a positive integer
• Sep 2nd 2008, 02:49 PM
ThePerfectHacker
Quote:

Originally Posted by rmpatel5
disprove the conjecture: There are infinitely many prime numbers expressible in the form n^3 +1 where n is a positive integer

Hint: $a^3+b^3 = (a+b)(a^2 - ab + b^2)$
• Sep 2nd 2008, 02:51 PM
rmpatel5
I have that part: n^3 +1= (n+1)(n^2-n+1). Just dont know where to go from. I know that 2 is the only integer that will work but i just dont know how to prove it.
• Sep 2nd 2008, 02:54 PM
ThePerfectHacker
Quote:

Originally Posted by rmpatel5
I have that part: n^3 +1= (n+1)(n^2-n+1). Just dont know where to go from. I know that 2 is the only integer that will work but i just dont know how to prove it.

Well because you can factor $n^3+1$ as a product of two integers.
So how can it be prime?
• Sep 2nd 2008, 04:53 PM
rmpatel5
so anything that factors can not be prime because that makes it a composite?
• Sep 2nd 2008, 07:23 PM
ThePerfectHacker
Quote:

Originally Posted by rmpatel5
so anything that factors can not be prime because that makes it a composite?

Yes if $a = bc$ and neither $b,c$ are $1$ then $a$ cannot be prime.