This is the second Challenge Problem I’ve made up. (Nerd)

Prove that is not prime for all positive integers with

Printable View

- Jun 4th 2009, 08:51 PMTheAbstractionistTA’s Challenge Problem #2
This is the second Challenge Problem I’ve made up. (Nerd)

Prove that is not prime for all positive integers with - Jun 4th 2009, 09:00 PMNonCommAlg
- Jun 4th 2009, 09:04 PMIsomorphism
NCA's identity has a name, its called

*Sophie Germain's Identity.* - Jun 4th 2009, 09:09 PMTheAbstractionist
Wow, that was really quick! (Rofl)

- Jun 4th 2009, 11:04 PMCaptainBlack
For real and and integer :

So the restriction to positive integers for and is unnecessary.

Also if we have which can never be prime for any integer (including ).

So the problem can be rewordrd to for all positive integers and integers show that

is never prime.

(we may have to be explicit about what we want to mean here or exclude the case where one or both of and are zero and is zero)

CB - Jun 4th 2009, 11:38 PMNonCommAlg
- Jun 5th 2009, 12:13 PMTheAbstractionist
That’s a very good point. However, the condition (or if you like) is absolutely vital, otherwise you could find a counterexample as

**NonCommAlg**pointed out.

The way I originally did it was to factorize as instead of writing the factors the way**NonCommAlg**did – and then it isn’t so clear that both factors are actually greater than 1. (One of them certainly is, but the other is iffy.) I then proceeded by considering the expressions as quadratics in Their determinant is and the vital condition ensures that the determinant is negative and therefore that the expressions are positive. - Jun 9th 2009, 02:35 AMCaptainBlack