This is the second Challenge Problem I’ve made up.
Prove that is not prime for all positive integers with
NCA's identity has a name, its called Sophie Germain's Identity.
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
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.