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Sophie Germain Identity

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The Sophie Germain Identity, credited to Marie-Sophie Germain, states that:

$a^4 + 4b^4 = (a^2 + 2b^2 + 2ab)(a^2 + 2b^2 - 2ab)$

The proof involves completing the square and then difference of squares.

$a^4 + 4b^4 = a^4 + 4a^2b^2 + 4b^4 - 4a^2b^2$
$= (a^2 + 2b^2)^2 - 4a^2b^2$
$= (a^2 + 2b^2 - 2ab) (a^2 + 2b^2 + 2ab)$

Problems

Introductory

Is $4^{545} + 545^{4}$ a prime?
Prove that if $n>1$ , then $n^4 + 4^n$ is composite.

Intermediate

Compute $\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$ . (1987 AIME, #14)

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