Sophie Germain Identity

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Introduction to the Sophie Germain Identiy and its applications in contests.

Author: Jaden Park

Formula

The Sophie Germain Identity basically says that

a^{4} + 4b^{4} = (a^{2} + 2b^{2} + 2ab)(a^{2} + 2b^{2} - 2ab)

Here are some practice problems.

(2020 AMC 10B Q22)

What is the remainder when $2^{202} + 202$ is divided by $2^{101} + 2^{51} + 1$ ?

For any integer $n$ greater than $1, 4^{n} + n^{4}$ is never a prime number.

Find the largest prime divisor of $5^{4} + 4 \cdot 6^{4}.$

(1969 IMO Q1)

Prove that there exist infinite natural numbers $m$ fulfilling the following property:

For all natural numbers $n, n^{4} + m$ is not a prime number.

Not that common and in my opinion not worth memorizing. Still a cool identity though.

Factor Theorem

Newton Sums