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Sophie Germain Identity

From AoPSWiki

The Sophie Germain Identity states that:

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

One can prove this identity simply by multiplying out the right side and verifying that it equals the left. To derive the factoring, first completing the square and then factor as a difference of squares:

$\begin{align*}a^4 + 4b^4 & = a^4 + 4a^2b^2 + 4b^4 - 4a^2b^2 \\& = (a^2 + 2b^2)^2 - (2ab)^2 \\& = (a^2 + 2b^2 - 2a...$

Contents

1 Problems
- 1.1 Introductory
- 1.2 Intermediate
2 See Also

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)
Find the largest prime divisor of $25^2+72^2$ . (Mock AIME 5 2005-2006 Problems/Problem 5)

See Also

MacTutor biography of Sophie Germain

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/Sophie_Germain_Identity"

Category: Algebra

Looking for a challenging geometry text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Geometry by Richard Rusczyk.

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