I decided to do some research on Sophie Germain, seeing as how in education classes, they emphasize getting students interested and involved. I thought it was nice to see a woman mathematician for once. Most of the time we deal with Euler, Euclid, Pythagoras, or another male mathematician. Upon doing my research on Sophie Germain and the Germain primes, I found a connection to today's lesson on Fermat's Last Theorem and found it relevant!

"Marie-Sophie Germain (April 1, 1776 – June 27, 1831) was a French mathematician who made important contributions to the fields of differential geometry and number theory, and to the study of Fermat's Last Theorem…."

"One of Germain's major contributions to number theory was the following theorem: if x, y, and z are integers, and $x^{5} + y^{5} = z^{5}$ then either x, y, or z has to be divisible by five. **This proof,** which she first described in a letter to Gauss, **became quite significant as it restricted the possible solutions of Fermat's Last Theorem**. One significant contribution is the concept of the Sophie Germain prime, which is a prime number p where 2p+1 is also prime. One of her most famous identities, commonly known as Sophie Germain's Identity, states that for any two numbers x and y:

$x^{4} + 4y^{4} = (x^{2} + 2y^{2} + 2xy)(x^{2} + 2y^{2} − 2xy)$.

I also found it interesting that Sophie worked with other great mathematicians like Gauss and Lagrange, so it is definitely true that collaboration is key in math!