#### Date of Award

Fall 2018

#### Document Type

Thesis

#### Degree Name

Master of Science

#### Department

Mathematics

#### First Advisor

Dr. Andrius Tamulis

#### Second Advisor

Dr. Dianna Galante

#### Third Advisor

Dr. J. Christopher Tweddle

#### Abstract

Fermat's Last Theorem sat unproven for more than 300 year. It all started around 1637, when Pierre de Fermat stated the theorem: *x ^{p}*

^{ }+

*y*=

^{p}*z*

^{p}^{ }has no positive integer solutions for

*x*,

*y*,

*z*when

*p*> 2. He wrote a note in the margin saying that he has the proof but it was bigger than the margin. Sophie Germain, a French mathematician tried her hand in proving Fermat's Last Theorem. She came up with a theorem that was later reference to as Germain Theorem. She took Fermat's Last Theorem

*x*

^{p}^{ }+

*y*=

^{p}*z*and suggested that if

^{p }*p*is a prime number greater than 2 and 2

*p*+ 1 is prime also, then

*p*must divide

*x*,

*y*, or

*z*. Germain's theory and proof changed the approach to proving Fermat's Last Theorem and divided it into two cases where the first case states that none of the three values

*x*,

*y*, or

*z*is divisible by

*p*, and the second case states that the exponent

*p*divides at least one of the three values

*x*,

*y*, or

*z*. The main concept of this paper is to explore Germaine's approach to solving Fermat's Last Theorem for the exponent

*p*> 2 and how her idea led to solving the problem by introducing a new and fresh approach.

#### Recommended Citation

Yosef, Amal Yaqoub, "The Role of Sophie Germain in Solving Fermat's Last Theorem" (2018). *Mathematics Theses*. 1.

https://opus.govst.edu/theses_math/1