**Marie-Sophie Germain**

History

Sophie Germain was born April 1, 1776-June 27, 1831

She was born in Rue Saint-Denis, Paris France

Women in the Time of Germain

Mathematics was not found to be appropriate for women in Germain's time period.

Interests

Germain was a French mathematician, physicist, and philosopher

**By: Nicki Sornson**

Her parents did not at all approve of her fascination; they would actually deny her warm clothes and a fire in her bedroom at night to try to keep her from studying.

However, when they left she would take out candles and wrap herself in quilts and begin mathematics.

When her parents found Sophie "asleep at her desk in the morning, the ink frozen and her slate covered with calculations," they realized that their daughter was serious and relented.

Eventually her mother even secretly supported her.

Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's library

She also gained education from correspondence with famous mathematicians such as:

Lagrange

Legendre

Gauss

&

Correspondence with Legendre:

Germain first became interested in number theory in 1978 when Adrien Legendre published

Essai sur la théorie des nombres.

After studying the work, she opened correspondence with him on number theory, and later, elasticity.

Legendre eventually showed some of Germain's work in the Supplément to his second edition of the

Théorie des Nombres

, where he calls it trés ingénieuse ("very ingenious")

Interests Continued

Germain's father was a wealthy silk merchant and goldsmith.

In 1789, he was appointed chief representative of the Etats-Généraux bourgeoisie. (in layman's terms-political figure!)

Through this, Sophie witnessed many discussions between her father and his friends on politics and philosophy.

Also through his status, the family remained well-off enough to support Germain throughout her endeavors as an adult.

GEOMETRY

Germain was very intrigued by Archimedes and the story of his death.

She decided that if geometry, which at that time referred to all of pure mathematics, could hold such fascination for Archimedes, it was a subject worthy of study.

Contributions to Mathematics

One of the pioneers of

Elasticity Theory

Won the grand prize from the Paris Academy of Sciences for her essay on the subject.

Her work on

Fermat's Last Theorem

provided a foundation for mathematicians

exploring the subject for hundreds of years after

ALSO...

Elasticity Theory

She took interest in a contest sponsored by the Paris Academy of Sciences concerning Ernst Chladni's experiments with vibrating metal plates.

The object of the competition was "

to give the mathematical theory of the vibration of an elastic surface and to compare the theory to experimental evidence

."

Even though her experiments "

presented ingenious results

", the first time she submitted her work, she did not win the prize.

She won an honorable mention the second time she entered, and finally at the third entry, she was awarded a prize.

She was the

first woman

to win a prize from the Paris Academy of Sciences!

Later Work in Number Theory

Germain's best work was in number theory

Her most significant contribution to number theory dealt with

Fermat's Last Theorem

.

In 1815, after the elasticity contest, the Academy offered a prize for a proof of Fermat's Last Theorem.

It reawakened Sophie's interest in number theory, and she realized the whole time she was studying elasticity,

it was at the back of her mind

.

"Sophie Germain's Theorem"

Germain proposed the following:

Her goal was to prove that for each odd prime exponent p, there are an infinite number of auxiliary primes of the form 2Np+1 such that the set of non-zero p-th power residues xp mod (2Np+1) does not contain any consecutive integers.

If there were a solution to xp + yp = zp, then Germain observes that any such auxiliary prime would have to necessarily divide one of the numbers x, y, or z. Her letter and manuscripts found in various libraries showed her analysis for the primes p less than 100 and for auxiliary primes with N from 1 to 10.

FOR EXAMPLE...

Suppose we take the case p = 5. Let's also consider N = 1. Then we need to look at the non-zero 5th power residues mod 11.

These are:

{15, 25, 35, 45, 55, 65, 75, 85, 95, 105} mod 11

= {1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000} mod 11

= {1, 10, 1, 1, 1, 10, 10, 10, 1, 10} mod 11

= {1, 10}

Those numbers are not consecutive, so 11 is an auxiliary prime that works for p = 5.

For Example:

Let p be an odd prime. If there exists an auxiliary prime P = 2Np + 1 such that:

if xp + yp + zp = 0, then P divides xyz, and

p is not a pth power residue.

Then the first case of

Fermat's Last Theorem

holds true for p.

Another Example:

Germain used this result to prove the first case of Fermat's Last Theorem for all odd primes p<100. L.E. Dickson later used Germain's theorem to prove Fermat's Last Theorem for odd primes less than 1700.

IN CONCLUSION

Did Sophie Germain ever overcome resistance to participating in mathematics?

Well, because of prejudice against her gender, she was unable to officially make a career out of mathematics. However, she worked independently throughout her life.

In recognition of her contribution towards advancement of mathematics, an honorary degree was also conferred upon her by University of Göttingen six years after her death.

At the Centenary of her life, a street and a girl's school were named after her, and the Academy of Sciences established

The Grand Prix Sophie Germain

in her honor.

Later...

And...

SOURCES:

• Del Centina, Andrea. “Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem.” Archive for History of Exact Sciences62.4 (2008): 349-392. Web. Sept. 2009.

• Gray, Mary (1978). "Sophie Germain (1776-1831)". In Louise S. Grinstein and Paul Campbell. Women of Mathematics: A Bibliographic Sourcebook. Greenwood. pp. 47–55. ISBN 978-0-313-24849-8.

• Osen, Lynn. Women in Mathematics. Cambridge: MIT Press, 1984. Print.

• Riddle, Larry. "Sophie Germain and Fermat's Last Theorem." Sophie Germain and FLT. Agnes Scott College, 21 July 2009. Web. 03 Dec. 2013.