Sophie Germain made significant contributions to the proof of Fermat's Last Theorem over 50 years working on the problem independently. She developed her own theorem aimed at proving certain cases of FLT for exponents less than 100. While initially credited by Legendre, modern research has shown the depth of her independent work and algorithm for applying her method. Her work formed a central part of later complete proofs and she made other contributions to mathematics, winning a grand prize from the French Academy of Sciences.

1. Sophie Germain and her Contribution
to Fermat’s Last Theorem
Dora E. Musiełak, Ph.D.
University of Texas at Arlington
In celebration of Sophie Germain Day

2.  For over three centuries mathematicians sought to prove
Fermat’s Last Theorem, a mysterious assertion by
number theorist Pierre de Fermat.
 Between 1630 and 1990, Euler, Dirichlet, Kummer, and
countless other mathematicians strived to find the
elusive proof.
 Fifty years after Euler’s partial proof, Sophie Germain
took the next step, conceiving a novel method aimed to
generalize it.
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3. Who was Sophie Germain?
How did she approach the proof
of Fermat’s Last Theorem?
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Let us start from the beginning …

4. 4
Sophie Germain was born on 1 April 1776 in Paris, France

5. French Revolution , Ignited
by Fall of La Bastille
July 1789
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6. 6
Sophie Germain came of age during
Reign of Terror 1793-1794

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Sophie Germain obtained lecture notes from
École Polytechnique.
In 1797, she submitted analysis to Lagrange,
using pseudonym “M. LeBlanc.”

8. Sophie Germain
 In 1804, Sophie Germain wrote to Gauss concerning
his Disquisitiones Arithmeticae (1801).
 She enclosed her own proofs, signing her letters “M.
LeBlanc.”
 Gauss praised her mathematical work.
 Gauss discovered Sophie’s true identity after French
occupation of his hometown in Germany (1806).
 At 28, Sophie Germain made first attempt to prove
Fermat’s Last Theorem. She called it a “beautiful
theorem” and said that it could be generalized.
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9. 9
17-18th Century Mathematics
Fermat
1601-1665
Newton
1643-1727
Leibniz
1646-1716
Jacob Bernoulli
1654-1705
L’Hopital
1661-1704
Johann Bernoulli
1667-1748
Moivre
1667-1754
Maclaurin
1698-1746

10. Pierre de Fermat
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 French lawyer by profession,
mathematician by his contributions.
 Fermat discovered analytic geometry
independently of Descartes.
 He founded theory of probability
with Pascal and discovered Least
Time Principle, a concept in calculus
of variations.
 Inspired by Diophantus, Fermat’s
work in number theory launched one
of history’s more challenging quests
for a mathematical proof.

11. Diophantus of Alexandria (200-284 AD)
 Diophantus wrote Arithmetica, a book on solution of
algebraic equations and number theory.
 Arithmetica, a collection of 130 problems, gives
numerical solutions of determinate equations (those
with a unique solution), and indeterminate equations.
 Diophantus studied three types of quadratics
ax2 + bx = c; ax2 = bx + c; ax2 + c = bx
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12. Problem 8 in Diophantus’
Arithmetica, Book II,
inspired Fermat’s Last
Theorem:
To divide a given square into
two squares,
z2 = x2 + y2
Note: this is a later edition.
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“To divide a cube into two other
cubes, or a fourth power, or in
general any power whatever into
two powers of the same
denomination above the second
is impossible.” Pierre de Fermat

13. Fermat’s Last Theorem (FLT)
 Fermat’s marginal note asserts that
zn = xn + yn
has no non-zero integer solutions for x, y and z when n > 2.
Indeed, no three integers x, y and z exist, such that xyz ≠ 0,
which satisfy equality above when exponent is greater than 2.
 Fermat added:
“I have discovered a truly remarkable proof which this
margin is too small to contain.”
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 In 1825, Legendre included Germain’s results when
he published partial proof of FLT for n = 5 in his
Théorie des nombres.
 In a footnote, Legendre credited Germain with first
general result toward FLT’s proof. This is known
today as Sophie Germain Theorem.
 Germain’s theorem aimed to prove Case I of
Fermat’s Last Theorem for all prime exponents less
than 100.
 Sophie Germain described her proofs in
correspondence with Legendre and Gauss.

15. Sophie Germain Theorem
 If n and 2n+1 are primes, then xn + yn = zn implies
that one of x, y, z is divisible by n.
Fermat’s Last Theorem splits into two cases:
Case 1: None of x, y, z is divisible by n.
Case 2: One and only one of x, y, z is divisible by n.
 Sophie Germain proved Case 1 of Fermat’s Last
Theorem for all n less than 100.
 Legendre extended her method to n < 197.
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Sophie Germain Primes
A prime number is a Sophie Germain prime if
when it’s doubled and we add 1 to that we get another
prime. In other words, p is a Sophie Germain prime if
2p + 1 is also prime.
Set of first Germain primes: 2, 3, 5, 11, 23, 29, 41, 53,
83, 89, 113, 131, 173, 179, 191, …
It includes 2, only even prime known to date.
How many Sophie Germain are there?
What is the largest found to date?

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 Until recently, mathematicians assumed that Sophie
Germain had a minor role collaborating with Legendre.
 In 1998, reevaluation of Germain’s manuscripts and her
correspondence with Legendre and Gauss changed that
perspective.*
 Germain developed a general version of her theorem
independently.
 She carried out substantial work to develop an algorithm
for applying her theorem to various exponents n of FLT.
* Laubenbacher, R. and Pengelley, D., 2010.“Voici ce que j’ai trouvé: Sophie Germain’s
grand plan to prove Fermat’s Last Theorem,” Historia Mathematica, Vol. 37.

18. Sophie Germain: Winner of
Mathematics Prize
 1811 – Competed for prize of mathematics awarded
by French Academy of Sciences. She derived a
mathematical theory to explain vibration patterns
on plates, as demonstrated by Ernest Chladni.
 1813 – Received honorable mention for her second
memoir that improved theory.
 1816 – Sophie Germain won a grand prize of
mathematics for her theory of vibrations of curved
and plane elastic surfaces. She was 40 years old.
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19. Last Years
Sophie Germain composed a philosophical essay
“Considérations générale sur l'état des sciences et
des lettres,” published posthumously in Oeuvres
Philosophique de Sophie Germain (1879)
She was stricken with breast cancer in 1829.
Undeterred by her illness and 1830 revolution,
Sophie published papers on number theory and
on curvature of surfaces.
Sophie Germain died on 27 June 1831. She was
55.
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20. Princess of Mathematics
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 Sophie Germain played major
role in development of acoustics
and elasticity theories, and in
number theory.
 Recognized contribution is partial
proof of Fermat’s Last Theorem
for case in which x, y, z are not
divisible by an odd prime, p.
 Sophie Germain Theorem and
Sophie Germain Primes.
 Generalizations of Germain’s
approach remain central to other
advances in proving Case I of
Fermat’s Last Theorem.

21. Mathematicians 18-19th Century
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Euler
1707-1783
Laplace
1749-1827
Germain
1776-1831
Gauss
1777-1855
Cauchy
1789-1857
D. Bernoulli
1700-1782
Fourier
1768-1830
Agnesi
1718-1799
Lagrange
1736-1813
Legendre
1752-1833

22. Andrew J. Wiles Proved FLT in 1994
 Andrew Wiles announced his proof of Fermat’s Last Theorem in
1993, but Richard Taylor showed proof was incomplete.
 In collaboration with Taylor, Wiles submitted a complete irrefutable
proof of FLT.
 Papers published in 1994 issue of Annals of Mathematics.
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23. Sophie Germain Last Home In Paris
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SOPHIE GERMAIN
PHILOSOPHE
ET MATHEMATICIENNE
NEE A PARIS EN 1776
EST MORTE DANS CETTE MAISON
LE 27 JUIN 1831
SOPHIE GERMAIN
PHILOSOPHER
AND MATHEMATICIAN
BORN IN PARIS IN 1776
DIED IN THIS HOUSE
ON 27 JUNE 1831
13 Rue de Savoi, across from the Seine River,
a few blocks from Rue St Denis where she
was born

24. Prime Mystery: The Life and Mathematics
of Sophie Germain
 This book paints a rich portrait of the
brilliant and complex woman, including
the mathematics she developed, her
associations with Gauss, Legendre, and
other leading researchers, and the
tumultuous times in which she lived.
 In Prime Mystery: The Life and
Mathematics of Sophie Germain,
author Dora Musielak has done the
impossible —she has chronicled
Germain’s brilliance through her life
and work in mathematics, in a way that
is simultaneously informative,
comprehensive, and accurate.
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Paperback: 294 pages
Publisher: AuthorHouse (January 23, 2015)
Language: English
ISBN-10: 1496965027
ISBN-13: 978-1496965028
Find it at AuthorHouse Books, Amazon, Barnes& Noble,
and other booksellers.

25. Sophie’s Diary
a mathematical novel
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A tale of a girl who discovers mathematics as
French Revolution raged in Paris.
Sophie finds refuge in her father’s library,
pursuing her studies with a mixture of
wonder, stubbornness, and resourcefulness.
Sophie’s Diary includes mathematics, what
the teen-ager taught herself, intermingled
with historically accurate accounts of history
of science and events that took place
between 1789 and 1793.
www.maa.org/publications/books/sophies-diary
MAA Book Catalog Code: SGD
Print ISBN: 978-0-88385-577-5
Electronic ISBN: 978-1-61444-510-4
291 pp., Hardbound, 2012
List Price: $42.50
Member Price: $34.00
Series: Spectrum