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

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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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 …

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

French Revolution , Ignited by Fall of La Bastille July 1789

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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.”

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 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

Pierre de Fermat 10

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.

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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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

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.

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.

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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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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Princess of Mathematics 20

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.

Mathematicians 18-19th Century 21

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

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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Sophie Germain Last Home In Paris 23

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

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.

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