Named after french mathematician sophie germain, who used them in her investigations of fermats last theorem. Germain s idea was based on the use of a particular typology of prime numbers that were subsequently called sophie germain s prime numbers. These primes were named after her when french woman mathematician sophie germain around 1825 proved that fermat s last theorem holds true for such primes, this is the first general result toward a proof of fermat s last theorem. Whole and colorful lives were devoted, and even sacrificed, to finding a proof. Apart from mathematics, sophie germain also studied philosophy and psychology. Aug 30, 2005 sophie s proof is the proof discovered by sophie germain that later led to the proof of fermat s last theorem for n 5. Fermats enigma by simon singh one of the best books ive read in a while. Before i picked up this book i had no idea about fermats last theorem or its significance. By the early 1800s, all of fermat s problems were solved except for flt thus justifying the name, fermat s last theorem. Although germain had at last been awarded the prix extraordinaire, the academy was still not fully satisfied. In the fictional work the last theorem by arthur c. Germains theorem is a major step toward proving fermats last theorem for the case where n equals 5 delmedico 119.
Because of prejudice against her gender, she was unable to make a career out of mathematics, but she worked independently throughout her life. This letter contained the first substantial progress toward a proof in 200 years. We then give a thorough exposition of kummers proof that no such solutions exist in the case that p does not divide the class number of qe2. Sophie germains theorem can be applied for many prime exponents. Her work on fermat s last theorem provided a foundation for mathematicians exploring the subject for hundreds of years after. Germain and her fearless attempt to prove fermats last theorem. Her goal was to prove that for each odd prime exponent p, there are an infinite number.
The first few sophie germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 1, and 1. Her plan was unsuccessful, but she did prove several interesting results, including that the first case of fermat s last theorem that there were no solutions with p. Such x,y and z cannot therefore be counterexamples to fermats last theorem for exponent p. Sophie germains early contribution to the elasticity theory 1 v. Abel claimed to have proven four partial results of ermatsf last theorem, but mentioned nothing about how he had. In the two centuries following its conjecture 16371839, fermat s last theorem was proven for three odd prime exponents p 3, 5 and 7.
The leading thought throughout the derivation is illustrated in fig. Sophie germains work on fermats last theorem was both innovative and in uential the case for n5 was immediately available due to germains proof. Rather than demonstrating that no solution exists, sophie shows for certain values of n, n must divide x,y, or z. Fermats last theorem became the holy grail of mathematics. Fermat s enigma contains some problems that offer a taste of the math, but it also includes limericks to give a feeling for the goofy side of mathematicians. In this unique biography, dora musielak has done the impossible. She was exposed to mathematics reading books in her fathers library.
A brief chronology of the progress made toward proving fermats last theorem prior to wiles work is listed below below. Sophie germain proved case 1 of fermats last theorem for all n less than 100 and. Germain and her fearless attempt to prove fermats last. Jul 21, 2009 sophie germain and fermat s last theorem larry riddle department of mathematics agnes scott college last revised. Sophie germains grand plan to prove fermats last theorem. Wiless proof of fermat s last theorem is a proof by british mathematician andrew wiles of a special case of the modularity theorem for elliptic curves.
Fermats last theorem for amateurs pp 109142 cite as. Sep 08, 1997 in a way this book fermats last theorem is a fantasy come true. She outlined a strategy for a general proof of fermats last theorem. Sophie germains mathematical works fermats last theorem. One of her most influential works in the field was the formulation of sophie germain s theorem, which partially solved a mystery that had puzzled all of mathematics, namely fermats last theorem, for a specific group of prime numbers see note 1.
In this thesis, we study what can be said about ermatsf last theorem using only elementary methods. Germain used this result to prove the first case of fermats last theorem for all odd primes p fermats last theorem reinhard laubenbachera, david pengelleyb, a virginia bioinformatics institute, virginia polytechnic institute and state university, blacksburg, va 24061, usa b mathematical sciences, new mexico state university, las cruces, nm 88003, usa available online 9 march 2010. When one supercube made up of unit cubes is subtracted from a. To be able to read about complex mathematics in a story book style is something that was possible only in this book by simon singh.
A new circulant condition for the first case, for primes of the form 6m1. Simon singhs book is a lively, comprehensible explanation of wiless work and of the star, trauma, and wackostudded history of fermat s last theorem. Sophie germain and fermats last theorem agnes scott college. The work of sophie germain and niels henrik abel on fermats last. A study of sophie germains extensive manuscripts on fermats last theorem calls for a reassessment of her work in number theory. Her contributions to number theory, solutions to special cases of the fermat theorem, are still remembered and referenced, along with a rieh harvest of. For these prime numbers germain succeeded in demonstrating that solutions of fermat s theorem probably did not exist. We then give a thorough exposition of kummers proof that no such solutions. We will study the work of sophie germain and niels henrik abel, both who worked with ermatsf last theorem in the early 1800s. Fermats last theorem in this respect is a good case study, because the work on the theorem started out as little more than the typical gameplaying, and it gradually grew beyond that to connect up with the great river of mathematics, right at its heart.
Simon singhs book is a lively, comprehensible explanation of wiless work and of the star, trauma, and wackostudded history of fermats last theorem. Sophie germains early contribution to the elasticity theory. Two centuries ago, sophie germain began to work on her grand plan to prove the theorem of fermat, the famous conjecture that is impossible for integral values of, and, when. The new book of prime number records, springerverlag, new york, 1995.
Germain s work on fermats last theorem was to be her greatest contribution to mathematics but initially she was not credited for her. This was the first significant breakthrough in the study of the theorem. Wiles 1, 2 announced a proof of fermats last theorem, which is stated as follows. Therefore, fermats last theorem can be proven for all n, if it can be proven for n 4 and for all odd primes p the only even prime number is the number 2. Truly, it is an amazing book, unique and wellwritten, based on research that involved original historical letters and papers cited in numerous footnotes. Fermats last theorem flt, 1637, states that if n is an integer greater than 2, then it is impossible to find three natural numbers x, y and z where such equality is met being x,y0 in xn. It seeks to maintain a simple approach but the proofs being correct, in some cases however it was.
The cases n 1 and n 2 have been known since antiquity to have an infinite number of solutions. Vandiver, the lehmers, computers, and number theory leo corry tel aviv university this article describes the work of harry schultz vandiver, derrick henry lehmer, and emma lehmer on calculations related with proofs of fermats last. Germain had shown numbers theorists how to destroy an entire section of prime cases and now it was up to the combined efforts of her colleagues to continue proving fermats last theorem one case at a time. In the two centuries following its conjecture 16371839, fermats last theorem was proven for three odd prime exponents p 3, 5 and 7. Germain, then 18 years old, first somehow obtained the lesson books of various. Fermats last theorem sometimes abbreviated as flt and also called fermats great theorem is one of the most famous theorems in the history of mathematics. The main article for this category is number theory wikimedia commons has media related to number theory subcategories. Her manuscripts reveal a comprehensive program to prove fermats last theorem in its entirety. Sophie germain 1 was the first woman known for important original research in mathematics.
The proof of the fermats last theorem will be derived utilizing such a geometrical representation of integer numbers raised to an integer power. Maths hidden woman following is the true story of sophie germain, an 18thcentury woman who assumed a mans identity in order to pursue her passionattempting to prove fermat s last theorem. Fermats enigma contains some problems that offer a taste of the math, but it also includes limericks to give a feeling for the goofy side of mathematicians. Germain and her fearless attempt to prove fermats last theorem dora musielak university of texas at arlington abstract. Therefore, fermats last theorem could be proved for all n if it could be proved for n 4 and for all odd primes p.
Aug 30, 2005 this theorem is also the foundation of sophie s proof see here for the details, an important result from the mathematician sophie germain. There is much in these manuscripts beyond the single theorem for case 1 for which she is known from a published footnote by legendre. This book should be read like a detective story that spans centuries, tiny nuggets of information collected all the way till the 20 th century when the last greatest theorem ever posed was solved. Although not actually a theorem at the time meaning a mathematical statement for which proof exists, the margin note became known over time as fermats last theorem, as it was the last of fermats asserted theorems to remain unproved. Pdf algebraic number theory and fermat s last theorem. Germains idea was based on the use of a particular typology of prime numbers that were subsequently called sophie germains prime numbers. Submitted on 7 apr 2019 v1, last revised 26 jun 2019 this version, v3. In the early 1820s sophie germain dramatically changed this picture by. This book will discuss one of the most famous theorems of mathematics.
Nigel boston university of wisconsin madison the proof. Leonhard euler, the greatest mathematician of the eighteenth century, had to admit defeat. Sophie germain and special cases of fermats last theorem. Fermat was a french lawyer and mathematician who is remembered for things like discovering an original method of finding the greatest and the smallest ordinates of curved lines, and researching number theory, analytic geometry, probability and optics. For these prime numbers germain succeeded in demonstrating that solutions of fermats theorem probably did not exist. Nigel boston university of wisconsin madison the proof of. It will talk about that which is commonly called fermats last theorem, the subject will be confronted from a principally historic point of view, the concepts and the theorems behind the proof being too complex even for the greater part of professional mathematicians. Sophie germain to the proof of fermats last theorem flt. Sophie germain took on the identity of a man to do research in a field forbidden to females, and made the most significant breakthrough of the nineteenth century. The proof is very important historically in that it represents a new approach to the problem. Fermats last theoremsophie germain wikibooks, open.
According to many, germains theories did not have solid proofs and she never got a response from gauss regarding this subject. Guardian to read it fermats last theorem is to realise that there is a world of beauty and intellectual challenge that is denied to 99. Getting involved in fermats mischievous conundrum set me on the path towards being an author and ignited an interest in mathematics that has continued ever since. Sophie germain biography childhood, life achievements. As has been mentioned, one of sophie germains most prominent achievements was her work with fermats last theorem. Fermats last theorem flt, the focus of all this commotion, is easily stated, saying that. Therefore, fermat s last theorem can be proven for all n, if it can be proven for n 4 and for all odd primes p the only even prime number is the number 2. This book will describe the recent proof of fermats last the orem by andrew wiles, aided by richard taylor, for graduate students and faculty with a reasonably broad background in algebra. Case 2 includes all p that divide at least one of x, y, or z. If an integer a is not divisible by a prime p, then a p1. In number theory, fermats last theorem sometimes called fermats. Fermats last theorem dominated my own life for four years, because i made a tv documentary, wrote a book and then lectured on the subject.
In this chapter we give the beautiful theorem of sophie germain for the first case of fermats theorem. Fermats last theorem project gutenberg selfpublishing. Sophie germain revolutionary mathematician 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. The exposition follows the historical development of the problem, beginning with the work of fermat and ending with kummers theory of ideal factorization, by means of which the theorem is proved for all prime exponents less than 37. Jan 23, 2015 prime mystery, the life and mathematics of sophie germain i just finished reading this book and i love it. Fermat s last theorem can be divided into two cases. Both abel and germain worked on fermats last theorem in the early. Therefore, fermat s last theorem could be proved for all n if it could be proved for n 4 and for all odd primes p.
This book is an introduction to algebraic number theory via the famous problem of fermats last theorem. Case 1 involves all powers p that do not divide any of x, y, or z. Around 1637, fermat wrote in the margin of a book that the more general. The work of sophie germain and niels henrik abel on fermat. Together with ribets theorem, it provides a proof for fermat s last theorem. Sophie germain and special cases of fermats last theorem colleen alkalayhoulihan 1 sophie germain 1. Fermats last theoremsophie germain wikibooks, open books. Marie sophie germain 1776 1831 from wikipedia, the free encyclopedia. A new musical about sophie germains life, entitled the limit, premiered at vault festival in london, 2019. After reading carl friedrich gauss book disquisitiones arithmeticae, sophie germain wrote to the author in 1804 regarding her own ideas in relation to fermats last theorem. She is the only woman who contributed to the proof of fermats last theorem. The work of sophie germain and niels henrik abel on fermats. Around 1825 sophie germain proved that the first case of fermat s last theorem is true for such primes. Her work on fermat s last theorem provided a foundation for mathematicians exploring the.
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