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This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it.

To explore other such papers go to the Math G Projects Page.
 

E. B. Burger

The evolution of Fermat's Last Theorem and the story of its solution is a prime example of the challenge faced by mankind in his attempt to better comprehend the world around him. The pursuit of methods of mathematics to solve scientific dilemmas is definitely more interesting than one might initially think. The following paper will highlight the pursuit of one of the better known mathematical fascinations of the last four hundred years. The fascination takes on an added interest as it has recently be solved and heralded as a major breakthrough of the decade.

Pierre De Fermat was born August 20, 1601 in Beaumont France and educated at the University of Toulouse. He was born to an affluent family, his father was a prosperous leather merchant and his mother was from a prominent family. This prominent status made a choice of a legal career typical of the time for someone of his social and financial rank. His career progressed successfully and he eventually obtained a position in the Parliament of Toulouse in 1648. During the sixteenth and seventieth century the uniqueness of mathematics was representative of the individuals, goals and methods each pursued. In the case of Fermat, with a career in law, the hobby of mathematics allowed him the freedom from the pressures to conform which law mandated. He did not depend on mathematics for his livelihood and held no career aspirations as a mathematician as there were no positions to be gained with its exploration or discoveries. Personal gratification was the only reward to be found. Perhaps as a result of this, Fermat was reluctant to publish any his mathematical findings for fear his hobby would no longer be enjoyable but a burden to prove. Fermat did correspond his mathematical theories with his other academic friends. If he had attempted to publish the letters that he circulated, he would have needed to labor over the methodology and procedures of proof. Undoubtedly this would have turned the hobby into work. During his lifetime he only published one mathematical paper and that was an anonymous article.

Upon his death in 1665 his family and friends were concerned his life long passion to gain further understanding of mathematics would be lost forever. His son Samuel, also interested in the study and hobby of mathematics took up the task of collecting and publishing his notes. This was the forum where the famous Fermat's Last Theorem came to be published. Around the year of 1637 Fermat was studying the work of Diophantus of Alexandria and its application to the area of number theory. His goal in his research was to "renew arithmetic as Plato had understood it, as the doctrine of whole numbers and their properties." To this end Fermat worked to develop a stricter method of mathematical solutions and disregarded the methods of Diophantus. Fermat's work in the area of number theory resulted in creating a more modern number theory. In the pursuit of number theory explanations he worked primarily alone which resulted in much of the confusion for future mathematicians. Because Fermat chose to work alone there was no one to decipher his methods or notes. His notes at best were sketchy, if at all.

As Fermat was studying the theorem of Pythagoras' it lead him to expand upon the idea of one of the best understood equations in mathematics:

In any right triangle with shorter sides of x and y and the longest side the hypotenuse z, the equation will hold true. Pythagoras's theorem is proof that there is a solution when n=2 .

While Samuel was searching through Fermat's copy of Diophantus he found Fermat's Last Theorem written as a marginal note. Fermat wrote "I have discovered a truly remarkable proof which this margin is too small to contain". His Last Theorem states that has no non-zero integer solutions for x, y and z when n2.

Fermat claimed to know the solution not only in cubes but any power greater that 2. This was the challenge Fermat left to future mathematicians to solve when n>2. Since the solutions was never written down the challenge was to remain unsolved for close to four hundred years. Future generations were left to prove that there are no whole number solutions for the following related equations:

The difficulty in proving these equations is in the fact that there are infinite possibilities for the values of x, y, and z , as well as infinite number of equations.

By the beginning of the 19th century, Fermat's Last Theorem had already become a challenging mathematical dilemma to be solved in the area of number theory. Large prizes were offered by Universities to anyone who could solve the infamous Fermat's Last Theorem. This theorem has the distinction of being the theorem with the largest number of published false proofs. Apparently, over 1000 false proofs were published between 1908 and 1912 alone.

The first major step forward in the solving of the Theorem was to the credit of Sophie Germain who was born on April 1, 1776 in France. Although her family was successful and financially able to enable her to pursue higher education her family did not belong to the aristocracy to permit a woman to formally study mathematics. Sophie Germain's interest in mathematics came after she found a copy of book History of Mathematics, by Jean-Etienne in her father's study. In particular she read about Archimedes and the legend of his death. Archimedes was so consumed with the study of a geometric figure in the sand he that he failed to answer a Roman soldier's question. His indifference was the cause of his death at the hands of the soldier. Sophie Germain felt that mathematics must truly be fascinating for this legend to happen. Her interest in mathematics alarmed her parents as it was not a subject to be studied by ladies in the 1700's. They attempted to discourage her pursuits but failed and eventually her father funded her research and efforts. Her interest in number theory and calculus led her to teach herself the works of Euler and Newton. For many years the only encouragement she would receive would be from her parents. In 1794, the academy for mathematical and scientific training was opened in Paris as a exclusively male school. Sophie resorted to posing as a former male student, Antoine-August Le Blanc, to enable her pursuit of her studies at the school. Each week she would submit the answers for the work under the name of Monsieur Le Blanc. She progressed well and unfortunately she was found out through her brilliant answers. It seems Monsieur Le Blanc was known to the teachers for his lack of mathematical abilities. The teacher, Lagrange, was astonished and pleased to meet Sophie and became her mentor and friend in the pursuit of her studies.

Through Lagrange's guidance Sophie came to learn and become interested in solving Fermat's Last Theorem. Sophie tried a new strategy to the problem. Her initial goal was not to prove that one particular equation had no solutions, but to make a conclusion about several equations. In Sophie's attempt to prove the equation had no solutions she developed what was to be know as the Germain primes. Germain primes are those prime numbers, n such that 2n+1 is also a prime number. Sophie's list of primes includes:

5 and 11 since

2(5)+1=11

2(11)+1=23.

Both 11 and 23 remain prime numbers (have no divisors other than themselves and 1). Sophie went on to deduce that if n and 2n +1 are primes then this would mean that implies that one of the x, y, z is divisible by n. With this conclusion Sophie Germain proved Fermat's Last Theorem for all n 100 and eventually this was extended to all number less that 197. The Second possibility proposed was that only one of x, y, z is divisible by n. Sophie and mathematicians for her time had not proven for even n=5 and this became the area that future mathematicians concentrated on proving.

In 1847 significant progress was made in the study of Fermat's Last Theorem when Lame announced to the Paris Academy that he had proved Fermat's Last Theorem. His proof was the inspiration of Mathematician Liouville and suggested that the answer involved factorizing into linear factors over the complex numbers.

Liouville became best known for his work in fractional calculus and discovered transcendental numbers that removed the dependence on continued fractions. Liouville built a infinite class of these numbers using continued fractions. The example he gave is now known as the Liouville number where there is a 1 in place of n! and 0 elsewhere:

.1100010000000000000000010000...

Liouville continued to work for the next several months to solve the problem of unique factoring for these complex numbers. Liouville failed to prove the uniqueness of factorization, however this failure was instrumental to discoveries made by Kummer.

In 1856 Kummer presented the concept that when uniqueness of factorization failed it could be recovered by using ideal complex numbers. With this new observation Kummer found examples which a prime number is regular and proved Fermat's Last Theorem. The condition for a prime to be regular would be if the prime number p does not divide the numerators of any of the Bernoulli numbers. The Bernoulli number Bn is defined by:

Kummer was able to prove that all primes up to 37 are regular with the exception of 37 as it divides the numerator of . These discoveries of Kummer were essential in the solving of Fermat's Last Theorem since all later work for many years was based upon the thoughts of an ideal complex number. This introduction by Kummer led to the development of ring theory and much of exploration of abstract algebra.

Many historically prominent mathematicians attempted the solution, only to fail to find the method to solve this mathematical riddle. In their attempts many other significant mathematical concepts were realized. For nearly another one hundred years nothing of major significance would be developed in this area.

In 1955, Yutaka Taniyama was questioning and studying elliptic curves and their relationship between properties of space he also made significant strides in solving the next piece of the puzzle. As a result of his work on elliptic curves in the form of for constants a and b. With assistance from mathematicians Shimura and Weil a conjecture was developed to be known as the Shimura-Taniyama-Weil Conjecture. In 1986 Frey made the announcement to the world that there was a connection between the 1955 conjecture and Fermat's Last Theorem and was more than just an unsolved mathematical puzzle but had fundamental importance to the properties of space.

To this point the solution to Fermat's Last Theorem had been verified for specific equations but proof to the a general method of solving the equation had not been presented. Princeton mathematician Andrew Wiles made the commitment to solve the equation! Dr. Wiles had been fascinated with Fermat's Last Theorem since childhood and this had been a deciding factor in the career choice of mathematics. In 1988 he began the task of providing the proof to the solution. He did not publicly make his search known but quietly worked away in his attic office at home only telling a few trusted friends of his intentions to solve the theorem.

In June 1993, Dr. Wiles gave a series of three lectures at Princeton claiming the proof. Ironically, all but one student had dropped the course so when Dr. Wiles started the first day of three lectures only one lone student student Nicholas Katz viewed his proof. By the third day of the lecture when word had spread about the forthcoming proof all the lecture hall was packed. Having written the Theorem on the blackboard he claimed to have solved the Last Theorem and would stop there. Dr. Wiles had not only proved the Shimura-Taniyama Conjecture he had provided examples to prove Fremat's Last Theorem. Dr. Wiles sent his careful research for review to a select group of experts in the field of mathematics. Having spent so much time on his work he wanted to ensure his proofs were secure and fix any problems before formal publication.

In the fall of 1993 some minor mistakes were fixed but a larger hole in the method was found and unable to easily be fixed. For the next six months Dr. Wiles worked in collaboration with Dr. Taylor of Cambridge University to find the missing pieces. The new insight came in a method to allow a infinite collection of mathematical objects called Hecke rings to be constructed. They used the idea to take one element of a set and use that to find the next element, then to use the second construct the third and so on.

Dr. Wiles work has been described as the greatest advances yet and decades beyond the predicted time-line for solving the Taniyama conjecture and proving Fermat's Last Theorem. After completing the written documentation to his proof Dr. Wiles retired from the Taniyama project.

The pursuit of Fermat's last Theorem has captivated thousands of professional and amateur mathematicians throughout the ages. In researching this paper and attemping to comprehend the mathematics involved has proved to be suprisingly interesting. This has provided me with a greater appreciation for the intensity and dedication with which these academics have devoted to solving the great mathmatical mysteries.

This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it.