Fermat’s Last Theorem

Fermat’s Last Theorem

Abstract

The works of the seventeenth century French mathematician Fermat de Pierre greatly shaped the field of pure mathematics that subsequently influenced various aspects of our present everyday lives. Being a classical genius, Fermat was extremely talented in multiple disciplines and made significant contributions to the world of math. Among his formulated theorems, the Fermat’s Last Theorem has perhaps been the most important and troublesome problem in the history of the subject. One of the most amazing and interesting facts about Fermat’s Last Theorem is its simple appearance and the complexity associated with its proof.  It can easily be described to anyone with a basic understanding of mathematics yet its proof took more than three centuries to be developed. Even after this proof was established, it challenged even the best mathematical minds. It is equally fascinating that Fermat’s Last Theorem facilitated the development of numerous mathematical techniques over the years as mathematicians from different generations attempted to provide its proof. This facilitated the development of several mathematic topics whose impact supersede the scope of their discipline. Fermat’s Last Theorem, therefore, provides the perfect example of the practicality of pure mathematics.Introduction

In 1994, an English professor offered satisfactory proof that laid the Fermat’s Last Theorem challenge to rest. Professor Andrew Wiles then went on to win Math’s Abel prize which is the disciplines highest award. He also received widespread acclaim and recognition from all over the world. This achievement was no easy task since he had dedicated seven years to working on his proof. His accomplishment was celebrated as a win for human intellect since generating the proof had taken successive generations over nearly 350 years. This warranted the question: What is Fermat’s Last Theorem and what are its implications on the world?

Formulated by Fermat de Pierre, one of the best mathematicians at the start of the seventeenth century, the Fermat Last equation is a simple mathematical statement denoted as  provided that . Simply put, the universally understood Pythagorean equation cannot hold true for powers higher than 2. At face value, it seems simple to understand. However, generating its mathematical proof would elude mathematicians for three and a half centuries. What’s even more astounding is that when Fermat formulated this famous Theorem, he probably did not anticipate the challenge it would present to generations of mathematicians and the impact this would have on the discipline.

The journey to proving Fermat’s Last Theorem led to the development of different topics and vindicated the human intellect. Number theory is perhaps the biggest beneficiary of Fermat’s Last Theorem as many new techniques were developed in this particular specialization. People developed a deeper understanding of topics such as factorization, complex numbers as well as polynomials. Since all these are topics that have been instrumental in bringing to life most of the present technological advancements, it is apparent that this Theorem is relevant to everyone hence warranting its understanding and appreciation.

Pierre de Fermat was born in 1601 in the South-West region of France. His father was a rich leather dealer which enabled Fermat enjoy quality education. Even though the specific details of his early schooling aren’t known, it is recorded that he studied law and graduated in 1631 from the University of Orleans. Due to influence from his family, he became a councilor at the parliament in Tolouse. As a councillor, he served as a link between the public and the king. Together with other councillor’s, he was mandated with the responsibility of conveying the locals’ wishes to the crown and ensure the implementation of royal decrees in the provinces. In addition to being a civil servant, Fermat also served in the judiciary. Even though he was an efficient and exemplary civil servant, it is his service in the legal profession that constituted a greater part of his professional career. He married his mother’s cousin in 1631 and had five children. His life as a career jurist was rather quiet and relatively uneventful compared to his other talents more so his contributions to mathematics that secured him a place in the annals of history.

Similar to other great inventors, Pierre de Fermat was a man of many talents. He had a passion for classical literature, foreign languages, science and mathematics. As a linguist, he is acclaimed to have fluently spoken six languages and offered valuable insights when people were undertaking Greek translations of various texts. He was a great amateur academic whose hobby involved trying to solve mathematical and scientific problems. Since he did not possess any grand political ambitions, he used much of his spare time reading the translated works of ancient scientists and mathematicians. His reading would commonly be accompanied by jotting of short noted and theorems on the texts’ margins. This characteristic would prove to be critical in recovering his works for the advancement of mathematics. Resulting from his hobby, he is credited with having developed the foundation for calculus way before Isaac Newton. He also contributed significantly to the fields of probability where his works still serve “as the cornerstone of modem insurance and other forms of risk management” (Bernstein, 1998, p.63) His scholarly exploits also involved analytical geometry and optics. Due to his unique scientific and mathematical talents Fermat had to collaborate with other renowned scientists of his time to achieve some of these feats.

Although Fermat was predominantly an academic hermit who preferred to work by himself, he is recorded at different instances to have enlisted the help of other prominent scholars in tackling different problems. Together with Pascal, they worked on probability theory and greatly appreciated each other’s work. Their joint efforts in developing probability theory was very practical and its principles were readily applicable to gambling and insurance. However, Fermat didn’t always enjoy cordial relations with his peers. His correspondences with his colleagues usually contained his latest theorems but always lacked proofs. He challenged his colleagues to find the proofs for his theorems causing great frustration among his peers. Singh notes how “Rene Descartes called Fermat a ‘braggart’ and the Englishman John Wallis referred to him as ‘That damned Frenchman’”(singh) to underscore the kind of relationship these scholars shared. Regardless of his antiques, he was highly esteemed and considered a brilliant scholar of high repute among his contemporaries. Among his achievements, it was his work on number theory that caused a stir in the world of mathematics since proving the famous Fermat’s Last Theorem defeated mathematicians for three and a half centuries after Fermat formulated it in the seventeenth century.

Fermat’s Final Theorem states that for all values of n greater than 2, there does not exist any value of x, y, and z such that, where x, y, and z are all positive integers. As noted by Amir, it was named as his ‘final’ theorem in jest since “all of his other theorems had either been proved or disproved by the early 1800s” (Aczel, 1997, p.10) leaving this seemingly simple statement unsettled. Since Fermat was an amateur mathematician, his theorem was just a marginal note on his copy of Diophuntus’ book: Arithmetica. He did not think it was a big deal therefore he never intended to publish it together with the proof he had done to his satisfaction. Similar to his other scribbled notes, this theorem was his answer to the problems posed by Diophuntus through his book. In classical Fermat style, he was not keen on documenting the proof to this theorem but pointed out that he had already established the proof. It was after his death in 1665 that one of his sons sought to compile the scribbled notes together with Fermat’s own proof and preserve his father’s genius for the advancement of mathematics. This marked the beginning of a long journey to prove Fermat’s theorem.

Upon the publishing of Fermat’s Final Theorem, great mathematicians in subsequent generations who were not satisfied with Fermat’s proof sought to provide proof to this theorem without success. This situation persisted up to 1993 when a Princeton University mathematics professor by the name Andrew Wiles proved it. His proof was contained in 200 pages and had taken seven years to generate. Even then, this proof was found to contain a gap that was filled towards the end of 1994. Even though the proof lacked any dramatic applications, it was another triumph of human intellect. (Cuoco & Rotman, 2013, p.xiv) The proof marked the end of what had perhaps been mathematics’ longest challenge. This momentous occasion was not only a victory for professor Wiles but a triumph for the world of math as well as the whole of humanity. Despite the fact that very few people can actually understand the provided proof, this feat still remains an impressive achievement. In order to appreciate Femart’s Last Theorem as well as professor Wiles’ accomplishment, it is paramount to better understand the theory itself.

As is the case with most developments in mathematics, Fermat’s theorem has its origins in a previously established mathematical concept. It owes its origin to the popular Pythagorean theorem:. Just when mathematicians had successfully understood the sets of triples (x, y, and z) that satisfied the Pythagorean equation, they realized that the equation had a more challenging dimension to it when the powers of x. y. and z were more than two. When the power was changed from ‘2’ to ‘3’, finding whole number solutions to satisfy the equation became impossible. Moreover, further increasing the power from ‘3’ to a higher number n (where n = 4, 5, 6, 7. . .) still made it impossible to find a solution to the problem. This was what made the Pierre de Fermat make the claim that “the reason why nobody could find any solutions was that no solutions existed.” (Singh, 1997) This could be summarized by what is referred to as Fermat’s Last Theorem expressed as:

1.1

for

Fermat himself proved that the equation 1.1 had no solution for. Despite the simplistic appearance of equation 1.1, finding the solution for this problem remained a big challenge for mathematicians for nearly 350 years. The quest for finding its proof also had the effect of developing other branches of mathematics such as the understanding of complex numbers, polynomials, and factorizations. This further explains why Fermat de Pierre is popularly referred to as the founder of the modern theory of numbers.

Proof of Fermat’s Last Theorem for

In order to prove Fermat’s Final Theorem, we start of by rewriting the original equation as follows:

2.0

One then proceeds to replace the  on the right hand side of equation 2.0 with a new value Z i.e . The procedure then follows to prove that there is no combination of positive integers of x, y, Z that can satisfy equation 2.1 below:

2.1

The proof makes use of infinite descent whereby given a triple of positive integers (x, y, Z) satisfying equation 2.1 above, it will be shown that there exists another combination of integers (u, v, w) where .

If equation 2.1 were to hold for (x, y, Z) as well as (u,v,w), then there would be a contradiction meaning that equation 2.1 cannot hold for any number since there can’t exist an infinite combinations of ‘smaller’ numbers that satisfy the equation 2.1.

Taking equation 2.1 to be the lowest possible lowest solution means that at least two of the terms are relatively prime lest their common factor is divided out and gives a new solution that would be much smaller than the initial solution. We then take x and y to be our relatively prime terms and re-write equation 2.1 in the form of equation 2.2 below:

2.2

This more recognizable form reveals that ( form a Pythagorean triple

It can also be established that all the three terms in equation 2.2 cannot be odd. For the equation to hold true, at least one of the terms has to be even. Also both   and  cannot be odd since if they were,  would take the form and  would assume the form . This could then re-write the form of equation 2.2 as:

2.3

Expanding equation 2.3 and collecting the terms would result to:

2.4

Which can further be summarized as equation 2.5 where h is an integer

With ,

2.5

But the square of any integer can only take the form  or . This proves both   and  cannot be odd. We therefore take  to be odd and  to be even. By extension x is an odd number and y is and even number since the square of an odd number is an odd number and the square of an even number is a resultant even number.

Using the method of Diophantus, there exists relative prime integers p and q where  such that

Rearranging  into  brings another set of Pythagorean triples (x, p,q)

Similarly, for this other set of triples, there exists relative prime integers c and d where

such that:

This form makes:
2.6

Since x was selected to be odd and y was even, equation 2.6 can be written as:

2.7

Since the term on the left hand side of equation 2.7 is a square, each term on the right hand side is also a square. This means that there must be integers u, v, w such that:

Substituting a and b in  yields

2.8

Equation 2.8 forms another ‘smaller’ solution of the initial equation 2.1

The initial hypotherical solution (equation 2.1) therefore implies the existence of a second smaller hypothetical solution (equation 2.8)

The process can be repeated for u, v, w over and over again proving that the solution for equation 2.1 does not exist.

It can be summarized that from Fermat’s Final Theorem, there exists no solution for

: since the provided proof establishes that  does not exist.

Similar to how Fermat’s Last Equation was re-written to equation 2.0 and subsequently equation 2.1 to prove its lack of a solution, the same can be applied for all values of .

By writing, where n and k are integers, with , this proof can extend to odd values of K and be used as a template for proving Fermat’s Last Theorem for all integral powers of the equation.

References

Aczel, A. (1997). FErmat’s Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem (p. 10). Dell Publishing.

Bernstein, P. (1998). Against the Gods: A Remarkable Story of Risk (p. 63). John Wiley & Sons, Inc.

Cuoco, A., & Rotman, J. (2013). Learning Modern Algebra: From Early Attempts to Prove Fermat’s Last Theorem (p. XIV, 15). The Mathematical Association of America.

Singh, S. (1997). Fermat’s Last Theorem: The Story of a Riddle That Confounded the World’s Greatest Minds for 358 Years (1st ed.). Fourth Estate.