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Fermat's Last Theorem... Solved!

Fermat’s Last Theorem was proposed by a French mathematician named Pierre de Fermat. While reviewing the work of Diophantus, the great Greek mathematician, Fermat became interested in the chapter on Pythagorean triples - sets of three numbers, a, b, and c, that satisfy the following equation: a² + b² = c². While many of us have learned the Pythagorean theorem, few if any have ever thought of what lies beyond the squared variables. Fermat, on the other hand, became deeply interested and formed a theorem, which read that a set of positive integers a, b, and c does not exist to satisfy aⁿ + bⁿ = cⁿ, where ⁿ is any number greater than 2 (a³ + b³ = c³ for example).

Fermat’s simple theorem turned out to be surprisingly difficult to prove (revise). Generations of mathematicians devoted their lives to prove Fermat’s statement true or to disprove it by presenting an exception. 350 years passed before a proof was found.

When he was 10 years of age, Andrew Wiles visited the local public library in London. There, he looked at a book on mathematics in which he read about Fermat’s last theorem. This theorem seemed so simple that even a child could understand it. In Wiles’ own words: It said that you will never find numbers, x, y, and z, so that x³ + y³ = z³. No matter how hard you tried, you will never, ever find such numbers. And it said the same was true for x⁴ + y⁴ = z⁴, x⁵ + y⁵ = z⁵ and so on… It seemed so simple. And it said that nobody has ever found a proof of this for over three hundred years. I wanted to prove it

In the 1970’s, after receiving his degree, Wiles was admitted as a research student in mathematics to Cambridge. Unfortunately, Wiles had to instantly drop his childhood dream of proving Fermat’s Last Theorem for a number of reasons. First, research on this problem that has been unsolved for generations, would take so much time that no graduate student could afford it. Besides, would a student working on such an ancient puzzle be accepted, a puzzle that had kept the world’s brightest minds from a solution? Also, Fermat was not in fashion. So Wiles abandoned his dream and spent all his time doing research on elliptical curves, which were the hot topic at the time. After receiving his Ph.D., he got a position in mathematics at Princeton University and moved to the United States, where he resumed his research on elliptical curves. Several years later, Wiles was accidentally reminded about Fermat’s theorem. Immediately, Wiles knew that his life was about to change. Now that he had established himself, he could devote time to finding the solution, his boyhood dream. Wiles decided to work in complete isolation, because too many spectators would ruin his concentration and other people are always willing to finish your work for you, especially at a place like Princeton, where gifted, able mathematicians are abound. Whatever the reason, Wiles isolated himself in his attic office and buried himself in work. He abandoned all other ongoing research and devoted himself completely to Fermat. After six years of working alone and making moderate strides, he felt a need for comparing notes and opinions with another person. In January of 1993, such a trustworthy person was found, Professor Nick Katz, a Princeton colleague. Katz was completely trustworthy; he would most certainly keep his mouth shut. To make their continuous meetings and talks seem innocent, Wiles devised a scheme. He would initiate a new course, called “Calculations with Elliptic Curves,” and Katz would enroll as a student. Soon, other students started to drift away because they found no interest in a challenging course that was not going anywhere. The only “student” who seemed to know anything and participate at all was Katz. After other breakthroughs in the related fields and 10 years of continuous work, Wiles was finally ready. The pieces of the puzzle fell together. Also, a number theory conference was to be hosted in Cambridge. The biggest names would unquestionably be present. Usually, after a discovery such as this, the material for the proof would be provided to journals, which in turn would contact their experts to confirm the accuracy of the proof. Wiles wished to avoid this process, where his proof would be in many hands (which could easily take his proof and publish it under his or her name), and therefore, decided to present his proof at the conference. After putting in enormous effort and writing 200 pages explaining his proof, Wiles bought a ticket to Cambridge, England. After being received with stunning the audience, Katz found a hole in the proof. However, Katz quickly solved it and now, Fermat’s last theorem is unproven no more! Mathematicians devoted their whole lives to this cause, yet no one, not a single soul was able to solve this theorem for 350 years, until a little boy named Andrew Wiles gave birth to a dream, a dream of solving Fermat’s Last Theorem. Also, Wiles’ solution paved the way towards further mathematical discoveries.