Author: Ted Isidor

Editor: Max Ye and He-Hanson Xuan

Artist: Lucy Chen

When you think of geometry, what equations come to mind? Some may say the formula for the area of a triangle, or the volume of a sphere. For students who have surpassed the superficial content of the subject, the first thing that pops into their minds is the Pythagorean theorem: the sum of the squares of the legs of a right triangle is equal to the square of the length of the hypotenuse. Although Greek mathematician Pythagoras derived this theorem during 1900-1600 BCE, its evolution did not stop there. Pierre De Fermat was a lawyer and an avid mathematician in France during the 1600s. After a long day's work of settling disputes and accepting clients, he would come home and work on mathematics. In his later life, Fermat began pondering if Pythagoras' Theorem could be applied to geometric figures other than triangles. Thus, a conjecture was born: the sum of two numbers raised to a power greater than two cannot equal another number raised to the same power. Fermat noted in his diary entry that he had a proof for such conjecture but died with most mathematicians believing that he was bluffing. Yet, an English mathematician by the name of Andrew Wiles was determined to prove that such a theorem exists and triumphed in doing so.

When it comes to understanding any theorem, mathematicians nearly always start by looking at examples. For instance, it is known that there exist particular combinations of integers like 3, 4, and 5 in which the equation holds true-- three squared plus 4 squared equals 5 squared. Another set of numbers can be 7, 24, and 25. These groups are termed Pythagorean Triples. Fermat wanted to use a similar approach. For the case of integers that were cubed or to the fourth power, he believed that there does not exist a special combination of integers in which the rule holds true. It is amusing to try it out for yourself, like seeing if 5 cubed plus 3 cubed can be rewritten as a cube. Fermat’s last theorem is relatively easy to tinker with but was extremely hard for mathematicians to prove, with Wiles taking 7 years of working in secret to discover the first proof. Due to such difficulty, the theorem was once listed as a millennium problem (a series of math problems that rewarded whoever solved them to be awarded a large sum of cash).

The beginning of Andrew Wiles’ journey in mathematics is quite the tale. According to a short biography written on the behalf of Wiles by the University of St. Andrews, Wiles became fascinated by Fermat’s theorem at the age of ten when he stumbled upon it while visiting a library. Ever since his first encounter with the problem that stumped mathematicians for centuries, Wiles became determined to be the one to solve it. While working in Paris at a university following getting his doctorate degree, the proof for a conjecture called the Shimura-Taniyama conjecture was made. This former conjecture connected elliptic curves to Fermat's last theorem. This turned a switch in the mind of Andrew Wiles. Due to this newfound connection between the Shimura-Taniyama conjecture (i.e. the modularity theorem) and Fermat’s last theorem, Wiles began to think that his childhood dream to prove it was more possible than ever. Thus began Wiles' secret mission, in which, over the course of 7 years, he worked on the proof in secret, going to the extent of not wanting to discuss the theorem in a casual manner with other mathematicians, but such dedication did pay off as in 1995 he announced his proof and it was confirmed to be true by other mathematicians. This felt like a case closed in mathematics, one more elusive conjecture finally proved and now onto the next problem; However, it turns out Wiles’ proof has caused a new tidal wave in mathematics. Wiles' proof of Fermat’s theorem meant that 2 entirely different areas of mathematics, number theory, and harmonic analysis, which feels like complete opposites, do have connections.

In an article written by the institute of advanced studies, Robert Langlands wrote to a friend about a wild but exciting idea he had on unifying mathematics. Mathematics as a whole feels somewhat like a group of cities. Take, for example, the discipline of number theory, if number theory were to be represented as a city it would look entirely different from another city dedicated to harmonic analysis. Sure some fields of mathematics might have connections with one another, but what Langlands suggested would be the golden gate bridge of mathematics, connecting number theory and harmonic analysis. Now, what connects Fermat’s theorem to Langland's project is not really the theorem itself but the proof instead. Wile’s method involved looking into harmonic analysis and using that subject to prove Fermat's last theorem. By completing his proof, Wiles made a bridge between the two cities, and now mathematicians are on the move to see what’s next. In sum, Fermat's last theorem and its proof are remarkable; a complex solution for what seemed to be an easy problem turned out to be one of the most incredible things our generation has produced in the field of mathematics.

**Citations**

Britannica, The Editors of Encyclopaedia. "Andrew Wiles". Encyclopedia Britannica, 7 Apr.

2022, https://www.britannica.com/biography/Andrew-Wiles. Accessed 21 June 2022.

Klareich, Erica. “‘Amazing’ Math Bridge Extended beyond Fermat’s Last Theorem.”

*Amazing’ Math Bridge Extended Beyond Fermat’s Last Theorem*, 2020,

https://www.quantamagazine.org/amazing-math-bridge-extended-beyond-fermats-

last-theorem-20200406/.

O'Connor, J J, and E F Robertson. “Andrew Wiles - Biography.” *Maths History*, MacTutor,

2009, https://mathshistory.st-andrews.ac.uk/Biographies/Wiles/. Quanta Magazine,

director. *The Biggest Project in Modern Mathematics*, YouTube, 1 June 2022,

https://www.youtube.com/watch?v=_bJeKUosqoY&t=362s. Accessed 22 June 2022.

Thomas, Kelly Devine. “Modern Mathematics and the Langlands Program.” *Institute for*

*Advanced Study*, 24 Oct. 2019, https://www.ias.edu/ideas/modern-mathematics-and-

langlands-program.

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