As students, you must have come across various mathematical theorems and formulas that you had to memorize and apply in problem-solving. However, there is one theorem that baffled the greatest mathematicians of their times for centuries until it was finally solved in the late 20th century by the legendary Andrew Wiles – the famous Fermat's theorem or the Fermat's Last Theorem. Considered one of the most significant breakthroughs in mathematics, proving Fermat’s theorem was one of the most unconventional yet fascinating endeavours of the time.
Try solving this :
Easy, right? You have infinite solutions like a = 2, b = 5, c = 7 or any end number of solutions
Try solving this :
Slightly harder but still doable. You have many solutions as you may recognise this as the Pythagorean Theorem. The most famous is a = 3, b = 4, c = 5
Now, try this:
Take some time experimenting with small numbers and the cubes. After some time, you may realise there is no solution that is obviously small (other than the variables equaling zero of course. That is a trivial solution that is uninteresting for our purpose). Here is where the theorem is born.
The Origin of the Theorem
The theorem came into the picture by a 17-th century mathematician, Pierre de Fermat.
The theorem is as follows:
In simpler terms, there are no integer solutions to the above equation if n is greater than two.
Amusingly, Fermat did not propose this theorem in a mathematical paper or book, rather he scribbled it just beyond the margin of his copy of Arithmetica, a book written by the ancient Greek mathematician Diophantus. In his scribblings, he claimed to have proved this theorem, however, he never shared said proof with anyone and teasingly joked that
I have a truly marvellous demonstration of this proposition that this margin is too narrow to contain
Fermat's theorem thus remained a mystery for the next 350 years, leading to numerous attempts by mathematicians to prove or disprove it. So, what makes Fermat's theorem so intriguing and why did it take so long to solve it? Let's dive deeper into the theorem and its proof to understand it better.
Fermat's Theorem and Its Significance
The Fermat's theorem is a part of number theory, which deals with the properties and behaviour of numbers. It is a special case of the more general equation known as Diophantine equation, which involves finding integer solutions to any kind of polynomial equations. The Diophantine equations are of no particular interest to our journey through Fermat's proof hence are not explained here, yet they are an interesting topic to read about as a hobbyist.
Fermat's theorem has far-reaching implications, and it has been studied extensively by mathematicians for centuries. Primarily, one of the most important applications of Fermat's theorem is in cryptography, where it is used to secure digital communication by encrypting messages using large prime numbers. If the Theorem's equation had a solution then modern cryptography's numerous applications would become weaker giving the most dedicated hackers a window of opportunity.
While the theorem itself contributed little to other proofs, the process used to prove Fermat’s Last Theorem’s became the foundation for an entirely new field of mathematics: Modularity Theorem. The unique approach of the proof was critical to further studies in Number Theory and mathematics, leading us now to the actual proof itself.
Proof of Fermat's Theorem
Proving a theorem requires a rigorous process that numerous mathematicians verify and approve. The attention behind the theorem caused every advancement in its proofs to be closely monitored and criticised such that any breakthroughs accepted and verified would lead to fame. Mathematicians were known to collaborate as they often shared ideas and proved or disproved each other’s theories and breakthroughs.
Thus, it was not a surprise when breakthroughs in the smallest communities of math quickly rippled to nationwide efforts. One of the largest breakthroughs in mathematics of the time that contributed to the proof was known as the “Shimura-Taniyama-Weil conjecture”. This conjecture was a novelty of the time as it brought together two fields of mathematics at opposite ends of the spectrum : modular forms and eliptic curves. These two fields had been well researched and explored, but always separately and never cohesively. Taniyama and Shimura had managed to go beyond the margins of both fields to truly advance mathematical thinking at the time, inspiring the research into other fields of mathematics that were mysteriously “connected”.
While the conjecture held little link to the Theorem at first, Gerhard Frey, a German mathematician, shared his development at a symposium where he transformed the Theorem’s equation into an elliptic equation. This transformation created the connection between the conjecture and Theorem, so that the task of proving the Theorem had been reduced to the task of proving the conjecture.
Wiles' entry
In the 1990s, Andrew Wiles’s isolated himself from the world and got to work to prove the conjecture. Wiles worked on the proof for over seven years, and his work involved developing entirely new mathematical concepts and techniques that had not been used before. He managed to show that if the theorem were false, then it would lead to a contradiction in the properties of modular forms, which would be impossible. His proof was riveting and exciting for the entirety of the mathematical community and marks him as one of the greatest mathematicians in history till date.
He presented it to the community in a series of three lectures at Cambridge where he built up the proof step by step for an audience crowded inside and outside the auditorium that waited patiently. His final statement, the equation of the Fermat’s Theorem, gave way to numerous rounds of applause and appreciation as the mathematical community received the greatest gift they could ask for.
A Small Hiccup
Unfortunately, it was not long before the community found a fundamental flaw in Wiles’s proof where his implementation of a conjecture had been faulty. Finding such a flaw in the world of mathematics typically means one of two things : an insignificant speed bumper or a devastating wall.
Fourteen months. Fourteen months of sheer devastation and misery as Wiles faced shame and dealt with the downfall of his fame. Newspapers and articles outlined Wiles’s mistake as he once again shut himself off to get to work on the theorem. Fourteen months was what it took for him and his brilliant mind to arrive at yet another beautiful insight. Wiles had used multiple methods over the course of the eight years working on the Theorem, hence it came to him as a moment of inspiration and realisation when he noticed that two methods of proof that had been inadequate so far happened to work perfectly well in unison. Wiles combined his techniques from two years ago with the techniques of his present to perfect the proof’s one flaw. Fourteen months had amounted to that aha moment of satisfaction and relief.
After working on this approach and completing the proof once and for all, Wiles released two manuscripts that detailed the approach and proved multiple theories alongside the elusive Fermat’s Last Theorem.
Hence, it was indeed proven that:
Conclusion
Fermat's Theorem is one of the most famous and important theorems in mathematics. It has fascinated mathematicians for centuries and has far-reaching implications in various fields, including cryptography and algebraic geometry. The theorem remained unsolved for over three centuries until Andrew Wiles finally cracked it using advanced mathematical concepts and techniques. His proof of the theorem was a significant breakthrough in the history of mathematics and inspired further developments in number theory and algebraic geometry.
As students, you may not yet have encountered the intricacies of Fermat's theorem, but it is an excellent example of the power and beauty of mathematics. It is a reminder that even the most complex problems can be solved with dedication, perseverance, and ingenuity.
Bibliography and Suggested Further Reading
John Wiles, Andrew. “Modular Elliptic Curves and Fermat’s Last Theorem.” Annals of Mathematics, vol. 141, 1995, pp. 443–551, scienzamedia.uniroma2.it/~eal/Wiles-Fermat.pdf.
Singh, Simon. Fermat’s Last Theorem : The Story of a Riddle That Confounded the World’s Greatest Minds for 358 Years. Harper Perennial, 2011.
Yves Hellegouarch. Invitation to the Mathematics of Fermat-Wiles. Elsevier, 2001.
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