It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
It is our old friend a2 + b2 = c2.
Now, as I’ve already spoken at length, there are many solutions to this equation. Many values of a, b, and c that are integers (whole numbers) and that make the equation true.
So let’s add another variable. Consider:
This is a Diphantian equation, a, b, and c, as well as n must be integers. Further, as we’ve already discussed, if n = 2, there are a lot of solutions for a, b, and c. But what if n = 3?
What integer solutions are there to a3 + b3 = c3? Well, no one has ever found such a, b, and c.
Same for n = 4, or 5 or 13, or any other number. There were no solutions for n > 2. Now, how to prove this? (This part here might require some mathematical thinking. Even though, for hundreds or thousands of years, no one had found solutions for n > 2, you can’t be sure there isn’t one, just, as yet, not discovered. A mathematical proof, on the other hand, would end the search. If it could be proven that no solution exists for any n > 2, that would be the end of the quest.)
Then, along came Fermat. That’s Pierre de Fermat, born around 1601 and died in 1665. He was a lawyer in Toulouse, France. He was also an amateur mathematician who is given credit for early developments that led to the infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for “Fermat’s Last Theorem,” which he described in a note at the margin of a copy of Diophantus’ “Arithmetica.”
It is actually more correct to call it “Fermat’s Conjecture,” since he did not publish a proof. That was the name used for most of the history of this conundrum. Recently it has become known as “Fermat’s Last Theorem,” and that is the term most likely to be familiar to non-mathematicians.
He communicated most of his work in letters to friends, often with little or no proof of his theorems. This allowed him to preserve his status as an "amateur" while gaining the recognition he desired. This naturally led to priority disputes with contemporaries such as Descartes and Wallis. He developed a close relationship with Blaise Pascal. These may not be household names, but they are some of the founding fathers of modern math.
Now he was not a braggart, nor a mathematical lightweight. So his statement was taken very seriously. Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century. Was he wrong?
(A reminder that Descartes discovered the Cartesian Coordinate System, that familiar graph of x and y that is used to merge algebra and geometry into Analytical Geometry, a forerunner of the Calculus. Descartes established the basic concepts of Calculus, but didn’t quite make it all the way. That came later when Sir Isaac Newton and Gottfried Wilhelm Leibniz independently finished the job.)
Lacking Fermat’s proof, over the next two centuries (1637–1839), the conjecture was proven for only the primes 3, 5, and 7, although the great female mathematician Sophie Germain proved a special case for all primes less than 100.
(Marie-Sophie Germain (1776 – 1831) was a French mathematician, physicist, and philosopher. Because of prejudice against her gender, she was unable to make a career out of mathematics, but she worked independently throughout her life.)
The list of mathematicians that contributed proofs for certain classes of numbers reads like a who’s who of important mathematicians including Euler, Legendre, and even Carl Friedrich Gauss, a man I consider the greatest mathematician of all time.
In the mid-19th century, Ernst Kummer proved the theorem for regular primes. Building on Kummer's work and using sophisticated computer studies, other mathematicians in the 20th century were able to prove the conjecture for all odd primes up to four million. But these were all limited proofs of what is called “special cases.”
The final proof of the conjecture for all n came in the late last century. In 1984, Gerhard Frey suggested the approach of proving the conjecture through a proof of the modularity theorem for elliptic curves. Building on work of Ken Ribet, Andrew Wiles succeeded in proving enough of the modularity theorem to prove Fermat's Last Theorem, with the assistance of Richard Taylor. Wiles' achievement was reported widely in the popular press, and has been popularized in books and television programs.
So, finally, in the final years of the twentieth century, a final proof was found and offered and reviewed. It is very complex, using powerful methods, some suggested by other work of Fermat’s. British mathematician Andrew Wiles and his former student, Richard Taylor, finished a long work on the problem and published two papers describing the proof in the Annals of Mathematics. It was 358 years after Fermat wrote in the margin of a book.
Fermat’s Conjecture was not just a puzzle to be solved. Like most of the progress of mathematics, it led to new discoveries, which led to more discoveries, which led to the overall advancement of the understanding of the deep mysteries locked in numbers and equations.
Still an open question is if Fermat really had the answer. Most doubt he had a complete answer. The final proof used many advanced methods that simply didn’t exist in Fermat’s day. It is more likely that he had solved the problem for a subset of all possible exponents. Recall we have his solution for n = 4. Perhaps he had other solutions in addition or a method to generalize that solution. But most modern mathematicians don’t believe he had a complete answer. But we will never know for sure.
Even if he hadn’t found the marvelous proof, Fermat’s Conjecture is considered one of the greatest mathematical problems of all time. That it took so long to solve is a testament to the energy of mathematicians through the ages. The problem seems so simple, and yet the solution so hard to produce.
It is a sort of revenge for all math students who struggled with the problems in textbooks. Nice to know even the authors of the textbooks had problems that they struggled with too. Even nerds can encounter unsolvable problems. Sort of humbling … if you are a nerd.
I find beauty in the deep thoughts, the powerful arguments, and creativity of the solutions, and the pure difficulty of these problems. I see the history of all struggles in the tales of these solutions. Even in science we see prejudice, but truth wins out ultimately. Like a runner who finishes well in a race, solving these problems is a thrill … if you can solve them. And studying the solution and the path taken to those solutions can be most uplifting.