Fermat’s Conjecture

My greatest passion is for math – not engineering, physics, computer science, programming, project management, nor – even – astronomy. No, math, often called the Queen of the Sciences, is my passionate love affair, my secret mistress, and a lasting infatuation of mine. Like most good love stories, the object of my affection is often beyond my grasp. Yet I continue to pursue her, flowers and chocolates in hand, as I chase her from café to café.

It was not always so, astronomy was my first love, followed by physics and electronics, and even geology caught my fancy at one time. But math, pure thought stuff, yet much more disciplined than programming (even though programming is of the mind too) that I keep coming back to time and time again.

“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” – John von Neumann.

One of the happiest times in my career was when I wrote two textbooks for use in IBM’s Programmer Retraining Program. I took about eight or ten months to focus on the creation of the texts. I had an intern to assist me, and I spent days at a terminal writing. I taught myself IBM’s Generalized Markup Language or GML, which was marketed as “IBM Script.” (I did take a one-week class on Script part way through the writing sessions.) Script had a mathematics markup extension, and I did all the editing on a mainframe computer. The mathematics book I wrote was a self-study manual on Discrete Mathematics, the branch of math associated with programming. I called the book “Logical Expressions.”

I also wrote a short tutorial on the use of IBM’s Personal Editor that contained some basic training on using the IBM PC. The students in Programming Fundamentals used these two textbooks in their training. The math textbook served as a self-study guide, and it was used before students took a preliminary test to qualify for the retraining program. I also taught a two-week-long math course based on my book in Boulder. I think IBM published around 5,000 copies of both texts, so they weren’t exactly best sellers, but it was a very joyous time for me to be immersed in math and writing. It may have been the genesis of what I’m doing right now!

It was a time of deep focus and daily successes. I loved to see my creation growing in the drafts produced by the printer, and, day-by-day, I added to the body of work that ultimately was read by hundreds, possibly thousands of students. They may not have shared the joy I felt in creation of the books, but they – at least – read them. This adventure increased my understanding of writing and printing and even introduced me to Donald E. Knuth, one of my long-distance mentors.

I’ve written about math before, and discussed the beauty of equations and described the deep connection I have with Maxwell’s Equations, the actual formulas that describe light and all electromagnetic radiation. However, to even read one of these fundamental equations requires a very advanced knowledge of math. Maxwell’s equations use Differential Calculus and a special short hand called Div, Grad, and Curl. I wrote about my discovery of these powerful mathematical tools before and how they influenced my thinking. Here are several links to those stories.

The Most Beautiful Formula in Mathematical Thought

A Treatise on Light (and Color)

A Note on Mathematics and the Importance of Gödel’s Theorem

A Monograph on the Special Number 36 and the Fundamental Theorem of Arithmetic

How to explain mathematics – especially to non-mathematicians, and that includes most out there? Sure, you all know about arithmetic, and even a smattering of algebra, geometry, trigonometry, and maybe even some calculus. When does the beauty appear? How much must you know to see that beauty? That is the question.

What if I told you about one of the hardest problems in all of mathematics? Would you say, “I can’t even understand the questions, much less the answer.”

This won’t be hard. It just involves some simple formulas with exponentiation … powers. Not hard. You know: squares and cubes, etc., plus some addition. It’s easy. Yet this problem has troubled mathematicians since it was first stated … and that started several thousand years ago.

The problem is very easy math. The solution turned out to be very difficult math; very difficult indeed.

A brilliant mathematician in the seventeenth century wrote in a margin of a mathematics book:

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.

The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most famous theorems in the history of mathematics and prior to its 1995 proof was in the Guinness Book of World Records for "most difficult mathematical problems".

So this is a tough problem … very tough … yet very beautiful in its simplicity. You don’t need advanced math at all to understand the problem. The solution … on the other hand … is very, very deep. From this simple problem, so much as been developed.

This problem involves Pythagorean Triples. You all remember ancient Pythagoras and his famous theorem regarding right triangles. This is related … and just as old.

Here’s the equation:

a² + b² = c²

This is also called “splitting a square.” “Split a given integer square into the sum of two integer squares.” It turns out that there are many, many … possibly an infinite number … of values for a, b, and c that will make the equation true.  The simplest, 3, 4, and 5.

3² + 4² = 5²,  9 + 16 = 25

When applied to a right triangle, with two sides equal to 3 and 4, then the hypotenuse will be 5. You remember, the square of the hypotenuse equals the sum of the squares of the other two sides. This is the famous “3, 4, 5 triangle” that demonstrates this theorem.

The Pythagorean triples must be integers. That doesn’t seem to limit things much. There are many, many Pythagorean Triples ... as I mentioned previously. Try 5, 12, and 13. Find some others.

This basic math goes back even further than the Greeks. We know the Babylonians knew of this theorem and the triples and the Egyptians used this formula to build the pyramids and make them “square.”

The ancient Chinese and Indian mathematicians knew these formulas too, and they used them to create right triangles for surveying, carpentry, masonry, and construction. It is very useful, practical math.

The Greeks were the ones that focused on the beauty of the math and its possibilities. There certainly is beauty in using the three integers, 3, 4, and 5, to create a right angle. What simple and powerful ideas … so often created and recreated by the mathematicians of many early cultures.

This leads us to the realm of Diphantian Equations. A Diophantine equation is a polynomial equation in which the solutions must be integers. Their name derives from the 3rd-century Alexandrian mathematician, Diophantus, who developed methods for their solution. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively.

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:

aⁿ + bⁿ = cⁿ

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.

He was reading a book on Diphantian mathematics and the particular problem I’ve been discussing. In the margin of the book he wrote, “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.” This became known as “Fermat’s Conjecture,” and later “Fermat’s Last Theorem.” He never provided his proof, although he did provide a proof that is equivalent to proving that solutions with n = 4 would be impossible. He used a method called “infinite descent” to prove that case. But there was no record of his, complete, “marvelous proof.”

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 20^th 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.