Fermat’s Last Theorem (FLT) set off over 350 years of mathematical frenzy

Pierre Fermat was a definite eccentric. He was a practising lawyer in 17th century France. Yet while his name is widely known today, you’d be hard-pressed to find a reference to him as a “17th-century French lawyer". That’s because he was also a serious amateur mathematician and is regarded now as one of the greats. That’s why he is invariably referred to as a “French mathematician." He was also something of a prankster. In his Fermat’s Enigma, the British mathematician Simon Singh writes: “The shy and retiring genius did have a mischievous streak which, when combined with his secrecy, meant that when he did sometimes communicate with other mathematicians, it was only to tease them. He would write letters stating his most recent theorem without providing the accompanying proof. [This] caused a great deal of frustration. René Descartes called Fermat a ‘braggart’ and the Englishman John Wallis referred to him as ‘that damned Frenchman.’"

But as Singh also points out, there was a method to this madness from Fermat. “It meant that he did not have to waste time fully fleshing out his methods; instead, he could rapidly proceed to his next conquest." And that’s what probably prompted his most famous tease, certainly the most famous margin note in mathematical history. After pondering—or so we must believe—a particular mathematical proposition, he wrote it out in the margin of his copy of Arithmetica by the ancient Greek mathematician Diophantus.

This was the proposition: xn + yn = zn has no solutions in which n is greater than 2 and x, y, and z are all integers. (Aside: If n is 2, then we have plenty of solutions, because that is the Pythagoras theorem about right-angle triangles, x2 + y2 = z2. For example,

82 + 152 = 64 + 225 = 289 = 172. Or 32 + 42 = 9 + 16 = 25 = 52. etc, and end of Aside.)

That simple statement, however, wasn’t Fermat’s tease. He also scribbled these words there: “I have discovered a truly remarkable proof which this margin is too small to contain."

That famous little note set off over 350 years of mathematical frenzy, as mathematicians the world over tried to prove what came to be known as Fermat’s Last Theorem (FLT). Professionals and amateurs alike were smitten by the Fermat bug. There must be few mathematics departments around the world that have never received claimed proofs that turned out to be duds. By the late 20th century, many mathematicians had started to think there was no proof at all, let alone one too big for a margin—or at any rate, that Fermat was mistaken about having found one.

All of which changed dramatically in 1993. That’s when a relatively unknown English mathematician called Andrew Wiles announced that he had, indeed, proved Fermat’s Last Theorem. Sensational news! It turned out that he had left some gaps, but he filled them over the next year, and in May 1995, he published his proof in the journal Annals of Mathematics. Not in a margin, though. Wiles’ proof accounted for about 130 pages. Recognition and plenty of awards followed, including being knighted by the Queen in 2000. Much of this Fermat story is now well-known mathematical history. (I once wrote about it, in more detail, here: https://bit.ly/3upj3d6) But the theorem, and Wiles’ work, is in the news again, for an unexpected reason: the doings of an oil baron in Texas.

That baron’s name is James Vaughn. His family made its fortune in the middle of the 20th century, riding the Texas oil boom. Vaughn, now 82, inherited the fortune and the business and is thus extremely wealthy. But his heart was really always in mathematics. Specifically, FLT, about which he read and obsessed. Maybe he even dreamed, as so many who learn of FLT do—present company included—about finding a proof himself. Vaughn didn’t have a degree in mathematics, so in that sense, his interest was strictly as an amateur, if an informed one. He must have realized soon enough that he was not going to prove FLT. But he had another thought: maybe he could use his wealth to help mathematicians prove it.

To that end, he set up a non-profit foundation in 1972. Its aim was to give research grants to mathematicians working on FLT - or, more correctly, working on mathematical themes that were presumed milestones on the way to a proof of FLT. For example, in the late 1970s, the foundation gave “encouragement and generous support" to Jay Goldman, a mathematician at the University of Minnesota. Goldman used this support to translate some celebrated lectures by the early 20th-century German mathematician Erich Hecke. In 1981, Goldman published his translations as Lectures on the Theory of Algebraic Numbers, a text for graduate students.

Earlier than that was a foundation grant to Harold Edwards, a mathematician at New York University. He used it to write his well-known text, Fermat’s Last Theorem. Also, in 1981, the Vaughn Foundation sponsored a major conference that both Harold Edwards and Andrew Wiles helped organize. The papers presented there were gathered into a book whose name says it all: Number Theory Related to Fermat’s Last Theorem: Proceedings of the conference sponsored by the Vaughn Foundation.

Five years later, Vaughn sponsored a much bigger and wider-ranging conference, the 1986 meeting of the International Congress of Mathematics (ICM). Wider-ranging though it was, FLT was a subject of plenty of discussions there. As the now well-known story goes, it was the spark Wiles was probably looking for. Then a professor at Princeton University, he returned from the ICM conference and spent the next seven years working on FLT on his own at Princeton.

The result, of course, was his famous announcement in 1993.

All in all, this is a story with lessons on what it takes to conduct research in mathematics. Because so many problems in the field are so hard, researchers need generous, long-term funding that doesn’t look for quick results, which gives them the time and support to work freely. Governments can offer such support, certainly, but there’s a role for generous private philanthropists like James Vaughn as well. It has lessons also because of what happened to the Vaughn Foundation at about the time Wiles was climbing the FLT mountain. The Internal Revenue Service (IRS)—the tax authorities in the USA - came down on the foundation heavily, alleging various irregularities. Among other things, they seized Vaughn’s collection of invaluable mathematics books.

He eventually fought off all the charges. But as he told the New York Times (“The Texas Oil Heir Who Took On Math’s Impossible Dare", 31 January 2022), he believed the IRS thought of mathematical research as “a boondoggle". Worried about more such measures and attitudes, the foundation withdrew from mathematical research and began funding the arts instead. In late 2021, the IRS returned the texts they had seized to Vaughn. In turn, he donated them to the University of Texas at Austin, his alma mater. As a grateful University press release points out, it was his experience at the university that put him “on the track of a mathematical enigma known as Fermat’s Last Theorem." There are plenty of other enigmas out there in the world of mathematics. Are there other generous philanthropists?

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.

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