Fermat’s last theorem, a riddle put forward by one of history’s great mathematicians, had baffled experts for more than 300 years. Then a genius toiled in secret for seven years to solve it, according to the usual narrative. That shy Englishman, Andrew Wiles, made his feat public in the early 1990s and amassed a glittering array of tributes. In 2016, he won the Abel Prize, math’s top award. It came with a $700,000 purse.
Now, a wealthy Texas philanthropist is recounting how his financial support created a community of Fermat innovators that, over decades, lent moral and mathematical support to Dr. Wiles. That patronage drew top mathematicians to the puzzle after great minds had given up, succeeded in bringing the moribund field back to life, and may have helped make Dr. Wiles’s breakthrough possible.
“We solved the problem,” the philanthropist, James M. Vaughn Jr., 82, president of the Vaughn Foundation Fund, said in an interview. “If we hadn’t put the program together as we did, it would still be unsolved.”
In interviews, top experts described Mr. Vaughn’s foundation and its early financial support as sparks that had lit an intellectual fire, although they stopped short of saying that his backing had been responsible for Dr. Wiles’s Fermat breakthrough. Dr. Wiles did not respond to inquiries.
Recently, Mr. Vaughn gave the University of Texas a collection of 125 rare and foundational books in the history of mathematics, and the gift has prompted him to speak publicly of other foundation projects that have gone largely unheralded.
While gregarious, Mr. Vaughn, heir to a Texas oil fortune, is an extremely private man who has never before claimed publicly that his philanthropy begot the mathematical feat. Even so, he takes immense pride in what he characterizes as his legacy. Mr. Vaughn said that he and his wife had no children and that the Fermat triumph was how he hoped he would be remembered.
“It was very important to have someone like Vaughn doing this,” said Dorian Goldfeld, a professor of mathematics at Columbia University who worked closely with Mr. Vaughn. “It made the problem more visible to more people.” Mr. Vaughn’s financial aid, he added, eventually led to wide Fermat collaborations, including an early gathering that Dr. Wiles helped organize.
“It’s really great to have people like Vaughn,” said Neal I. Koblitz, a professor of mathematics at the University of Washington who edited a Fermat book for the Vaughn Foundation. “Nobody was working on the problem. Vaughn had the money and the interest.”
It seems unlikely that Mr. Vaughn had a direct impact on what turned out to be the Byzantine math of the Fermat proof. But in science, private donors often act as pathfinders for government investment in difficult research. So it was with Mr. Vaughn. He led, and Washington followed. In the 1970s and 1980s, he directed millions of dollars to Fermat conferences, authors and researchers, giving the old field new life and social acceptability.
Subsequently, in the early 1990s, the National Science Foundation, a federal agency, lent its support. It provided Dr. Wiles, then at Princeton University, with math research grants totaling nearly a half-million dollars.
It turns out that both men sought to advance an arcane branch of mathematics known as elliptic curves. The field’s equations build up sets of simple geometric forms that can lead to infinite runs of subsidiary equations and, when solved, solutions to problems of blinding complexity. The exotic forms led Dr. Wiles to the Fermat breakthrough.
Dr. Wiles, now 68 and known as Sir Andrew after being named a knight commander of the British Empire, made no response to repeated emails asking his view on Mr. Vaughn’s claim of having made his feat possible. According to his biographer, Dr. Wiles comes across as a “diffident” man who dislikes publicity.
While historians of science may one day debate whether Mr. Vaughn should share a measure of credit for the Wiles breakthrough, the body of developing evidence already makes the Texan’s blunt declaration seem less like a yarn than a reasonable possibility.
“He really deserves the recognition,” Dr. Goldfeld said of Mr. Vaughn.
Pierre de Fermat was a French lawyer of the 17th century who pursued math as a hobby. After his death, appraisals of his work revealed him to be a giant. He helped lay the foundations of calculus and probability theory.
Fermat also left behind a large body of what he called theorems. The general claims rest on chains of logic. Fermat, however, was something of a tease. He often asserted the truth of a proposition but gave no details. Skeptical experts found to their surprise that many of his sketchy claims were, in fact, true.
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The exception came to be known as Fermat’s last theorem. Early in his career, around 1637, he had scribbled the equation in a book’s margins, claiming a marvelous proof, but called the space too small for particulars. Over time, his reputation drove thousands of mathematicians to take on the problem.
Part of the appeal was basic. The simple equation had just three elements but an infinite number of possible solutions. The challenge was to find ways of bounding the infinite.
Leonhard Euler was an 18th-century Swiss mathematician who solved hundreds of knotty problems in acoustics, finance, navigation and many other fields. In 1753, he announced that he had solved an aspect of Fermat’s theorem. It was the first such advance in a century of grueling effort.
After that, little progress was made, and top mathematicians came to see the riddle as irrelevant. Carl Gauss, a German savant of the 19th century, called it an isolated claim of “very little interest.”
Fascination with the Fermat riddle nonetheless lingered among a subset of mathematicians, professional and amateur.
Mr. Vaughn was born on July 11, 1939. His family lived in Tyler, Texas, near one of the world’s great oil strikes — the East Texas gushers of 1930.
The Vaughns grew rich from the resulting economic boom. Mr. Vaughn’s father and paternal grandfather were both physicians who became successful businessmen and investors in oil and its associated fields, such as steel pipes for drilling into layers of bedrock.
Mr. Vaughn was the kind of boy who liked nothing better than scanning the sky with his family’s telescope. In third grade, he recalled, his fascination with the starry dome caught the attention of his future wife, Salle Werner. She eventually became an artist.
In 1961, Mr. Vaughn graduated from the University of Texas at Austin, and he soon became a Fermat devotee. A main catalyst was a book published that year, “The Last Problem,” by Eric Temple Bell, a mathematician at the California Institute of Technology. Boldly, Dr. Bell declared that if civilization ended, the riddle would in all likelihood remain unsolved.
In 1963, Andrew Wiles, 10 years old, picked up the same book at a small public library in his hometown, Cambridge, England. As with Mr. Vaughn, it changed his life.
Mr. Vaughn studied math in graduate school at the University of Texas but never got an advanced degree. Back in Tyler, looking for direction, he decided to establish a foundation that would fund basic research on the Fermat question. He did so in 1972.
To his surprise, no mathematician would take his financial aid or even admit to being interested in working on the conundrum. “It was seen as throwing money away,” he recalled. The reputation of the riddle, he added, “was that fierce; it was two years before I could give any grants.”
During the same years, Dr. Wiles was pursuing math studies in graduate school at the University of Cambridge and had put his Fermat dream on hold. His adviser, John Coates, urged him take up research on elliptic curves. Unknown to either man, the obscure field would eventually help the young mathematician solve the famous puzzle.
In 1977, Dr. Wiles went to Harvard, where he helped perfect a new take on elliptic curves. It addressed a problem known as Iwasawa theory, after Kenkichi Iwasawa, a Princeton mathematician.
By then, Mr. Vaughn was busy revitalizing the hunt for a Fermat solution and starting to heed his advisers on the wisdom of pursuing elliptic-curve studies.
An early grantee was Harold M. Edwards, a mathematician at the Courant Institute of New York University who died in 2020. His 1977 book, “Fermat’s Last Theorem,” became a classic that was often reprinted and cited, and quite likely inspired other titles.
Mr. Vaughn also promoted the basics. He gave so generously to a 1978 fund-raising drive of the Mathematical Association of America that the society named its new national headquarters in Washington, D.C., after his grandfather. It became the Edgar H. Vaughn Building. Later, the group republished “The Last Problem,” Dr. Bell’s inspirational book.
In September 1981, Mr. Vaughn funded the world’s first big conference on the Fermat riddle. It took place at Endicott House, an M.I.T. meeting center near Boston set in a French manor-style mansion on leafy grounds. The organizers were Dr. Goldfeld of Columbia, Dr. Edwards of N.Y.U., Dr. Koblitz of the University of Washington, Nicholas Katz of Princeton University and two Harvard mathematicians: Barry Mazur and Dr. Wiles.
The conference drew 76 participants, 16 from abroad, and the mathematicians presented 25 research papers. It was a dramatic shift from the early lack of interest. The attendees included Dr. Coates, the doctoral adviser to Dr. Wiles; Dr. Iwasawa, the Princeton professor; and Atle Selberg, a giant of mathematics who later won the Abel Prize.
In 1982, the proceedings were published as “Number Theory Related to Fermat’s Last Theorem.” It was part of a series, “Progress in Mathematics,” for which Dr. Coates was a co-editor.
In the book’s preface, Dr. Goldfeld credited Mr. Vaughn with the idea for the conference and thanked him for supporting it and “pure mathematics in general.” A half dozen of the book’s chapters, including one Dr. Wiles co-wrote, addressed Iwasawa theory and elliptic curves.
Some attendees complained to Mr. Vaughn that direct attacks on the Fermat question had been sidelined by the elliptic-curve focus, Dr. Goldfeld recalled Mr. Vaughn as saying. But Mr. Vaughn, he added, “was right in the end” to have embraced the esoteric subspecialty.
Shortly after the Boston conference, Mr. Vaughn aimed higher. As a “grand benefactor,” he helped fund a gathering in 1986 of the International Congress of Mathematicians, the world’s largest math body. The weeklong math fest was held in Berkeley, Calif.
On its sidelines, a discovery hinted at possible Fermat progress. It occurred over cappuccino as Dr. Mazur of Harvard met with Ken Ribet, a Berkeley math professor. As Dr. Ribet described his most recent work on the Fermat question, Dr. Mazur, fresh from the Boston conference, gaped at him in wonder. “But don’t you see?” he asked, according to “Fermat’s Enigma,” the author Simon Singh’s account of its solving. “You’ve already done it!”
What Dr. Ribet had done — in outline and unknowingly — was tie a possible Fermat solution to an elliptic-curve puzzle known as the Taniyama-Shimura conjecture. He united worlds. If the conjecture could be proved true, so would the famed theorem. Now, after centuries of failure, the surprise linkage offered new hope.
As news of the breakthrough spread, many mathematicians began to doubt that it would spur an advance. The conjecture, after all, had resisted proof for decades. Why would anyone succeed now?
In contrast, Dr. Wiles was electrified. “I knew that moment that the course of my life was changing,” he recalled in an interview with Dr. Singh. Thus began seven years of solitude in which he traded math conferences and departmental colloquia for his cluttered office at Princeton and, when possible, his attic study. It was a risky move in a field that thrived on the open exchange of ideas.
In contrast, Mr. Vaughn in the late 1980s was becoming more social, not less. He was on math visiting committees at Harvard and Princeton.
Then, quite suddenly, at the peak of his influence, Mr. Vaughn’s world fell apart. The reason was an audit of his foundation by the Internal Revenue Service.
It ran from 1988 to 1992 and accused the charity of major improprieties. In the interview, Mr. Vaughn called the audit “very unpleasant” and “extremely traumatic.” Ultimately, he said, the tax agency threatened to impose $15 million in fines and to incarcerate him and his wife, who was on the foundation’s board.
According to Mr. Vaughn, the root problem was the ignorance of the I.R.S. auditor, who felt that mathematical research was “a boondoggle.” In the end, Mr. Vaughn added, “We won every one of the charges,” but the Vaughns had nonetheless come to a turning point. The foundation decided to end its funding of basic mathematics.
“We figured it was too dangerous,” Mr. Vaughn recalled. “You don’t have that kind of trouble if you give to a ballet company.”
In an interview, Bruce I. Friedland, an I.R.S. spokesman, said the agency was prohibited by law from commenting on individual tax matters and audits.
In the audit’s aftermath, the Vaughns kept up their philanthropy but emphasized the arts. They gave to the Metropolitan Museum of Art, the Morgan Library & Museum, the Museum of Fine Arts, Boston, the National Gallery of Art and the Museum of Fine Arts, Houston. They also donated some of Mrs. Vaughn’s artwork.
In June 1993, as the Vaughns reinvented themselves, Dr. Wiles came out of seclusion to announce that he had solved the problem. The New York Times ran its story atop the front page: “At Last, Shout of ‘Eureka!’ In Age-Old Math Mystery.”
Mr. Vaughn was thrilled. “We think he’s got it,” he told a reporter in one of his rare comments in the news media. The public response was so enthusiastic that a clothing chain asked Dr. Wiles to endorse its new line of men's wear.
Then disaster struck. The proof turned out to harbor a major flaw, and Dr. Wiles once again withdrew, this time seeking to rectify the error. As mathematicians clashed over the blunder’s import, he called in Richard Taylor, a former student, for assistance.
Finally, in May 1995, nearly two years after the Wiles announcement, the revised proof was published. The maze of equations ran to 130 pages. After more than three centuries of effort, the Fermat infinities had finally been surmounted, and civilization, amazingly, was still intact. Not only that, but experts hailed the proof as establishing a series of unexpected finds that promised to open new frontiers.
Despite Mr. Vaughn’s role in the field’s resurrection, the subsequent tributes paid him little if any note. At that point, it had been several years since his foundation had funded research on the theorem, reducing the charity’s visibility. And Mr. Vaughn in any case tended to be inaccessible, especially to the news media.
As a result, only a small number of math experts knew of his early patronage. Their numbers are dwindling today as mathematicians who once worked with Mr. Vaughn begin to pass away.
It has been decades since Mr. Vaughn ceased his math philanthropy, but he recently donated his collection of rare math books to his alma mater. They include a first edition of Newton’s masterwork, “Principia,” or “The Principles,” as well as dozens of volumes by such distinguished figures as Euler and Gauss.
“They’re incredible,” Aaron Pratt, curator of early books at the University of Texas, said of the volumes, which now reside at the school’s Harry Ransom Center, a collection of collections.
How the Vaughn library got to the university is another twist in a story full of surprises. Mr. Vaughn said the books, after being seized by the I.R.S. during the audit, had been lost for a time. They ended up in the hands of the Texas Comptroller of Public Accounts, a resting place for unclaimed property.
Chris Bryan, an agency spokesman, said the agency had eventually succeeded in tracking down Mr. Vaughn and had helped arrange for the library to be transferred to his alma mater. “They could have been lost to history,” Mr. Bryan said of the books.
Mr. Vaughn’s re-emergence as a patron of mathematics seems to have rekindled his early interest in science. Before turning to math, he had dreamed of being an astronomer, and he said he was now considering whether to direct some of his philanthropy to that field.
Still, he expressed no regrets about his mathematical past and seemed confident that history would see him as playing a decisive role in the Fermat breakthrough.
“Things worked out just very well,” he said. “We solved the problem.”
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