More than years later, mathematician Andrew Wiles finally closed the book on Fermat's Last Theorem. The most famous note ever scribbled in a book may very well be, "I have a truly marvelous demonstration of this proposition that this margin is too narrow to contain.
In the s, French mathematician Pierre de Fermat jotted that unassuming statement and set a thorny challenge for three centuries' of mathematicians. Fermat never got around to writing down his "marvelous" proof, and the margin note wasn't discovered until after his death. For years, Fermat's statement was known in mathematical circles as Fermat's Last Theorem, despite remaining stubbornly unproved. However, a proof of the more general case -- that is, for all integer values of n greater than 2 -- eluded professionals and amateurs alike until Princeton University's Andrew Wiles brought years of mathematical advances to bear on the problem.
Better known as the Pythagorean theorem, this equation immortalizes the ancient Greek mathematician Pythagoras, who proved that "the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. No one ever found three integers that worked with n greater than 2, and most mathematicians suspected there weren't any.
But no one could prove it. Fermat's Last Theorem first intrigued Wiles as a teenager and inspired him to pursue a career in mathematics, but it wasn't until that a key piece to the puzzle fell into place.
That year, mathematician Ken Ribet showed that solving a modern problem in math, called the Taniyama-Shimura conjecture, would allow you to prove Fermat's Last Theorem. There was one small hitch: No one was really sure how to approach the Taniyama-Shimura conjecture, or even whether it was provable at all. But number theorists were not the only ones electrified by this story. I was reminded of this unexpectedly in when, in the space of a few days, two logicians, speaking on two continents, alluded to ways of enhancing the proof of FLT — and reported how surprised some of their colleagues were that number theorists showed no interest in their ideas.
The logicians spoke the languages of their respective specialties — set theory and theoretical computer science — in expressing these ideas. Over the last few centuries, mathematicians repeatedly tried to explain this contrast, failing each time but leaving entire branches of mathematics in their wake. These branches include large areas of the modern number theory that Wiles drew on for his successful solution, as well as many of the fundamental ideas in every part of science touched by mathematics.
The computer scientist had recently been excited to learn about progress in automated proof verification , an ambitious attempt to implement the formalist approach to mathematics in practice. For formalists, a mathematical proof is a list of statements that meet strict requirements:. Mathematical logic was developed with the hope of placing mathematics on firm foundations — as an axiomatic system, free of contradiction, that could keep reasoning from slipping into incoherence.
Mathematicians never write proofs this way, however. Automated proof verification seems to offer a solution. It entails reformulating the proof as a series of discrete statements, each written in an unambiguous language that can be read by a computer, which then confirms that every step deserves to be stamped with a constitutional certification.
This painstaking method has been applied with success to many long and difficult proofs, most famously by Thomas Hales and his collaborators to the proof of the Kepler conjecture on the densest way to pack spheres. In , Gerhard Frey pointed out that a , b and c could be rearranged into. More precisely, it had long been known how to leverage such an elliptic curve into.
The links between these three steps were all well-understood in Biggest mystery in mathematics in limbo after cryptic meeting Dec The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof Oct Chaos-theory pioneer nabs Abel Prize Mar Mathematician wins award for shaping algebra Mar Fermat's theorem proves elusive to the last. Reprints and Permissions. Castelvecchi, D. Fermat's last theorem earns Andrew Wiles the Abel Prize. Nature , Download citation.
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Mathematician receives coveted award for solving three-century-old problem in number theory. Andrew Wiles in poses next to Fermat's last theorem — the proof of which has won him the Abel prize. The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof.
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