Rational or Not? This Basic Math Question Took Decades to Answer.

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Rational or Not? This Basic Math Question Took Decades to Answer.:

In June 1978, the organizers of a large mathematics conference in Marseille, France, announced a last-minute addition to the program. During the lunch hour, the mathematician Roger Apery would present a proof that one of the most famous numbers in mathematics - "zeta of 3," or (3), as mathematicians write it - could not be expressed as a fraction of two whole numbers. It was what mathematicians call "irrational."

Conference attendees were skeptical. The Riemann zeta function is one of the most central functions in number theory, and mathematicians had been trying for centuries to prove the irrationality of (3) - the number that the zeta function outputs when its input is 3. Apery, who was 61, was not widely viewed as a top mathematician. He had the French equivalent of a hillbilly accent and a reputation as a provocateur. Many attendees, assuming Apery was pulling an elaborate hoax, arrived ready to pay the prankster back in his own coin. As one mathematician later recounted, they "came to cause a ruckus."

The lecture quickly descended into pandemonium. With little explanation, Apery presented equation after equation, some involving impossible operations like dividing by zero. When asked where his formulas came from, he claimed, "They grow in my garden." Mathematicians greeted his assertions with hoots of laughter, called out to friends across the room, and threw paper airplanes.

But at least one person - Henri Cohen, now at the University of Bordeaux - emerged from the talk convinced that Apery was correct. Cohen immediately began to flesh out the details of Apery's argument; within a couple of months, together with a handful of other mathematicians, he had completed the proof. When he presented their conclusions at a later conference, a listener grumbled, "A victory for the French peasant."

Once mathematicians had, however reluctantly, accepted Apery's proof, many anticipated a flood of further irrationality results. Irrational numbers vastly outnumber rational ones: If you pick a point along the number line at random, it's almost guaranteed to be irrational. Even though the numbers that feature in mathematics research are, by definition, not random, mathematicians believe most of them should be irrational too. But while mathematicians have succeeded in showing this basic fact for some numbers, such as and e, for most other numbers it remains frustratingly hard to prove. Apery's technique, mathematicians hoped, might finally let them make headway, starting with values of the zeta function other than (3).

"Everyone believed that it [was] just a question of one or two years to prove that every zeta value is irrational," said Wadim Zudilin of Radboud University in the Netherlands.

But the predicted flood failed to materialize. No one really understood where Apery's formulas had come from, and when "you have a proof that's so alien, it's not always so easy to generalize, to repeat the magic," said Frank Calegari of the University of Chicago. Mathematicians came to regard Apery's proof as an isolated miracle.

But now, Calegari and two other mathematicians - Vesselin Dimitrov of the California Institute of Technology and Yunqing Tang of the University of California, Berkeley - have shown how to broaden Apery's approach into a much more powerful method for proving that numbers are irrational. In doing so, they have established the irrationality of an infinite collection of zeta-like values.

Jean-Benoit Bost of Paris-Saclay University called their finding "a clear breakthrough in number theory."

Mathematicians are enthused not just by the result but also by the researchers' approach, which they used in 2021 to settle a 50-year-old conjecture about important equations in number theory called modular forms. "Maybe now we have enough tools to push this kind of subject way further than was thought possible," said Francois Charles of the Ecole Normale Superieure in Paris. "It's a very exciting time."

Whereas Apery's proof seemed to come out of nowhere - one mathematician described it as "a mixture of miracles and mysteries" - the new paper fits his method into an expansive framework. This added clarity raises the hope that Calegari, Dimitrov and Tang's advances will be easier to build on than Apery's were.

"Hopefully," said Daniel Litt of the University of Toronto, "we'll see a gold rush of related irrationality proofs soon."

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