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Fnord666 writes:

Teenager Solves Stubborn Riddle About Prime Number Look-Alikes

Over a century ago, in that quest for a fast, powerful primality test, mathematicians stumbled on a group of troublemakers — numbers that fool tests into thinking they're prime, even though they're not. These pseudoprimes, known as Carmichael numbers, have been particularly difficult to grasp. It was only in the mid-1990s, for instance, that mathematicians proved there are infinitely many of them. Being able to say something more about how they're distributed along the number line has posed an even greater challenge.

Then along came Larsen with a new proof about just that, one inspired by recent epochal work in a different area of number theory. At the time, he was just 17.

[...] Mathematicians saw the potential for a perfect test of whether a given number is prime or composite. They knew that if N is prime, b^N – b is always a multiple of N. What if the reverse was also true? That is, if b^N – b is a multiple of N for all values of b, must N be prime?

Alas, it turned out that in very rare cases, N can satisfy this condition and still be composite. The smallest such number is 561: For any integer b, b⁵⁶¹ – b is always a multiple of 561, even though 561 is not prime. Numbers like these were named after the mathematician Robert Carmichael, who is often credited with publishing the first example in 1910 (though the Czech mathematician Václav Šimerka independently discovered examples in 1885).

In particular, he (Andrew Granville) and his co-authors hoped to prove a statement that reflected this idea — that given a sufficiently large number X, there will always be a Carmichael number between X and 2X. "It's another way of expressing how ubiquitous they are," said Jon Grantham, a mathematician at the Institute for Defense Analyses who has done related work.

But for decades, no one could prove it. The techniques developed by Alford, Granville and Pomerance "allowed us to show that there were going to be many Carmichael numbers," Pomerance said, "but didn't really allow us to have a whole lot of control about where they'd be."

Then, in November 2021, Granville opened up an email from Larsen, then 17 years old and in his senior year of high school. A paper was attached — and to Granville's surprise, it looked correct. "It wasn't the easiest read ever," he said. "But when I read it, it was quite clear that he wasn't messing around. He had brilliant ideas."

Pomerance, who read a later version of the work, agreed. "His proof is really quite advanced," he said. "It would be a paper that any mathematician would be really proud to have written. And here's a high school kid writing it."

The key to Larsen's proof was the work that had drawn him to Carmichael numbers in the first place: the results by Maynard and Tao on prime gaps.

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