Saas

4 weeks ago

21

Mathematicians wanted to better understand these numbers, which so closely resemble the most fundamental objects of number theory, the prime numbers. It turned out that in 1899 – ten years before Carmichael’s result – another mathematician, Alvin Corselt, proposed an equivalent definition. He just didn’t know if there were any numbers that fit the bill.

According to the Corselt criterion, the number *H* is a Carmichael number if and only if it satisfies three properties. First, it must have more than one prime factor. Second, no prime factor can be repeated. And thirdly, for every prime *P* what separates *H*, *P* – 1 also divides *H* – 1. Consider again the number 561. It is equal to 3 × 11 × 17, so it explicitly satisfies the first two properties in the Corselt list. To show the last property, subtract 1 from each prime factor to get 2, 10, and 16. Also, subtract 1 from 561. All three smaller numbers are divisors of 560. So 561 is a Carmichael number.

Although mathematicians suspected that there were infinitely many Carmichael numbers, they were relatively few compared to prime numbers, making them difficult to determine. Then in 1994 Red Alford, Andrew Granvilleand Karl Pomerans published a breakthrough paper in which they finally proved that there are indeed infinitely many such pseudoprimes.

Unfortunately, the methods they developed did not allow them to say anything about what these Carmichael numbers looked like. Did they appear in groups along the number line with large gaps between them? Or can you always find Carmichael’s number in a short amount of time? “You might think that if you can prove that there are infinitely many of them,” said Granville, “you can certainly prove that there are no large gaps between them, that they should be relatively well spaced.”

In particular, he and his co-authors hoped to prove a statement reflecting this idea that for a sufficiently large number *X*there will always be a Carmichael number between *X* and 2*X*. “This is another way of showing how ubiquitous they are,” said John Grantham, a mathematician at the Institute for Defense Analysis who has done similar work.

But for decades no one could prove it. The methods developed by Alford, Granville, and Pomerance “allowed us to show that there must be many Carmichael numbers,” Pomerance said, “but didn’t really give us full control over where they would be. ”

Then, in November 2021, Granville opened an email from Larsen, then 17 and in high school. AND paper was attached – and, to Granville’s surprise, it looked right. “It wasn’t the easiest reading,” he said. “But when I read it, it was pretty clear that he wasn’t messing around. He had brilliant ideas.”

Pomerance, who had read a later version of the work, agreed. “His proof is really quite advanced,” he said. “This would be an article that any mathematician would be proud to write. And this is a high school student writing.