Mathematicians needed to higher perceive these numbers that so carefully resemble probably the most elementary objects in quantity idea, the primes. It turned out that in 1899—a decade earlier than Carmichael’s consequence—one other mathematician, Alwin Korselt, had provide you with an equal definition. He merely hadn’t identified if there have been any numbers that match the invoice.

According to Korselt’s criterion, a quantity N is a Carmichael quantity if and provided that it satisfies three properties. First, it will need to have multiple prime issue. Second, no prime issue can repeat. And third, for each prime p that divides N, p – 1 additionally divides N – 1. Consider once more the quantity 561. It’s equal to three × 11 × 17, so it clearly satisfies the primary two properties in Korselt’s listing. To present the final property, subtract 1 from every prime issue to get 2, 10 and 16. In addition, subtract 1 from 561. All three of the smaller numbers are divisors of 560. The quantity 561 is subsequently a Carmichael quantity.

Though mathematicians suspected that there are infinitely many Carmichael numbers, there are comparatively few in comparison with the primes, which made them tough to pin down. Then in 1994, Red Alford, Andrew Granville, and Carl Pomerance revealed a breakthrough paper wherein they lastly proved that there are certainly infinitely many of those pseudoprimes.

Unfortunately, the strategies they developed didn’t enable them to say something about what these Carmichael numbers appeared like. Did they seem in clusters alongside the quantity line, with giant gaps in between? Or may you all the time discover a Carmichael quantity in a brief interval? “You’d think if you can prove there’s infinitely many of them,” Granville stated, “surely you should be able to prove that there are no big gaps between them, that they should be relatively well spaced out.”

In specific, he and his coauthors hoped to show an announcement that mirrored this concept—that given a sufficiently giant quantity X, there’ll all the time be a Carmichael quantity between X and a couple ofX. “It’s another way of expressing how ubiquitous they are,” stated Jon Grantham, a mathematician on the Institute for Defense Analyses who has finished associated work.

But for many years, nobody may show it. The strategies developed by Alford, Granville and Pomerance “allowed us to show that there were going to be many Carmichael numbers,” Pomerance stated, “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 e-mail from Larsen, then 17 years outdated and in his senior yr of highschool. A paper was hooked up—and to Granville’s shock, it appeared appropriate. “It wasn’t the easiest read ever,” he stated. “But when I read it, it was quite clear that he wasn’t messing around. He had brilliant ideas.”

Pomerance, who learn a later model of the work, agreed. “His proof is really quite advanced,” he stated. “It would be a paper that any mathematician would be really proud to have written. And here’s a high school kid writing it.”