Mathematicians wanted to better understand these numbers, which are very similar to prime numbers, the most basic objects of number theory. Carmichael’s results turn out that ten years earlier in his 1899, another mathematician, Alwyn Coselt, had come up with an equivalent definition. He simply didn’t know if there were any numbers to fit the bill.
According to Coselt’s criterion, the number N. is a Carmichael number only if it satisfies three properties. First, we need multiple prime factors. Second, prime factors cannot be repeated.And third, for each prime number p it divides N., p – also divide by 1 N. – 1. Consider the number 561 again. This equals 3 x 11 x 17, so it clearly meets Korselt’s first two traits on his 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 the number 561 is a Carmichael number.
Mathematicians thought there were an infinite number of Carmichael numbers, but they were few in number compared to primes and difficult to identify. Then, in 1994, Red Alford, Andrew Granville, and Carl Pomerance published a landmark paper that finally proved that there are infinitely many of these pseudoprimes.
Unfortunately, the method they developed could not say anything about what the Carmichael number would look like. Did they appear as clusters along the number line, with large gaps between them? Or could you always find Carmichael numbers at short intervals? We should be able to prove they’re relatively well apart, with no big gaps in between,” Granville said.
In particular, he and his co-authors hoped to prove statements that reflected this idea. Xthere is always a Carmichael number between X and twoX“It’s just another way of describing how ubiquitous they are,” says John Grantham, a mathematician at the Institute for Defense Analysis who has done related work.
But for decades, no one has been able to prove it. The method developed by Alford, Granville, and Pomerance “allowed us to show that there are many Carmichael numbers,” Pomerance said. “
Then, in November 2021, Granville opened an email from Larsen, who was 17 years old and a senior in high school at the time. A document was attached, and to Glanville’s surprise, it appeared to be correct. “It wasn’t the most readable ever,” he said. “But when I read it, it was clear he wasn’t messing around. He had a great idea.”
Pomerans, who had read a later version of the work, agreed. “His proof is very advanced,” he said. “It would be a paper that any mathematician would be really proud to have written. And this is being written by a high school student.”
