Mathematicians want to better understand numbers that closely resemble the most fundamental object in number theory, the prime numbers. It turned out that in 1899—a decade before Carmichael’s results—another mathematician, Alwin Korselt, had come up with an equivalent definition. He simply didn’t know if any of the numbers would fit the bill.
According to Korselt’s criteria, some WOMEN is a Carmichael number if and only if it satisfies three properties. First, it must have more than one prime factor. Second, there are no repeatable prime factors. And third, for all primes P share WOMEN, P – 1 is also divisible WOMEN – 1. Consider the number 561. It is equal to 3 × 11 × 17, so it clearly satisfies the first two properties in the Korselt list. To show the final property, subtract 1 from each prime factor to get 2, 10, and 16. Alternatively, subtract 1 from 561. All three smaller numbers are divisors of 560. Therefore, the number 561 is the Carmichael number.
Although mathematicians suspect that there are infinitely many Carmichael numbers, they are relatively few compared to primes, making them difficult to determine. Then in 1994, Red Alford, Andrew Granvilleand Carl Pomerance Breakthrough announcement paper where they finally proved that there really are an infinite number of these pseudo-primes.
Unfortunately, the techniques they’ve developed don’t allow them to say anything about what those Carmichael numbers look like. Do they appear in clusters along the number line, with a large gap in between? Or can you always find the Carmichael number in a short amount of time? “You would think that if you could prove that there are infinitely many of them,” Granville said, “surely you would be able to prove that there is no great distance between them, that they must be relatively well spaced.”
Specifically, he and his co-authors hope to prove a proposition that reflects this idea – that for a sufficiently large number Xthere will always be some Carmichael between X and 2X. “It’s another way to show how common they are,” said Jon Grantham, a mathematician at the Institute for Defense Analysis who has done related work.
But for decades, no one has proven that. The techniques developed by Alford, Granville and Pomerance “allow us to show that there will be many Carmichael numbers,” Pomerance said, “but do not really allow us to control their entire position. “
Then, in November 2021, Granville opened an email from Larsen, then 17 and in his senior year of high school. One paper was attached – and to Granville’s surprise, it seemed correct. “It’s not the easiest book to read,” he said. “But when I read it, it was clear that he wasn’t messing around. He has great ideas.”
Pomerance, who has read the later version of the work, agrees. “His evidence is actually quite advanced,” he said. “It would be a paper that any mathematician would be really proud to write. And here is a high school student who wrote it.”
