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.

When Larsen first set out to show that you can always find a Carmichael number in a short interval, “it seemed that it was so obviously true, how hard can it be to prove?” he said. He quickly realized it could be very hard indeed. “This is a problem which tests the technology of our time,” he said.

In their 1994 paper, Alford, Granville, and Pomerance had shown how to create infinitely many Carmichael numbers. But they hadn’t been able to control the size of the primes they used to construct them. That’s what Larsen would need to do to build Carmichael numbers that were relatively close in size. The difficulty of the problem worried his father, Michael Larsen. “I didn’t think it was impossible, but I thought it was unlikely he’d succeed,” he said. “I saw how much time he was spending on it … and I felt it would be devastating for him to give so much of himself to this and not get it.”

Still, he knew better than to try to dissuade his son. “When Daniel commits to something that really interests him, he sticks with it through thick and thin,” he said.

So Larsen returned to Maynard’s papers—in particular, to work showing that if you take certain sequences of enough numbers, some subset of those numbers must be prime. Larsen modified Maynard’s techniques to combine them with the methods used by Alford, Granville, and Pomerance. This allowed him to ensure that the primes he ended up with would vary in size—enough to produce Carmichael numbers that would fall within the intervals he wanted.

“He has more control over things than we’ve ever had,” Granville said. And he achieved this through a particularly clever use of Maynard’s work. “It’s not easy … to use this progress on short gaps between primes,” said Kaisa Matomäki, a mathematician at the University of Turku in Finland. “It’s quite nice that he’s able to combine it with this question about the Carmichael numbers.”