Larsen was a high school student in 2022 when he proved a result about a certain kind of number that had eluded mathematicians for decades. He proved that Carmichael numbers — a curious kind of not-quite-prime number — could be found more frequently than was previously known, establishing a new theorem that will forever be associated with his work. So, what are Carmichael numbers? To answer that, we need to go back in time.
Pierre de Fermat has his name on one of the most famous theorems in mathematics. For over 300 years, Fermat’s Last Theorem stood as the ultimate symbol of unachievable mathematical greatness. In the 1600s, Fermat scribbled a note about his proposed theorem in a book he was reading, claiming to know how to prove it without providing any details. Mathematicians attempted to solve the problem themselves until the 1990s, when Andrew Wiles finally proved it using new techniques discovered hundreds of years after Fermat died.
But it’s Fermat’s less famous “little theorem” that relates to Carmichael numbers. Here’s one way to state it: