Pierre de Fermat's Link to a High School Student's Prime Math Proof

> When you start exploring the converse of Fermat’s little theorem, you’ll discover that it’s true for a lot of numbers. For example, for any integer a, the number a^2–a is divisible by 2. You can see this by factoring a^2–a as a×(a−1). Since a and a−1 are consecutive integers, one of them has to be even, and so their product must be divisible by 2.

This is Fermat's little theorem itself, NOT its converse. The latter would say that if a^p-a is divisible by p, then p is prime. To explore that, you would of course try non-prime p, as it holds vacuously for prime p.

So the the author continues to "explore the converse" by trying p=3 and p=5.

> In the 1800s, Bertrand’s postulate showed that for any n>3, there is always a prime number between n and 2n.

Why not state the weakest possible condition of n>1 ?!

> a new result about how Carmichael numbers are distributed.

I was a bit surprised to see the article end here, as I was expecting to learn a little about just what this distribution is like. Fortunately, the other "story of Daniel Larsen" article that this one links to does say more on that:

> In fact, Larsen’s argument didn’t just allow him to show that a Carmichael number must always appear between X and 2X. His proof works for much smaller intervals as well.

Carmichael numbers are interesting indeed. Their very existence indicates that the "true nature" of primes is more obscure and complicated than people of the earlier ages thought. Some applies to regular and irregular primes. If all primes were regular, Fermat's Last Theorem would have been a lot easier to prove. It is possible that Fermat himself was mistaken and saw the outlines of the proof for regular primes.

Also, maths is one of the last remaining sciences where a fresh insight from a relative newbie can still move major, century-old obstacles.