The first proof that many people ever learn, early in high school, is the ancient Greek mathematician Euclid’s proof that there are infinitely many prime numbers. It takes just a few lines and uses no concepts more complicated than integers and multiplication.

His proof relies on the fact that, if there were a finite number of primes, multiplying them all together and adding 1 would imply the existence of another prime number. This contradiction implies that the primes must be infinite.

Mathematicians have a curiously popular pastime: proving it over and over again.

Why bother to do this? For one thing, it’s fun. More importantly, “I think the line between recreational math and serious math is very thin,” said William Gasarch, a professor of computer science at the University of Maryland and author of a new proof posted online earlier this year.

Gasarch’s proof is only the latest in a long succession of novel proofs. In 2018, Romeo Meštrović of the University of Montenegro compiled nearly 200 proofs of Euclid’s theorem in a comprehensive historical survey. Indeed, the whole field of analytic number theory, which uses continuously varying quantities to study the integers, arguably originated in 1737, when the mathematical giant Leonhard Euler used the fact that the infinite series 1 + 1/2 + 1/3 + 1/4 + 1/5 + … diverges (meaning that it doesn’t sum to a finite number), to again prove that there are an infinite number of primes.

Christian Elsholtz, a mathematician at Graz University of Technology in Austria and author of another recent proof, said that instead of proving hard results from many smaller results — what mathematicians do when they systematically assemble lemmas into theorems — he did the opposite. “I use Fermat’s Last Theorem, which is really a nontrivial result. And then I conclude a very simple result.” Working backward like this can reveal hidden connections between different areas of math, he said.

“There’s a little competition out there for people to have the most ridiculously difficult proof,” said Andrew Granville, a mathematician at the University of Montreal and author of two other proofs. “It has to be amusing. Doing something technically awful is not the point. The only way you want to do something difficult is that it’s amusing.”

Granville said that there is a serious point to this friendly one-upmanship. Researchers aren’t just fed questions that they try to solve. “The creation process in mathematics is not about, you just set a task to a machine and the machine resolves it. It’s about somebody taking what they’ve done in the past and using that to create a technique and create a way to develop ideas.”

As Gasarch puts it, “All the papers, they segue from a cute new proof that primes are infinite into serious math. One day you’re just looking at primes, and the next day you are looking at densities of squares.”