There’s a New Way to Count Prime Numbers

A new proof has brought mathematicians one step closer to understanding the hidden order of those “atoms of arithmetic,” the prime numbers.

The primes—numbers that are only divisible by themselves and 1—are the most fundamental building blocks in math. They’re also the most mysterious. At first glance, they seem to be scattered at random across the number line. But of course, the primes aren’t random. They’re completely determined, and a closer look at them reveals all sorts of strange patterns, which mathematicians have spent centuries trying to unravel. A better understanding of how the primes are distributed would illuminate vast swaths of the mathematical universe.

But while mathematicians have formulas that give an approximate sense of where the primes are located, they can’t pinpoint them exactly. Instead, they’ve had to take a more indirect approach.

Around 300 BCE, Euclid proved that there are infinitely many prime numbers. Mathematicians have since built on his theorem, proving the same statement for primes that meet additional criteria. (A simple example: Are there an infinite number of primes that don’t contain the number 7?) Over time, mathematicians have made these criteria stricter and stricter. By showing that there are still infinitely many primes that satisfy such increasingly rigid constraints, they’ve been able to learn more about where the primes live.

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