Welcome to Quantum Magazine's podcast. Each episode we bring you stories about developments in science and mathematics. I'm Susan Vallett. As the atoms of arithmetic, prime numbers have always occupied a special place on the number line. Now, a graduate student has proved a long-standing conjecture relating prime numbers to a broad class of primitive sets.

And it came as a shock. That's next.

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Jared Duker Lickman is a 26-year-old graduate student at the University of Oxford.

But he's not just any grad student.

Lickman has resolved a well-known conjecture establishing another facet of what makes the primes special

and in some sense even optimal.

It perhaps gives you a larger context

to see how the primes are, in what ways they are unique and special, in what ways they kind of relate to kind of the larger universe of sets.

The conjecture deals with primitive sets, sequences in which no number divides any other.

Since each prime number can only be divided by one and itself, the set of all prime numbers is one example of a primitive set.

So is the set of all numbers that have exactly two or three or 100 prime factors.

Primitive sets were introduced by mathematician Paul Erdisch in the 1930s.

At the time, they were simply a tool that made it easier for him to prove something about a certain class of numbers.

This class is called perfect numbers, and they have roots in ancient Greece.

But they quickly became objects of interest in their own right,

ones that Erdisch would return to time and again throughout his career.

That's because, though their definition is straightforward enough, primitive sets turned out

to be strange beasts. That strangeness could be captured by simply asking how big a primitive

set can get. Consider the set of all integers up to 1,000. All the numbers from 501 to 1,000,

half of the set, form a primitive set, as no number is divisible by any other.

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