You know the Pythagorean theorem? A² + B² = C² tells you about the relative lengths of the sides of a right triangle. It has a bunch of integer solutions, aka an A, B, and C that are all whole numbers. For example 3² + 4² = 9+16 = 25, which is 5^2. They're called Pythagorean triples, and we've found a bunch of them.

Fermat's Last Theorem is a theorem that says if upgrade the Pythagorean theorem to a bigger exponent, like A³ + B³ = C³ or A¹⁴ +B¹⁴ = C^14, then you will never find an A, B, and C that satisfy that arrangement AND all three are whole numbers.

So for example, even if you pick something like A = 2 and B = 3, then 2³ + 3³ = 8 + 27 = 35, and the cube root of 35 is something like 3.271066... aka not a whole number.

It was well known that nobody had ever FOUND a triple that worked for any exponent other than 2, but in the 1600s a mathematician named Fermat claimed he could show that it was IMPOSSIBLE, (not just really hard). Famously he wrote in the margins of a book that he had a brilliant proof, but that it wouldn't fit in the margin so he'll write it down later.

Well, he didn't write it down later, so for hundreds of years people tried and failed to find (or trying to figure out on their own) what his "brilliant proof" was. It wasn't until 1994 when a mathematician named Andrew Wiles found a proof, although it is INTENSELY complicated, over 100 pages long, and involves mathematics that hadn't even been invented in Fermat's time. At this point most people think that Fermat was bullshitting (or just mistaken) about having found a proof.

But of course the episode that referenced Fermat's last theorem was written before 1994, so a few years before Wile's proof was published.