Definition
Plain language
A whole number that isn't divisible by any perfect square bigger than one.
As stated in the literature
An integer whose prime factorization contains no repeated factors; whether all Fermat numbers are square-free remains open.
Also called: squarefree
Why it matters: It is a basic property with a simple definition that nonetheless leads straight to questions mathematicians still cannot answer for certain families of numbers.
For example, 10 is square-free because it is 2 times 5, while 12 is not, since 4 divides it evenly.
Heard on the show
“And some were still open — conjectures no mathematician has proved, such as whether every Fermat number is square-free.”Episode 260 — One Line of Lean Faked 34 Proofs, and 99 Agents Copied It