Today I learned that if you don't know about primes but you do know about even and odd, then you can prove that the square root of 5 is irrational. Similarly for all non-square integers up to 17. At 17 you need something more. See this twitter thread by AndrĂ©s E. Caicedo for details.
I want to start telling people that there exists a thing that might be called "non-deterministic linear algebra" and then point them at Sheldon Axler.
Slightly off of pure mathematics, here is a paper on the longest possible chess game. It's kind of great.