The Collatz Conjecture!
Take any positive integer. If it’s divisible by two, divide it by two. Otherwise, multiply by 3 and add 1.
The Collatz Conjecture states that no matter what number you start with, you will eventually reach 1.
A first grader knows enough math to understand it, but it’s still unproven.
Edit: relevant xkcd
Edit 2.0: I’m getting a lot of comments and messages that you can do it by induction or that a computer can check if it’s true.
A computer can’t check infinitely many integers. If a counterexample exists, it might be found by checking by computer, but you can’t prove it true by brute force.
It has been checked through 5∗2605∗260 with no counterexamples found, but that still isn’t sufficient to prove it true for all numbers. After all, that’s only half the numbers below 5∗2615∗261 and less than 0.10.1% of all numbers below 5∗2705∗270.
If you think induction will work, try it. You’ll find a problem in constructing your induction hypothesis.
And showing that after applying 3n+13n+1 you get an even number doesn’t prove anything either…
This problem has been unsolved for about 80 years. You aren’t going to solve it in a day.