Million Dollar Problems (2000) [pdf]

A quick skim suggests https://en.m.wikipedia.org/wiki/Euler_brick#Perfect_cuboid : it sounds like a simple generalisation from Pythagorean triangles that we are all shown as young children, but it turns out to be intractable so far.

No "small" solutions exist, thanks to exhaustive search, but the prospect that one does exist at a horrifyingly large number remains.

A lot of nice "approachable"-sounding problems here: https://mathoverflow.net/questions/66084/open-problems-with-...

Landau's problems are pretty simple in their statement. I believe that Goldbach's conjecture is the oldest, dating to 1742. So I wouldn't exactly call it approachable in the sense of easy to solve, but the statement is quite simple. The full list of Laundau's problems, from the Wikipedia page ( https://en.wikipedia.org/wiki/Landau's_problems ), is:

1. Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?

2. Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?

3. Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?

4. Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n^2 + 1?

I don't think any of them has a million dollar prize, but tenure at a decent university seems like a fairly reasonable expectation for solving one of these.