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.
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.