Landau's problems

Edmund Landau, German mathematician

Landau's problems are four basic problems about prime numbers. They were listed at the 1912 International Congress of Mathematicians and presented by Edmund Landau.

Problems

These problems were presented in his speech as "unattackable at the present state of mathematics". They are as follows:

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 n² + 1?

As of August 2022, all four problems are unresolved.