Landau's problems

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.^[1]

Edmund Landau, German mathematician

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.