Landau's problems

At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau's problems. 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 February 2024, all four problems are unresolved.

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