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.

References
  1. Weisstein, Eric W. "Landau's Problems". mathworld.wolfram.com. Retrieved 2022-08-27.
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