At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about primes. These problems were characterised in his speech as "unattackable at the present state of science" and are now known as Landau's problems. They are as follows:
- Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?
- Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?
- Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?
- 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? (sequence A002496 in OEIS).
As of 2012, all four problems are unresolved.
Famous quotes containing the word problems:
“The truth of the thoughts that are here set forth seems to me unassailable and definitive. I therefore believe myself to have found, on all essential points, the final solution of the problems. And if I am not mistaken in this belief, then the second thing in which the value of this work consists is that it shows how little is achieved when these problems are solved.”
—Ludwig Wittgenstein (1889–1951)