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:
““What we know, is a point to what we do not know.” Open any recent journal of science, and weigh the problems suggested concerning Light, Heat, Electricity, Magnetism, Physiology, Geology, and judge whether the interest of natural science is likely to be soon exhausted.”
—Ralph Waldo Emerson (1803–1882)