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 mathematics" and are now known as Landau's problems. They are as follows:
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- 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 the OEIS).
As of 2017, all four problems are unresolved.
Goldbach's conjecture
Vinogradov's theorem proves Goldbach's weak conjecture for sufficiently large n. In 2013 Harald Helfgott proved the weak conjecture for all odd numbers greater than 5.
Chen's theorem proves that for all sufficiently large n,
Tomohiro Yamada proved an explicit version of Chen's theorem: every even number greater than
Twin prime conjecture
Yitang Zhang showed that there are infinitely many prime pairs with gap bounded by 70 million, and this result has been improved to gaps of length 246 by a collaborative effort of the Polymath Project. Under the generalized Elliott–Halberstam conjecture this was improved to 6, extending earlier work by Maynard and Goldston, Pintz & Yıldırım.
Chen showed that there are infinitely many primes p (later called Chen primes) such that p+2 is either a prime or a semiprime.
Legendre's conjecture
It suffices to check that each prime gap starting at p is smaller than
A result due to Ingham shows that there is a prime between
Near-square primes
The Friedlander–Iwaniec theorem shows that infinitely many primes are of the form
Iwaniec showed that there are infinitely many numbers of the form
Ankeny proved that, under the extended Riemann hypothesis for L-functions on Hecke characters, there are infinitely many primes of the form
Deshouillers & Iwaniec, improving on Hooley and Todd, show that there are infinitely many numbers of the form
In the opposite direction, the Brun sieve shows that there are