Landau's problems

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:

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? (sequence A002496 in the OEIS).

As of 2009, all four problems are unresolved.

Progress toward solutions

Goldbach's conjecture

Vinogradov's theorem proves Goldbach's weak conjecture for sufficiently large n. Deshouillers, Effinger, te Riele and Zinoviev conditionally proved the weak conjecture under the GRH. The weak conjecture is known to hold for all n outside the range $(10^{20},e^{3100}).$

Chen's theorem proves that for all sufficiently large n, $2n=p+q$ where p is prime and q is either prime or semiprime. Montgomery and Vaughan showed that the exceptional set (even numbers not expressible as the sum of two primes) was of density zero.

Twin prime conjecture

Goldston, Pintz and Yıldırım showed that the size of the gap between primes could be far smaller than the average size of the prime gap:

$\liminf {\frac {p_{n+1}-p_{n}}{{\sqrt {\log p_{n}}}(\log \log p_{n})^{2}}}<\infty .$

Earlier, they conditionally proved a weaker version of the twin prime conjecture, that infinitely many primes p exist with $\pi (p+20)-\pi (p)\geq 1$, under the Elliott-Halberstam conjecture. $\pi (x)$ is the prime-counting function. The twin prime conjecture replaces 20 with 2.

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 $2{\sqrt {p}}.$ A table of maximal prime gaps shows that the conjecture holds to 10¹⁸. A counterexample near 10¹⁸ would require a prime gap fifty million times the size of the average gap.

A result due to Ingham shows that there is a prime between $n^{3}$ and $(n+1)^{3}$ for every large enough n.

Near-square primes

The Friedlander–Iwaniec theorem shows that infinitely many primes are of the form $x^{2}+y^{4}$.

Iwaniec showed that there are infinitely many numbers of the form $n^{2}+1$ with at most two prime factors.

External links

• Weisstein, Eric W. "Landau's Problems". MathWorld.