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 mathematics" 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 OEIS).

As of 2016, all four problems are unresolved.

Progress toward solutions

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, $2n=p+q$ 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.

Tomohiro Yamada proved an explicit version of Chen's theorem: every even number greater than $e^{e^{36}} \approx 1.7\cdot10^{1872344071119348}$ is the sum of a prime and a product of at most two primes.

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. 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 $2 \sqrt p$. A table of maximal prime gaps shows that the conjecture holds to 4×10¹⁸. A counterexample near 10¹⁸ would require a prime gap fifty million times the size of the average gap. Matomäki shows that there are at most $x^{1/6}$ exceptional primes followed by gaps larger than $\sqrt{2p}$; in particular,

$\sum_{\stackrel{p_{n+1}-p_n>x^{1/2}}{x\le p_n\le 2x}}p_{n+1}-p_n\ll x^{2/3}.$

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.

Ankeny proved that, under the extended Riemann hypothesis for L-functions on Hecke characters, there are infinitely many primes of the form $x^2+y^2$ with $y=O(\log x)$.

Deshouillers & Iwaniec, improving on Hooley and Todd, show that there are infinitely many numbers of the form $n^2+1$ with greatest prime factor at least $n^{1.2}$. Replacing the exponent with 2 would yield the conjecture.

In the opposite direction, the Brun sieve shows that there are $O(\sqrt x/\log x)$ such primes up to x.

External links

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