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:

As of 2017^[update], all four problems are unresolved.

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, ${\displaystyle 2n=p+q}$ 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 ${\displaystyle e^{e^{36}}\approx 1.7\cdot 10^{1872344071119348}}$ is the sum of a prime and a product of at most two primes.

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