Definition:Landau's Problems

Definition

Landau's problems are a set of $4$ (still) unsolved problems about prime numbers listed by Edmund Georg Hermann Landau in an address at the $1912$ International Congress of Mathematicians.

They are as follows:

Goldbach's Conjecture

Every even integer greater than $2$ is the sum of two primes.

Twin Prime Conjecture

It is conjectured that there exist infinitely many pairs of twin primes: that is, primes which differ by $2$.

Legendre's Conjecture

It is not known whether:

$\exists n \in \N_{>1}: \map \pi {n^2 + 2 n + 1} = \map \pi {n^2}$

where $\pi$ denotes the prime-counting function.

That is:

Is there always a prime number between consecutive squares?

Is there an Infinite Number of Primes of Form $n^2 + 1$?

Is there an infinite number of prime numbers of the form $n^2 + 1$?

Historical Note

Landau's problems were listed by Edmund Georg Hermann Landau in an address at the $1912$ International Congress of Mathematicians.

As of November $2019$ they remain unsolved.