# Definition:Landau's Problems

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## 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.