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.