Legendre's Conjecture

Open Question

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?


Source of Name

This entry was named for Adrien-Marie Legendre.


Landau's Problems

This is the $3$rd of Landau's problems.


Sources

  • 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Example $2.1.1$
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