# Legendre's Conjecture

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## 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 $2.1.1$