Legendre's conjecture

Still open as of 2026.

The conjecture is attributed to Legendre, and in 1912 Edmund Landau included it among four basic unsolved problems about primes. If true, the gap after a prime $p$ would be at most of order $\sqrt{p}$. Even the Riemann hypothesis only gives gaps of order $\sqrt{p}\log p$, which is not quite enough. The prime number theorem predicts about $n/\ln n$ primes in each interval, and almost all such intervals are known to contain about that many. A result of Ingham shows that for large $n$ there is always a prime between consecutive cubes $n^3$ and $(n+1{)}^3$, but squares are still out of reach.

References

• Legendre's conjecture (Wikipedia)