Dice Question Streamline Icon: https://streamlinehq.com

Oppermann’s Conjecture: Primes in Intervals Around Squares

Establish whether, for every positive integer x, there exists at least one prime in each interval x(x−1) < p < x^2 and x^2 < q < x(x+1).

Information Square Streamline Icon: https://streamlinehq.com

Background

Oppermann’s conjecture strengthens Legendre’s by demanding primes in both multiplicative intervals immediately below and above each square. It is a prominent open problem closely tied to the size of prime gaps.

The paper references this conjecture while discussing classic questions on prime gaps and later derives interval-prime existence results from its prime-gap bound.

References

For example it has been conjectured by Legendre that given a positive integer $x$, there is at least one prime number between $x2$ and $(x+1)2$, Oppermann(1877) made a slightly stronger conjecture that given a positive integer $x$, there is at least one prime number between $x(x-1)$ and $x2$, and a prime between $x2$ and $x(x+1)$.

— On the Maximal Gap between Primes (2510.17065 - Wang, 20 Oct 2025) in Section 1 (Introduction)