Landau 3: Legendre's conjecture

Slug: landau-legendre
List: Landau 3 in this notebook's ordering
Status: open; conditional exponent dent $0.22525\to0.22524401991935$
Area: Analytic and computational number theory
Sources: Pintz 2009; Sorenson--Webster 2024; Chamberland--Straub 2026
Started: 2026-08-27

Statement

For every integer $n\geq 1$ , there is a prime $p$ such that

n^2<p<(n+1)^2.

Equivalently,

\pi((n+1)^2)-\pi(n^2)>0

for every $n\geq 1$ . A global bound

p_{k+1}-p_k<2\sqrt{p_k}

would imply Legendre's conjecture, but it is stronger and is not an equivalent formulation.

The conjecture remains open, even assuming the Riemann hypothesis. The unconditional Baker--Harman--Pintz exponent $0.525$ for prime gaps gives primes between sufficiently large consecutive powers $x^\alpha$ and $(x+1)^\alpha$ only when $\alpha>1/(1-0.525)\approx2.1053$ . Ingham's work gives a prime between sufficiently large consecutive cubes.

Published computational record

Sorenson and Webster verified the stronger Oppermann conjecture for every integer

n\leq N=70{,}500{,}000{,}000{,}000.

Thus both $(n^2,n(n+1))$ and $(n(n+1),(n+1)^2)$ contain primes throughout that range. In particular, Legendre is checked through

N^2=4{,}970{,}250{,}000{,}000{,}000{,}000{,}000{,}000{,}000.

This supersedes the older shorthand that the conjecture was checked only through $2^{64}$ by maximal-prime-gap tables. The Prime Gap List community now reports exhaustive gap analysis through $10^{20}$ , but that is still far below the square-height reached by Sorenson and Webster.

What would count as a new result

A verified finite extension past $N=7.05\cdot10^{13}$ , with witnesses or primality certificates and an independent verifier.
A smaller explicit exponent than the published conditional threshold $\delta=0.2253$ in the Chamberland--Straub theorem on primes between $x^{2+\delta}$ and $(x+1)^{2+\delta}$ for all real $x\geq1$ , still under RH.
An independently replayed documented range, or a reproducible near-miss table, with its scope stated exactly.

An incomplete upstream-data reconstruction or unfinished search is not a finite lower bound. None of these outcomes proves Legendre's conjecture.

Results through 2026-09-04

Assume RH. For every real $x\geq1$ and every

\delta\geq\frac{4504880398387}{20000000000000} =0.22524401991935,

the interval $[x^{2+\delta},(x+1)^{2+\delta}]$ contains a prime. This improves the earlier notebook value $901/4000=0.22525$ , which had tightened the $0.2253$ printed by Chamberland and Straub. The new exact certificate retains the positive quadratic Taylor term in their inequality (6). Their simpler condition (5) fails at the new rational value, but the strengthened splice inequality holds and is increasing from the splice onward. This does not remove RH and does not settle the exponent $2$ case.

The computation also independently verifies both Oppermann halves for the 100,000 integers

2^{32}-100000\leq n<2^{32},

ending at square-height $2^{64}$ . This replay is far below the published record and is not a record extension.

Finally, the checked-in public OLC worker logs at the pinned upstream commit have four holes and stop at $n=31{,}894{,}400{,}000{,}352$ . This is an incomplete public-data reconstruction only. It does not revise the peer-reviewed $N=7.05\cdot10^{13}$ result.

Replay the original finite certificates with problems/landau-legendre/compute/q1/run_all.sh and the new exponent certificate with problems/landau-legendre/compute/q3/run_all.sh.