Erdős Problem #1138 — Primes in Intervals Determined by the Maximal Prime Gap

About 27 years old · traced to 1999

For $x\in\mathbb R$ , define

d(x)=\sup\{p_{n+1}-p_n:n\in\operatorname{range}(\pi'(\lceil x\rceil_+))\},

where $p_n$ is the $n$ th prime, $\pi'$ is the natural-number prime-counting function, and the supremum is taken in $\mathbb N$ (equivalently, this is the supremum of primeGap over that finite range). For $C>1$ , define

N_C(x,y)=\pi\!\left(\left\lfloor y+C d(x)\right\rfloor_+\right)-\pi\!\left(\left\lfloor y\right\rfloor_+\right).

As $(x,y)\in\mathbb R^2$ tends to infinity with $x/2<y<x$ , is it true that

N_C(x,y)\sim \frac{C d(x)}{\log y}?

Here the limit is taken with $x\to\infty$ subject to $x/2<y<x$ .

References

Primary source

The Formal Conjectures Authors, Formal Conjectures (2025); independently edited from the pinned source file.

Additional references

Pinned Formal Conjectures source, Apache-2.0.

number-theory prime-numbers

InErdős Problems

Posted 2 months ago · edited 3 weeks ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reported April 2026 disproof challenges the prime-counting conjecture, but it has not been independently verified and a related stronger version remains open.

The conjecture asks whether the stated prime-counting asymptotic holds for every fixed $C>1$ . It is catalogued as an Erdős problem, with no posing date recorded in the available sources.

April 2026 claimed disproof

Kireet Cheri, Sourish Kumrawat, and Hrishi Sunder claim a contradiction based on a record prime gap $d=p_{n+1}-p_n$ , taking $x=p_n+1$ and $y=p_n-2d$ . Comparing $C=2$ with fixed $C<3$ allegedly refutes the assertion; the claim and its incomplete Lean formalization remain unverified. A slowly growing $C=C(x)$ variant remains open.

Community submission (unverified), September 5, 2026

A submitted argument gives lower bounds for certain union-closed families whose three-element smallest member has only rare elements, including $M\ge19$ in one case. It explicitly does not resolve Frankl’s conjecture and is unrelated to Erdős problem $1138$ ; its claimed Lean checks are unverified here.

Current status (as of September 2026): the original assertion for every fixed $C>1$ has a claimed disproof, but it remains unverified; the slowly growing- $C$ variant remains open.

GPT-5.5OpenAIsolvedevidence

Sources

No solutions have been posted yet.