A quadratic bound for Jacobsthal’s function

1 · The essentials

Choose a positive integer, then look for an integer that shares no divisor with it other than 1. How many consecutive integers must you inspect to be certain of finding one? The answer depends on how many different prime numbers divide the chosen integer, rather than on its size alone. If there are at most $k$ such primes, a single fixed number $C$ guarantees success in every block of at least $C k^{2}$ consecutive integers. The stated result gives an even sharper limit than this expression. The guarantee holds regardless of the chosen integer or where the block starts. The main result has a Lean formalization.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a row of numbered squares: each prime dividing the chosen integer crosses out every square numbered with one of its multiples. The result guarantees an uncrossed square in every sufficiently long block, with the required length controlled by the number of primes used. The image does not, however, explain why such a guarantee is possible.

Why it matters

The manuscript gives an affirmative answer to Jacobsthal’s quadratic-bound question. The guarantee is uniform: it requires no further restrictions on the primes involved or on the interval’s position. The repository summary describes the result as removing the classical logarithmic loss.

2 · Going further

Let $h (k)$ be the least integer such that every interval of $h (k)$ consecutive integers contains an integer coprime to any prescribed positive integer with at most $k$ distinct prime divisors. The problem is therefore to control the required length using only the number of these divisors, while covering every choice of primes and every interval position.

The main result is $h (k) ≪ k^{2} / (lo g lo g (3 k))^{2}$ . The notation $≪$ means that the left-hand side is bounded above by the right-hand side multiplied by an absolute constant. The manuscript presents this estimate as an affirmative answer to Jacobsthal’s quadratic-bound question; the repository summary also states its consequence that a length of $C k^{2}$ suffices.

The supplied abstract states the result but does not describe the idea of the argument, so it does not support an explanation of the method used. Only one manuscript is supplied for this family, with no companion manuscript to connect to it. The main result has a Lean formalization, but the available information does not detail the scope of that formalization beyond the main result.

Useful notions

Divisibility of integers
Primes and distinct prime divisors
Coprime integers
Logarithms and growth bounds

What was known before / what is new

Before

The source presents the existence of a quadratic bound as Jacobsthal’s question, which the manuscript answers. The repository summary mentions a classical logarithmic loss, without specifying the earlier estimate or its history.

New

The manuscript gives the bound $h (k) ≪ k^{2} / (lo g lo g (3 k))^{2}$ , answering the quadratic-bound question affirmatively. The guarantee covers all relevant sets of prime divisors and all interval positions. The main result has a Lean formalization.

Limits and assumptions

The statement concerns positive integers with at most $k$ distinct prime divisors and intervals of consecutive integers. The constant is absolute, but no numerical value is supplied. The main result is formalized in Lean; the source does not specify which other details or claims are also covered. It describes neither the argument’s method nor any remaining open questions.

Mini glossary

Coprime integers: Two integers are coprime when they share no positive divisor other than 1.
Distinct prime divisors: These are the prime numbers dividing an integer, with each prime counted only once.
Function $h (k)$: This is the minimum length guaranteeing, in every interval of that length, an integer coprime to any prescribed positive integer with at most $k$ distinct prime divisors.
Quadratic bound: This is an upper bound given by a fixed constant multiplied by $k^{2}$ .
Absolute constant: This is a constant independent of $k$ , the chosen primes, and the interval’s position.

Keywords : Jacobsthal’s function · Quadratic bound · Coprime integers

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Answers Jacobsthal's quadratic-bound question: every interval of $C k^{2}$ consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/021.md

Manuscripts in this family

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A quadratic bound for Jacobsthal's function

25/09/2026

Verified by Lean

Let $h (k)$ be the least integer such that every interval of $h (k)$ consecutive integers contains an integer coprime to any prescribed positive integer having at most k distinct prime divisors. We prove $h (k) ≪ k^{2} / (lo g lo g (3 k))^{2}$ , giving an affirmative answer to Jacobsthal's quadratic-bound question.

PDF GitHub folder Lean formalization