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Statement

Jacobsthal's function $j(n)$, for a positive integer $n$, is the least $m$ with an integer coprime to $n$ in every block of $m$ consecutive integers, that is, one more than the largest gap between integers coprime to $n$; $\omega(n)$ counts the distinct prime factors of $n$, $\omega(1)=0$; and for $k\ge1$ the manuscript puts

$$h(k)=\sup_{n\ge1,\ \omega(n)\le k}j(n).$$

The interval may start anywhere, and $j(n)$ depends only on the distinct primes dividing $n$, so $h(k)-1$ is the longest interval that the divisibility classes of at most $k$ primes can cover; the manuscript notes that Erdős's $C(k)+1$, the maximum over exactly $k$ distinct prime divisors, equals $h(k)$ because adjoining primes cannot decrease $j(n)$.

Theorem 1.1. For some absolute constant $C>0$,

$$h(k)\le C\,\frac{k^2}{(\log\log(3k))^2}\qquad\text{for all integers }k\ge1.$$

Logarithms are natural. The manuscript presents this as "an affirmative answer" to Jacobsthal's question whether $h(k)\ll k^2$, which it attributes to Erdős's 1962 paper (p. 163, equations (1) and (3)), uniform over arbitrary sets of prime divisors and over the interval's position; the constant is not made explicit and, by the stated conventions, need not be effective.

Source. OpenAI, A quadratic bound for Jacobsthal's function, OpenAI Math Release preprint of 25 September 2026, folder preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026; TeX file sections/introduction.tex, label thm:main (lines 24--30), PDF p. 2; proof in sections/assembly.tex, subsection "Removing the remaining divisor primes" (PDF pp. 72--74). The card openai_2026_quadratic_bound_jacobsthal_function records the provenance, the release's attestations and its Lean listing.

Read depth. Claims checked: the statement, the definitions of $j$, $\omega$ and $h$ and the remark identifying $h(k)$ with Erdős's $C(k)+1$ were read clause by clause in the TeX source. The proof was read for its structure (below) and not checked step by step; its input, Theorem 1.2, is an argument of about 65 pages read for structure only. Nothing here is independently reviewed.

Proof pointer

Section 11.2. By Theorem 1.2 together with Mertens' formula, some absolute $c_0>0$ has the property that, for all large $z$ and any one forbidden class at each prime $p\le z$, at least $c_0Y\log w/L^2$ integers of $[1,Y]$ avoid all the classes, where $L=\log z$, $Y=\lfloor z^2/L^2\rfloor$ and $w=L/(\log L)^2$. For large $k$ take $z=A_0k\log k/\log\log k$ with an absolute $A_0>1$ fixed later, so that $Y\sim A_0^2k^2/(\log\log k)^2$ and $X:=Y/z\sim A_0k/(\log k\log\log k)$. Given $n$ with $\omega(n)\le k$ and an interval of $Y$ consecutive integers starting at any integer $a$, prescribe at each prime $p\le z$ dividing $n$ the class that marks the multiples of $p$ in the interval, and any class at the other primes $p\le z$; the survivors are coprime to every divisor prime up to $z$. Each remaining divisor prime $q>z$ has at most $1+X$ multiples in the interval; enclosing them in a progression of step $q$ and length about $X$ and sieving it by the primes up to $X^\theta$ (which lie below $z$ and whose classes the survivors already avoid), the upper bound of Lemma 2.3 with Mertens' formula shows that $q$ removes at most $C_1(1+X/\log X)$ survivors, uniformly in $q$, the classes and the translation, including $q>Y$. With at most $k$ such primes the total removed is at most $C_1(1/A_0+o(1))\,Y\log w/L^2$, against the survivor lower bound $c_0Y\log w/L^2$, so choosing $A_0>2C_1/c_0$ leaves a survivor coprime to $n$. Hence $h(k)\le Y\ll k^2/(\log\log k)^2$ for large $k$, and $\log\log(3k)/\log\log k\to1$ gives the displayed form. For the finitely many remaining $k$, inclusion--exclusion over the $r\le k$ distinct primes of $n$ gives at least $m\prod_{p\mid n}(1-1/p)-2^r\ge m/(k+1)-2^k$ coprime integers in any $m$ consecutive integers (using $p_j\ge j+1$), so $m_k=(k+1)2^k+1$ bounds $h(k)$ there, and one absolute $C$ covers every $k\ge1$. The argument uses no information about prime multiplicities and allows negative starting points.

Dependencies

Theorem 1.2 of the manuscript (the survivor count; see its page for the inputs behind it); Lemma 2.3 (the small-prime sieve in a progression, derived in the manuscript from the fundamental lemma, Lemma 2.1, which is cited to Sofos 2023 and Friedlander--Iwaniec, Opera de Cribro, Corollary 6.10); Mertens' formula. The deduction itself uses only an upper sieve. External premises are taken at statement level; none was checked here.

Bears on

• Problem 970: the statement is the problem's displayed question $h(k)\ll k^2$, claimed with the extra factor $(\log\log 3k)^{-2}$, for the same function (the page's maximum of $j(n)$ over $\omega(n)\le k$). The order of magnitude, to which the label attaches, stays open between $k(\log k)^2\log_3k/\log_2k$ and this bound. The claim is unverified here; the page's status rests on its acceptance evidence.
• Problem 687: context. The manuscript bounds $h(k)$, which relates to the problem's $Y(x)$ through $Y(x)=j(P(x))-1$ (Ford, Green, Konyagin, Maynard and Tao), where $P(x)$ is the product of the primes up to $x$. The manuscript does not state a bound for $Y$ or name this problem; the page's status rests on its acceptance evidence.
• Problem 929: context. The problem's $S(k)$ is tied to the same covering quantity $Y(x)$; the manuscript does not name this problem, and the page's status rests on its acceptance evidence.
• Iwaniec's Corollary: the bound $C(r)\ll r^2\log^2r$, that is $h(k)\ll(k\log k)^2$, that the manuscript names as the previous best uniform estimate and claims to improve; the claim is unverified here, and it stays the best bound from a refereed source held here.