---

Call an interval $[u,v]$ bad when the greatest prime factor of $\prod_{u\le m\le v}m$ divides that product to a power higher than the first, and let $B(x)$ count the $n\le x$ lying in at least one bad interval. Tao [Ta26c] proves that the $n\le x$ lying in a bad interval but not satisfying $P(n)^2\mid n$ number at most $\#\{n\le x:P(n)^2\mid n\}/(\log x)^{1-o(1)}$ (Theorem 1.7 of the preprint), so that

$$B(x)=\bigl(1+O((\log x)^{-1+o(1)})\bigr)\,\#\{n\le x:P(n)^2\mid n\},$$

which is the asymptotic the problem asks for, with an explicit relative error. The same preprint settles the site's companion remark on very bad intervals, those whose product is powerful: the $n\le x$ lying in a very bad interval without being powerful themselves number $O(x^{2/5+o(1)})$ (Theorem 1.8), so the count of integers in very bad intervals is asymptotic to the count of powerful numbers, $(\zeta(3/2)/\zeta(3))\sqrt{x}$. The proof of Theorem 1.7 (Section 6 of the preprint) sorts the bad intervals $\{N+1,\dots,N+H\}$ with $H>1$ by length. Such an interval contains no prime and contains an element $p_0^2m$, where $p_0>H$ is the largest prime factor of its product. Intervals with $H\ge x^{0.14}$ cover few integers, because almost every interval of that length contains a prime (Proposition 2.3(iii), an application of the Guth–Maynard zero-density estimate); once $H<x^{0.14}$, the intervals with $p_0>x^{0.15}$ are few as well. Intervals with $\log^{20}x\le H<x^{0.14}$ are counted by the simplified large sieve (Corollary 2.9, which follows from Montgomery's finite Fourier uncertainty principle, Lemma 2.7, through the large sieve of Corollary 2.8): no element of such an interval is divisible by a prime $p$ in $(p_0,2p_0)$, so $m$ avoids $H$ residue classes modulo each such $p$, and the sieve can combine two or more of these primes because $p_0\le x^{0.15}<x^{1/6}$. A footnote at that step says that older zero-density estimates, such as Huxley's, would not quite have sufficed, Remark 2.4 says that Proposition 2.3(iii) with an exponent below $1/6$ became available only with Guth and Maynard, and Tao's forum announcement calls that input crucial. The short intervals, $H<\log^{20}x$, are the main difficulty and are handled by an anti-sieve: each further element $p_0^2m+l$ is $p_0$-smooth, so the primes below $p_0$ that divide it must make up its whole size, far more than for a typical integer, and moment estimates over the random large prime factors of $p_0^2m$ show that this is rare, through bounds for character sums over primes proved in Section 5 from the Burgess bound, the Bombieri–Halász–Montgomery inequality and the fundamental lemma of sieve theory. The hyperbola counting of Lemma 2.10 and Corollary 2.11 serves the results on very bad and type $F_3$ intervals (Theorems 1.8 to 1.10), and the large sieve serves Theorem 1.9 as well as Theorem 1.7. The source card digests the preprint, whose primary home is its factorial equation. Tao announced the result and the preprint in the problem's forum thread on 2026-03-31 (the discussion link).

Submission note. Posted to the site's forum by Terence Tao on 31 March 2026:

I have a new arXiv preprint positively resolving this problem as well as the analogous problem for very bad numbers. In fact I show that

$$> {\mathcal B}(x) = (1 + O(1/\log^{1-o(1)} x)) \# \{ n \leq x : P(n)^2 | n \} >$$

while the number of numbers up to $x$ that lie in a very bad interval but are not powerful is only $O(n^{2/5+o(1)})$, in contrast to the powerful numbers up to $x$ of which there are $(\frac{\zeta(3/2)}{\zeta(3)}+o(1)) x^{1/2}$ many.

One amusing feature of the proof is that the recent zero density estimate of Guth and Maynard plays a crucial role; the older zero density estimates of Huxley and others just barely fail to close the argument.

(The site has been updated to address this comment.)

Acceptance. Reviewed: the site's curator, Thomas F. Bloom, labels the problem PROVED and names this preprint as the proof, stating its asymptotic with the error term above (problem page last edited 10 April 2026). The preprint's third version (2026-09-26) says that referee comments and corrections were incorporated, but no journal publication is recorded, so the evidence listed is the curator's acceptance alone. This corpus has not reviewed the argument. No Lean proof of this preprint's argument is recorded: the Lean 4 development in the lean-proofs repository that proves the qualitative asymptotic follows its own route, formalizes Lemma 2.7 and Corollaries 2.8 and 2.9 of the preprint as tools, and declares no informal author, so it is recorded as the independent claim Alexeev 2026 rather than as a formalization of this one.

Depends on. No page of this wiki: the result rests on the cited preprint alone.