Erdős problem #942 — live audit, exact computation, and analytic wall

Access date: 2026-07-28 UTC. All mathematical assertions below carry one of the requested labels:

• (a) elementary-rigorous — a proof is supplied here.
• (b) rigorous-modulo-named-theorem — this report checked the cited primary

source, but does not reprove its theorem.

• (c) plausible/structural-unverified — a comment or preprint claim that this

report did not validate completely.

• (d) computational-only — an exact finite computation, not an asymptotic

theorem.

0. Mandatory live-page gate

I fetched the rendered live page and its discussion through the Bright Data browser path, not datacenter curl.

The live page is OPEN, was last edited 15 June 2026, shows 11 comments, 0 claimed proofs, and the following markers: “Interested in collaborating: None” and “Currently working on this problem: None”. Its other reaction markers (likes, difficult, tractable, formalisable, and currently formalising) are also all “None”. Thus neither stop condition fired. (d: browser-verified page metadata)

Verbatim live statement

The /latex/942 view gives:

Let $h(n)$ count the number of powerful (if $p\mid m$ then $p^2\mid m$) integers in $[n^2,(n+1)^2)$. Estimate $h(n)$. In particular is there some constant $c>0$ such that\[h(n) < (\log n)^{c+o(1)}\]and, for infinitely many $n$,\[h(n) >(\log n)^{c-o(1)}?\]

This half-open interval convention is authoritative for this run.

Results listed in the main body

The page records the following.

1. Erdős says that $\limsup h(n)=\infty$, that each fixed-level density

$\delta_\ell$ exists, and that $\sum_\ell\delta_\ell=1$. (b: original Erdős source and later distribution theorems)

1. The comments contain van Doorn's Kronecker/Besicovitch proof of

unboundedness. (b: named classical approximation theorems)

1. De Koninck--Luca prove, infinitely often,

\[ h(n)\gg\left(\frac{\log n}{\log\log n}\right)^{1/3}, \] and compute the density $\approx0.275$ of the live-normalized event $h(n)=1$. (b: De Koninck--Luca, Theorems 1 and 2, after the normalization shift explained below)

1. The page now records the optimized De Koninck--Luca--Shparlinski

construction \[ h(n)\gg \frac{\log n} {\log\log n\,\log\log\log n} \quad\hbox{infinitely often}. \] (b: the Dirichlet/primorial optimization supplied in the page discussion; no claim of novelty)

All 11 discussion comments checked

The comments, including nested replies, were read in full:

1. Scott Hughes (15 Jun 2026, post 7013) claims a frequency lower bound

and an upper bound $h(n)\ll_\varepsilon n^{6/25+\varepsilon}$ via Filaseta--Trifonov, links a repository/paper, and reports an empirical open-interval maximum of 9 through $10^7$. (c)

1. Will Sawin (15 Jun) gives the elementary parameter-split upper bound

$O(n^{2/5})$. (a)

1. Hughes (13 Jun, post 6956) presents the optimized

$(\log n)^{1-o(1)}$ lower construction and links the formalization. The bound is now adopted in the main page. (b) Its initial novelty claim was subsequently withdrawn. (d: comment history)

1. Nat Sothanaphan asks for a write-up and notes that the then-small Lean

files did not formalize the whole argument. (d: comment metadata)

1. Hughes replies that a four-page note was added. (d)
2. Sothanaphan reports a screening check finding no error, distinguishes

the $\kappa=2$ formalization from the general claim, and requests AI disclosure. (c)

1. Hughes says the disclosure, general-$\kappa$ scope, and literature

discussion were updated. (d)

1. Hughes adds that the complete $\kappa=2$ rate theorem is now

machine-checked. (c: repository claim, not independently rebuilt here)

1. Thomas Bloom (15 Jun) observes that the construction is essentially

the DLS argument; using a primorial and direct Dirichlet approximation, rather than the inefficient squarefree product and Roth detour, yields the displayed $(\log n)^{1-o(1)}$ bound. (b)

1. Wouter van Doorn (23 Nov 2025, post 1782) supplies the De

Koninck--Luca and Chan references and explains what each proves. (b)

1. Van Doorn (23 Nov, post 1781) gives a

Besicovitch--Kronecker proof that $h$ is unbounded. (b)

The site itself warns that comments are not verified. In particular, this report does not promote the $n^{6/25+\varepsilon}$ or frequency claims from comment 1 to established literature results. (c)

1. Primary-source literature check

The following sources were opened and their relevant theorem/abstract was checked.

• Erdős, *Problems and results on number theoretic properties of consecutive

integers and related questions*, Proc. Fifth Manitoba Conference on Numerical Mathematics, pp. 25--44: scan. Page 35 states the question using the open interval. (b)

• De Koninck--Luca, Sur la proximité des nombres puissants, Acta

Arith. 114 (2004), 149--157, DOI / official PDF. Theorem 1 gives the constant $9/20$ times $(\log n/\log\log n)^{1/3}$ infinitely often; Theorem 2 gives \[ \prod_{m\ge2}\left(1-\frac{\mu^2(m)}{m^{3/2}}\right)\approx0.275 \] as the density of open intervals containing no powerful integer. (b)

• De Koninck--Luca--Shparlinski, Powerful numbers in short intervals,

Bull. Aust. Math. Soc. 71 (2005), 11--16, official PDF. Theorem 1 gives $(\log N)^{1/3+o(1)}$ many $\kappa$-full integers infinitely often between successive $\kappa$th powers. (b)

• Shiu, On the number of square-full integers between successive squares,

Mathematika 27 (1980), 171--178, DOI, and Xiong--Zaharescu, K-full integers between successive k-th powers, Indag. Math. 22 (2011), 77--86, DOI, establish fixed-level distribution results; the latter's primary abstract explicitly says it generalizes Shiu and supplies explicit error terms. (b)

• Chan, A note on powerful numbers in short intervals,

arXiv:2207.08874, published in Bull. Aust. Math. Soc. 108 (2023), proves uniformly $Q_k(x+y)-Q_k(x)\ll y/\log(y+1)$. With $x=n^2$ and $y\asymp n$ this remains a power-of-$n$ bound, not the requested polylogarithmic bound. (b)

• Narumi--Tachiya,

On the number of $k$-full integers between three successive $k$-th powers, arXiv:2512.07438v2, proves explicit joint densities for every fixed pair $(\ell,m)$. It does not give a uniform maximal-order bound as $\ell\to\infty$. (b)

Exact-title, exact-phrase, arXiv, and 2025--2026 searches found no primary source claiming the desired uniform polylogarithmic upper bound or a matching exponent $c$. This is a search miss, not proof that no such paper exists. (d)

I also inspected the current Hughes repository at commit 37feb687a3b2a75e5cae00297b73532da4b088e9. Its own README says the $n^{6/25+\varepsilon}$ theorem is conditional on an explicit ft_curve_count axiom encoding the Filaseta--Trifonov input. I did not independently verify that the stated axiom follows with the required uniform constants in every dyadic regime, so this report leaves that upper bound at (c).

2. Normalization correction

Most of the original literature defines

The authoritative live page instead includes the left endpoint. The only square in its interval is $n^2$, hence

(a) This explains both the page's density statement ($h(n)=1$ corresponds to $g(n)=0$) and the recent comment saying that the empirical maximum is 9: the linked code actually computes $g$. Under the live definition its quoted $n=524827$ has $h(n)=10$. The shift is elementary (a) and the numerical value was recomputed exactly (d).

3. Exact reduction used by the verifier

Unique representation

Every positive powerful integer has a unique representation

Indeed, if $m=\prod p^{e_p}$ with every $e_p\ge2$, put $p$ in $b$ exactly when $e_p$ is odd, and give $p$ exponent $(e_p-3\mathbf1_{e_p\ {\rm odd}})/2$ in $a$. Parity makes this choice unique. (a)

For squarefree $b>1$, $a b\sqrt b$ is irrational, so

Consequently the live-page function is exactly

This is a union-of-Beatty-sequences multiplicity formulation. (a)

Finite cutoff

To compute all $1\le n\le N$, it is necessary and sufficient to enumerate

with $b$ squarefree, and increment $h(\lfloor\sqrt{a^2b^3}\rfloor)$. (a)

The implementation uses floating point only to propose the integer square root in large vector batches. It then corrects and asserts, using signed 64-bit integer arithmetic,

for every event. At $N=10^9$ all such products are below $10^{18}+O(10^9)$, well below $2^{63}-1$. (a)

The complete standalone code is erdos942_wave7v_verify.py. It also:

• independently factors every integer in all intervals through $n=1000$

using a smallest-prime-factor sieve and obtains identical counts;

• trial-factors every displayed witness via its independently factored

$a$ and $b$, reconstructs $m$, and asserts every prime exponent is at least 2;

• verifies exact cube-root and per-$b$ endpoint inequalities;
• emits a portable SHA-256 of the byte sequence

$(h(1),\ldots,h(N))$.

Run:

python3 runs/erdos942_wave7v_verify.py

On this VM (Python 3.12.3, NumPy 2.4.4), the final run took 87.74 seconds and peaked at 2,222,036 KiB RSS. A separate full run reproduced every headline field and the same digest. (d)

4. Exact results through $10^9$

The two full runs give

uniquely at

This is an exact exhaustive finite result, not evidence sufficient to settle the asymptotic question. (d)

The digest of the one-byte encoding of $h(1),\ldots,h(10^9)$ is

d04a9b29d9b1d18d62aeae6a78be5c5d029039f1f2b12f0f52b935dc2fb9e865

The enumeration processed exactly 1,171,766,332 nonsquare powerful integers. As consistency checks,

(d)

Full distribution

(d)

First threshold records

The simultaneous first occurrence for thresholds 9 and 10 is not a typo: $h(524827)=10$. (d)

5. Explicit 12-term witness

For $n=472\,532\,614$, the live interval is

The following are 12 distinct powerful integers in it. The offset is $m-n^2$.

Every displayed exponent is at least 2 and every offset lies in $[0,2n+1)$, so the list alone proves $h(472532614)\ge12$. (a) The statement that these are all the powerful integers in the interval comes from the exhaustive squarefree-$b$ scan through $b=606672$ and is therefore (d).

6. What remains and why standard machinery stalls

The optimized lower bound is $(\log n)^{1-o(1)}$ infinitely often, so if a constant $c$ of the form asked for exists, then $c\ge1$. (b) Proving

would settle the question with $c=1$.

In the exact reduction (1), the missing statement is a uniform polylogarithmic multiplicity estimate for the integer pairs

At the balanced scale $a\asymp b\asymp n^{2/5}$, elementary splitting only gives $O(n^{2/5})$. (a) Chan's uniform short-interval theorem gives $O(n/\log n)$ here. (b) The page comment's Filaseta--Trifonov reduction claims $O_\varepsilon(n^{6/25+\varepsilon})$, still a power rather than a polylogarithm. (c)

Thus the precise analytic wall is a positive-width lattice-point theorem at the balanced scale strong enough to replace the power saving in (4) by $(\log n)^{1+o(1)}$, uniformly in the central square $n^2$. Neither fixed-$\ell$ equidistribution nor existing uniform short-interval counts supply that uniformity. (b) The stronger near-curve assertion in the page comment was not validated here. (c)

More brute force cannot repair this finiteness gap. The present enumerator is linear in the roughly $1.17N$ nonsquare events and stores two bytes per $n$. Scaling the observed run to $N=10^{10}$ would cost about 0.25 core-hours and 20--25 GB RAM; it could find another record but would still say nothing uniform for all $n$. (d: measured-cost extrapolation)

PARTIAL: Exact live-normalized computation gives a unique maximum h(n)=12 for n<=10^9 at n=472532614, with 12 factored witnesses and a reproducible verifier; the uniform polylogarithmic upper bound remains open.