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)**
2. The comments contain van Doorn's Kronecker/Besicovitch proof of
unboundedness. (b: named classical approximation theorems)
3. 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)**
4. 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)
2. Will Sawin (15 Jun) gives the elementary parameter-split upper bound
$O(n^{2/5})$. (a)
3. 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)
4. Nat Sothanaphan asks for a write-up and notes that the then-small Lean
files did not formalize the whole argument. (d: comment metadata)
5. Hughes replies that a four-page note was added. (d)
6. Sothanaphan reports a screening check finding no error, distinguishes
the $\kappa=2$ formalization from the general claim, and requests AI
disclosure. (c)
7. Hughes says the disclosure, general-$\kappa$ scope, and literature
discussion were updated. (d)
8. Hughes adds that the complete $\kappa=2$ rate theorem is now
machine-checked. **(c: repository claim, not independently rebuilt
here)**
9. 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)
10. Wouter van Doorn (23 Nov 2025, post 1782) supplies the De
Koninck--Luca and Chan references and explains what each proves.
(b)
11. 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,
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,
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
\[ g(n)=\#\{m\text{ powerful}:n^2only square in its interval is $n^2$, hence
\[ \boxed{h(n)=g(n)+1.} \](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
\[ m=a^2b^3,\qquad a\ge1,\quad b\ge1\text{ squarefree}. \]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
\[ n^2This 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
\[ 2\le b\le\left\lfloor((N+1)^2-1)^{1/3}\right\rfloor,\qquad 1\le a\le \left\lfloor\sqrt{\frac{(N+1)^2-1}{b^3}}\right\rfloor, \tag{2} \]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,
\[ n^2\le a^2b^3<(n+1)^2 \]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
\[ \boxed{\max_{1\le n\le10^9}h(n)=12,} \]uniquely at
\[ \boxed{n=472\,532\,614.} \]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,
\[ \sum_j\#\{n:h(n)=j\}=10^9,\qquad \sum_j j\,\#\{n:h(n)=j\}=2\,171\,766\,332 =10^9+1\,171\,766\,332. \](d)
Full distribution
| $j$ | $\#\{1\le n\le10^9:h(n)=j\}$ |
|---:|---:|
| 1 | 276,407,642 |
| 2 | 395,711,787 |
| 3 | 231,044,200 |
| 4 | 76,838,805 |
| 5 | 16,938,003 |
| 6 | 2,697,055 |
| 7 | 328,069 |
| 8 | 31,690 |
| 9 | 2,557 |
| 10 | 178 |
| 11 | 13 |
| 12 | 1 |
(d)
First threshold records
| $j$ | first $n$ with $h(n)\ge j$ |
|---:|---:|
| 1 | 1 |
| 2 | 2 |
| 3 | 5 |
| 4 | 31 |
| 5 | 234 |
| 6 | 1,822 |
| 7 | 3,611 |
| 8 | 17,329 |
| 9 | 524,827 |
| 10 | 524,827 |
| 11 | 180,469,424 |
| 12 | 472,532,614 |
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
\[ [223\,287\,071\,293\,672\,996,\; 223\,287\,072\,238\,738\,225). \]The following are 12 distinct powerful integers in it. The offset is
$m-n^2$.
| offset | $m$ | $(a,b)$ with $m=a^2b^3$ | prime factorization |
|---:|---:|---:|:---|
| 0 | 223,287,071,293,672,996 | (472,532,614, 1) | $2^2\,10289^2\,22963^2$ |
| 48,972,380 | 223,287,071,342,645,376 | (881,284, 66) | $2^7\,3^3\,11^3\,53^2\,4157^2$ |
| 173,875,676 | 223,287,071,467,548,672 | (90,938,944, 3) | $2^{12}\,3^3\,1420921^2$ |
| 182,253,812 | 223,287,071,475,926,808 | (23,379, 742) | $2^3\,3^2\,7^3\,53^3\,7793^2$ |
| 338,163,548 | 223,287,071,631,836,544 | (32,151,772, 6) | $2^7\,3^3\,8037943^2$ |
| 412,711,516 | 223,287,071,706,384,512 | (167,065,508, 2) | $2^7\,37^2\,1128821^2$ |
| 442,099,837 | 223,287,071,735,772,833 | (6,741,529, 17) | $17^3\,619^2\,10891^2$ |
| 547,088,604 | 223,287,071,840,761,600 | (22,960, 751) | $2^8\,5^2\,7^2\,41^2\,751^3$ |
| 593,422,007 | 223,287,071,887,095,003 | (346,397, 123) | $3^3\,41^3\,346397^2$ |
| 714,502,004 | 223,287,072,008,175,000 | (806,835, 70) | $2^3\,3^2\,5^5\,7^3\,19^4\,149^2$ |
| 859,114,812 | 223,287,072,152,787,808 | (4,579,286, 22) | $2^5\,11^3\,2289643^2$ |
| 872,110,504 | 223,287,072,165,783,500 | (2,282,074, 35) | $2^2\,5^3\,7^3\,53^2\,21529^2$ |
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
\[ h(n)\le(\log n)^{1+o(1)}\quad\hbox{for every }n \tag{3} \]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
\[ \mu^2(b)=1,\qquad |a^2b^3-n^2|\le 2n+1. \tag{4} \]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.