Erdős problem #935 — wave 7u report

Access date: 2026-07-28 (UTC). Companion verifier: runs/erdos935_wave7u_reverify.py.

Claim labels

• [a] elementary-rigorous: a proof is given here from definitions.
• [b] rigorous modulo the explicitly named theorem or conjecture.
• [c] plausible/structural-unverified.
• [d] computational-only or source-checked metadata/transcription.

No claim below silently promotes a computation or conjecture to a theorem.

0. Mandatory live-page check

[d] I fetched the rendered live page erdosproblems.com/935 through the Bright Data browser route, then separately fetched its LaTeX view and 12-comment thread. Direct datacenter curl was not used as a substitute for this check.

[d] At access time the page said:

• status OPEN;
• last edited 08 February 2026;
• 12 comments;
• 0 claimed proofs for this problem;
• “Currently working on this problem: None”;
• “Interested in collaborating: Woett”;
• “Likes this problem: Woett”;
• the difficult, tractable, formalisable, and working-on-formalisation markers

were all None.

[d] Thus the literal stop trigger on page 935 did not fire. The linked, essentially equivalent Problem 367 page now separately lists sproutseeds as a current worker; this is recorded as a collaboration warning, but it was not the page-935 status specified by the task. No external claim, message, or outreach was made in this run.

Verbatim current statement

The following is copied verbatim from the live LaTeX view (only placed in a code block so that its source characters are preserved):

For any integer $n=\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that\[Q_2(n) = \prod_{\substack{p\\ k_p\geq 2}}p^{k_p}.\]Is it true that, for every $\epsilon>0$ and $\ell\geq 1$, if $n$ is sufficiently large then\[Q_2(n(n+1)\cdots(n+\ell))<n^{2+\epsilon}?\]If $\ell\geq 2$ then is\[\limsup_{n\to \infty}\frac{Q_2(n(n+1)\cdots(n+\ell))}{n^2}\]infinite?

If $\ell\geq 2$ then is\[\lim_{n\to \infty}\frac{Q_2(n(n+1)\cdots(n+\ell))}{n^{\ell+1}}=0?\]

Results and discussion already on the page

[d] The page attributes the question to Erdős [Er76d], reports Erdős's comment that the first assertion “seems very difficult to prove”, and states that Mahler's result gives, for every fixed \(\ell\geq1\),

[d] The page now records the second question as answered affirmatively: the Pell equation \(x^2-8y^2=1\), via the construction on Problem 367 and the independent rediscovery in Feng et al. [Fe26], gives

It also records that the \(abc\) conjecture implies a positive answer to the third question, and notes the analogous \(Q_r\) questions.

[d] I read all 12 comments. In chronological order they are by Woett, Nat Sothanaphan, TerenceTao, Kevin Barreto, Woett, Thomas Bloom, fengt, Woett, fengt, BorisAlexeev, fengt, and old-bielefelder (2–5 February 2026). Their mathematical content concerns: recognizing the near-identity with Problem 367; priority versus independent rediscovery of the Pell construction; the fact that Pell equations are standard here; moving the Aletheia result to an “Independent Rediscovery” category; and the fact that Aletheia incorrectly described its response as solving the whole problem even though only its second part survived review. None of the comments claims a proof of the remaining first or third question.

1. Primary-source literature audit

[d] The original source exists and was inspected: P. Erdős, “Problems and results on number theoretic properties of consecutive integers and related questions”, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (1976), 25–44. The relevant discussion is on PDF page 8 (printed page 32). It contains the three directions represented on the live page and the “seems very difficult” sentence.

[d] Feng et al., “Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems”, arXiv:2601.22401, was downloaded and checked at §4.2, pp. 21–23. The reviewed paper explicitly says that Aletheia solved the second question, gives the Pell proof, and only remarks that \(abc\) implies the third. It does not publish an unconditional proof of the first or third question.

[d] I also read the newer Problem 367 discussion, because page 935 itself says the questions are equivalent up to fixed constants. A 10 June 2026 comment by ScottHughes claims a stronger Pell lower bound for the already-solved second direction, an \(r\)-full extension, and the \(abc\)-conditional first direction. The site labels comments as unverified; I used none of those claims as a theorem and did not duplicate the Pell construction.

[d] A dangerous false lead was rejected. The repository's raw, unreviewed model response claims a 2008 paper by J. Cilleruelo, The powerful part of the product of consecutive integers, Bull. London Math. Soc. 40 (2008), 873–877. Exact-title searches found no such paper. More decisively, a Crossref query for ISSN 0024-6093, volume 40 (2008), shows that pages 863–875 are Deshouillers' A refined bound for sum-free sets in groups of prime order and pages 876–886 are Ortega-Cerdà's Interpolating and sampling sequences in finite Riemann surfaces. The alleged 873–877 article cannot occupy that page range. Nothing in this report uses it.

[d] Terence Tao, “Products of consecutive integers with unusual anatomy”, arXiv:2603.27990v2 (22 April 2026), was checked because it postdates the last edit of page 935. Its “very bad” intervals are intervals whose whole product is powerful. Theorem 1.8 proves a counting asymptotic for integers lying in such intervals, but the paper explicitly says the conjecture that a very bad interval has length at most two remains open. It does not give the uniform lower bound on exponent-one primes needed below.

[d] Carlo Sanna, “On the exponents in the factorizations of \(r\) consecutive numbers”, Quaestiones Mathematicae 45 (2022), 1221–1228, was checked. Its Lemma 3.3 gives the same basic \(O_r(x/y^{1/2})\) tail estimate for large powerful parts that appears in the elementary density argument below. I reprove the needed estimate rather than importing it.

[d] For the conditional radical bound I checked P. Ribenboim, “Finite sets of binary forms”, Publ. Math. Debrecen 68 (2006), 261–282. Theorem 4.1 attributes the binary form radical theorem to Granville (earlier, with extra endpoint hypotheses, to Langevin), and Lemma 4.2 gives the \(n-1-\epsilon\) exponent after fixing the second variable to \(1\).

[d] Searches for the exact problem, “powerful part” plus consecutive integers, the alleged Cilleruelo title, and later citations found no verified primary source proving the remaining uniform assertions. This is a reported search miss, not a proof that no such literature exists.

2. Exact reformulation

For fixed \(\ell\), put

and

[a] These are exact integer identities, directly from the definition of \(Q_2\). Consequently

The third question is therefore exactly \(W_\ell(n)\to\infty\). The first question is exactly the eventual inequality

Thus its asymptotic content is the uniform lower bound \(W_\ell(n)\ge n^{\ell-1-o(1)}\).

Whole-product versus individual powerful parts

Define the fixed constant

[a] Lemma.

[a] Proof. If \(p>\ell\), then \(p\) divides at most one of the \(n+i\), so the \(p\)-contribution agrees on the two sides before the factor \(C_\ell\). Let \(p\leq\ell\), write \(e_i=v_p(n+i)\), let \(E=\sum e_i\), and let \(s=\#\{i:e_i=1\}\). If \(E<2\), there is no missing factor. If \(E\geq2\), the whole-product powerful part contains \(p^E\), whereas the product of the individual powerful parts contains \(p^{E-s}\); the ratio is \(p^s\). An interval of \(\ell+1\) integers contains at most \(\lceil(\ell+1)/p\rceil\) multiples of \(p\). Multiplying over \(p\leq\ell\) proves (2.4). \(\square\)

[a] This constant is precisely why computing \(\prod_i Q_2(n+i)\) alone would not be a valid checker for the stated problem.

3. Unconditional almost-everywhere progress

[a] Powerful-number count. Every powerful integer has a unique representation \(a^2b^3\) with \(b\) squarefree. Hence the number \(A(Y)\) of powerful integers at most \(Y\) satisfies

Splitting into dyadic intervals then gives

[a] Tail lemma. Uniformly for \(1\leq T\ll X\),

[a] Proof. If \(Q_2(m)=d>T\), then \(d\) is a powerful divisor of \(m\). A union bound over powerful \(d\) gives

by (3.1)–(3.2) and \(T\ll X\). \(\square\)

[a] Theorem (density version of the first question). Fix \(\ell\geq2\) and \(0<\epsilon\leq\ell-1\). Then

For \(\epsilon>\ell-1\), the desired inequality is elementary for every sufficiently large \(n\), simply because \(Q_2(P_\ell(n))\leq P_\ell(n)\) and the exponent \(2+\epsilon\) then exceeds \(\ell+1\). For \(\ell=1\), the original first question is also elementary: \(Q_2(n(n+1))\leq n(n+1)<n^{2+\epsilon}\) eventually.

[a] Proof of (3.4). A violation and (2.4) imply

At least one factor is at least

Apply (3.3) to each of the \(\ell+1\) shifts. Since \(\epsilon\leq\ell-1\), \(T\ll X\), and the stated exponent follows.

\(\square\)

[a] Theorem (density version of the third question). For every fixed \(\ell\geq2\) and \(\eta>0\),

Thus the third limit is unconditionally true in natural density, with a power-saving exceptional set.

[a] Proof. A member of the set in (3.5), together with (2.4), forces some \(Q_2(n+i)\gg_{\ell,\eta}X\). Use (3.3) with \(T\asymp_{\ell,\eta}X\), then sum over the fixed number of shifts.

\(\square\)

[a] Scope warning. Neither (3.4) nor (3.5) controls every \(n\). An exceptional set of size \(O(\sqrt X)\) can still contain an infinite sparse sequence, so these results do not settle either remaining uniform question.

4. The \(abc\) route, checked precisely

Let \(R=\operatorname{rad}(P_\ell(n))\). Because the primes in \(W_\ell(n)\) and \(Q_2(P_\ell(n))\) are disjoint,

[a] Since every exponent in \(Q_2\) is at least two,

Using \(P_\ell=W_\ell Q_2(P_\ell)\) gives the useful exact inequality

[b] Assume the \(abc\) conjecture through the Granville–Langevin radical consequence (Ribenboim, Lemma 4.2): for a squarefree polynomial \(f\) of degree \(d\),

For \(f(x)=P_\ell(x)\), \(d=\ell+1\). Equations (4.1)–(4.2) yield

Choosing \(\delta\) arbitrarily small proves both the first question and, for \(\ell\geq2\), the third question conditionally on \(abc\). This derivation identifies the exact exponent supplied by the named theorem; it is not an unconditional claim.

5. Exact finite computation

What was computed

[d] The standalone verifier exhaustively scans every \(1\leq n\leq10^9\) for \(\ell=2\) and \(\ell=3\), computes the integer \(W_\ell(n)\), and finds its exact minimum in each displayed range. All minimizers in the table are unique.

[d] In particular, the following are sharp over the stated finite range:

with equality only at \(n=128961799\), and

with equality only at \(n=129800448\).

[d] Combining the exact finite minima with the elementary inequalities \(\prod_{i=1}^{\ell}(1+i/n)\leq \prod_{i=1}^{\ell}(1+i/10^8)\) gives, throughout the same range,

These decimal bounds are corollaries of the exact rational bounds, not floating-point search criteria.

Independent extremizer factorizations

[d] For the final \(\ell=2\) range, scalar trial division independently gave

Thus the exponent-one primes of the whole product are exactly \(6871\) and \(67\), and \(6871\cdot67=460357\).

[d] For the final \(\ell=3\) range, the independent factorization was

The \(2\)- and \(3\)-valuations aggregate across terms, so the exponent-one primes of the whole product are \(89,211,15361,32507\), whose product is \(9377106367033\).

Algorithm and reproducibility

[a] For a segment \([L,R]\), the verifier initializes an array with the integers themselves. For every prime \(p\leq\sqrt R\), it divides the \(p^2\)-progression by \(p^2\), then the \(p^j\)-progression by \(p\) for each \(j\geq3\). A number with \(v_p(m)=e\geq2\) is therefore divided by \(p^2p^{e-2}=p^e\), while a prime occurring exactly once remains. This computes every individual \(w(m)\) exactly.

[a] A prime \(p>\ell\) cannot divide two entries of an \((\ell+1)\)-term block. For \(p\leq\ell\), whether it must be removed from the product of individual \(w(n+i)\) depends only on \(n\bmod p^2\). The verifier constructs this periodic correction from the valuations, rather than assuming multiplicativity. Its maximum correction is \(2\) for \(\ell=2\) and \(18\) for \(\ell=3\).

[a] Values are saturated at \(10^{15}\) only after a guarded calculation: the uncorrected product is first capped at \(10^{15}C_{\max}\). Since the eventual correction is at most \(C_{\max}\), a capped value cannot become a false value below \(10^{15}\). All certified minima are below \(10^{15}\).

[d] Before the full scan, the script compares every individual sieve value through 10,003 and every block value for both \(\ell=2,3\) through \(n=10,000\) against independent scalar trial division. After the scan it scalar-refactors every decade extremizer, reconstructs \(P_\ell=W_\ell Q_2\), and asserts the hard-coded table.

Run:

cd /home/exedev/MathDyad
/usr/bin/time -f 'wall=%E maxrss=%MKB' \
  python3 runs/erdos935_wave7u_reverify.py

[d] The clean default run ended with:

scalar/vector cross-check: PASS (all starts n<=10,000)
FULL CERTIFICATE: PASS
elapsed_seconds=183.113
wall=3:03.30 maxrss=439812KB

[d] Verifier SHA-256: d7fe8a5cbe418d87cfe9e28b5ff5fdbe211604d64bb865314a8bc2a4983793bf. The complete 484-line code is in the companion .py; it contains the prime sieve, segmented computation, periodic correction, independent trial division, range assertions, and command-line options. A quick audit can use --limit 10000000, while the default is required to reproduce the full certificate.

[d] Scope warning. The finite table proves no eventual asymptotic. Its early minima are not even monotone by decade (for example \(61\) is followed by \(29\) for \(\ell=2\)), so no monotonic extrapolation was made.

6. Exact wall for a uniform proof

[a] By (2.1)–(2.2), the two remaining targets are precisely:

1. \(W_\ell(n)\geq n^{\ell-1-o(1)}\) uniformly in \(n\);
2. \(W_\ell(n)\to\infty\) uniformly in \(n\).

Average estimates such as (3.3) allow an \(O(\sqrt X)\) exceptional set and therefore do not reach either target.

[a] The obstruction is already concrete for \(\ell=2\). Write

Every \(a_i\) is squarefree, \(\gcd(a_i,u_i)=1\), and every \(u_i\) is powerful. The only prime that can occur exactly once in one term but be absorbed by another term of the three-term block is \(2\). Hence

If \(W_2(n)\leq B\) infinitely often, one of finitely many squarefree coefficient triples with \(a_0a_1a_2\leq2B\) must support infinitely many powerful solutions of

[a] A theorem giving finiteness of (6.2) for every fixed bounded coefficient triple would prove the third question for \(\ell=2\). Its special coefficient choice \(a_0=a_1=a_2=1\) is the problem of three consecutive powerful integers.

[d] Tao's 2026 paper explicitly records the corresponding finiteness/no-triple phenomenon as still open unconditionally (and finite under \(abc\)). Thus the checked literature supplies no generic Pell, \(S\)-unit, or powerful-number-count theorem that proves the missing simultaneous finiteness lemma.

[b] Alternatively, the near-optimal polynomial radical estimate

would close the first question through (4.1), but the verified source gives this strength conditional on \(abc\), not unconditionally. This is the exact named-theorem gap in the radical approach.

[d] Extending the same exhaustive scan from \(10^9\) to \(10^{10}\) would cost roughly 30–40 core-minutes at the measured throughput, with about the same 440 MB segmented memory footprint. I did not run it because it exceeds the requested few-minute budget and, regardless of outcome, cannot provide the required uniform finiteness step.

PARTIAL: elementary density-one bounds and exact \(n\le10^9\) tables for \(\ell=2,3\) are verified; the uniform first and third questions remain open at the explicit radical/simultaneous-powerful finiteness wall.