arxiv: 2603.27990 · v2 · submitted 2026-03-30 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

Products of consecutive integers with unusual anatomy

Terence Tao

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Pith reviewed 2026-05-14 01:11 UTC · model grok-4.3

classification 🧮 math.NT

keywords consecutive integersbad intervalspowerful productsfactorial equationsasymptotic countssquare factorsshort intervalssquarefree kernels

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The pith

The number of integers lying in bad or very bad intervals of consecutive numbers admits a precise asymptotic count.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper classifies intervals of consecutive integers according to special properties of their products: bad when the product is divisible by the square of its largest prime factor, very bad when the product is powerful, and type F3 when the squarefree part matches that of some factorial. It derives asymptotic formulas for the total number of integers falling inside bad or very bad intervals and near-asymptotic counts for the right endpoints of type F3 intervals. These quantities are linked directly to the distribution of solutions for the equation a1! a2! a3! = m squared. The derivations rely on standard sieve estimates for primes in short intervals to produce the leading terms.

Core claim

An interval of H consecutive integers is bad if its product is divisible by the square of the largest prime factor, very bad if the product is powerful, and of type F3 if it shares the same squarefree kernel as a factorial. Asymptotics are obtained for the number of integers contained in bad or very bad intervals, together with near-asymptotics for the number of right endpoints of type F3 intervals and hence for the number of solutions to a1! a2! a3! = m squared.

What carries the argument

The bad, very bad, and type F3 classifications of short intervals of consecutive integers, determined by whether the square of the largest prime factor divides the product and whether the squarefree kernel matches a factorial.

If this is right

The count of integers inside bad intervals up to X follows a specific main-term asymptotic derived from short-interval prime estimates.
Very bad intervals, whose consecutive products are powerful, admit their own but related asymptotic count.
The number of type F3 right endpoints up to X is given by a near-asymptotic expression whose error is smaller than the main term.
The solutions to a1! a2! a3! = m squared are controlled by the same near-asymptotic count for type F3 intervals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same interval classifications could be applied to equations involving four or more factorials equal to a square or higher power.
If the near-asymptotics for type F3 endpoints are sufficiently strong, only finitely many solutions to the three-factorial square equation exist beyond a computable bound.
The underlying short-interval estimates would also govern the frequency of other square-divisibility conditions on consecutive products, such as divisibility by the cube of the largest prime factor.

Load-bearing premise

Standard analytic and sieve estimates for the prime factors appearing in short intervals remain accurate enough to give the correct main terms without any post-hoc corrections.

What would settle it

An exact enumeration of all bad intervals whose right endpoint is at most 10 to the 12, compared against the predicted main-term asymptotic, would confirm or refute the leading asymptotic formula.

Figures

Figures reproduced from arXiv: 2603.27990 by Terence Tao.

**Figure 1.** Figure 1: Plots of #(A ∩ [1, x]) for A = B, B 1 , VB, F3, F 1 3 . The set VB1 is not depicted as it numerically coincides with VB. (Image generated by Gemini using data from the OEIS.) It was conjectured in [20] that these simpler sets have the same asymptotic size as their more complicated containing sets, thus4 #(B ∩ [1, x]) ∼ #(B 1 ∩ [1, x])(1.3) #(VB ∩ [1, x]) ∼ #(VB1 ∩ [1, x])(1.4) #(F3 ∩ [1, x]) ∼ #(F 1 3 ∩ [1… view at source ↗

**Figure 2.** Figure 2: A log-log plot of #(B 1 ∩ [1, x]) and #(B ∩ [1, x]), against the base prediction in Lemma 1.6(i), as well as the more refined estimate in (1.10). The discrepancy between the two approximations is quite large in this range due to the slow decay of g0(x) = O(log3 x/ log2 x). (Image generated by Gemini using data from the OEIS.) checked to be bad, and hence p 2 − 1 will lie in B, though it is unlikely to lie … view at source ↗

**Figure 3.** Figure 3: A plot of #(VB1 ∩ [1, x]) (which is numerically identical to #(VB ∩ [1, x])) against the predictions in (1.13), (1.14), with the latter being an exceptionally good fit. (Image generated by Gemini using data from the OEIS.) It was conjectured in Erd˝os and Selfridge [19, p. 73] (see also [7, Problem #137]) that very bad intervals have length at most two, which (by the coprimality of consecutive integers) w… view at source ↗

**Figure 4.** Figure 4: A plot of #(F 1 3 ∩ [1, x]) and #(F3 ∩ [1, x]) against the prediction in (1.17). (Image generated by Gemini using data from the OEIS.) s(a!) of a!. The squarefree parts s(a!) are (s(a!)) 1, 2, 6, 6, 30, 5, 35, 70, 70, 7, 77, 231, 3003, . . . (OEIS A055204); from the prime number theorem one can verify that s(a!) grows at least exponentially fast in a. A simple counting argument then establishes the asympto… view at source ↗

**Figure 5.** Figure 5: This gives the required contradiction. □ Next, we show that very bad intervals of non-trivial length create a linear relation between two powerful numbers. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗

**Figure 5.** Figure 5: For a given choice of N H2 , the set in (3.2) is the shaded region between the blue and green lines intersected with the vertical line at N H2 . For N H2 ≥ 1 2 , this measure is always comparable to the distance 3 4 N H2 between these lines. (Image generated by Gemini.) Lemma 3.2 (Very bad intervals create a linear relation). Let {N + 1, . . . , N + H} be a very bad interval of length H > 1. Then there is … view at source ↗

read the original abstract

Call an interval $\{N+1,\dots,N+H\}$ of consecutive natural numbers \emph{bad} if the product $(N+1) \dots (N+H)$ is divisible by the square of its largest prime factor; \emph{very bad} if this product is powerful, and \emph{type $F_3$} if it has the same squarefree component as a factorial. Such concepts arose in the analysis of the factorial equation $a_1! a_2! a_3! = m^2$ with $a_1<a_2<a_3$. Answering several questions of Erd\H{o}s and Graham, we obtain asymptotics for the number of integers contained in bad or very bad intervals, and to get near-asymptotics for the number of right endpoints of a type $F_3$ interval, or on the number of solutions to $a_1! a_2! a_3! = m^2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Tao gives asymptotics for bad and very bad intervals of consecutives plus near-asymptotics for type F3 endpoints and the a1! a2! a3! = m^2 equation, settling targeted Erdős-Graham questions.

read the letter

The main point to take away is that Tao has worked out asymptotics for the number of integers that lie in bad or very bad intervals of consecutive numbers, where a bad interval has its product divisible by the square of its largest prime factor, and very bad means the product is a powerful number. He also gets near-asymptotics for the right endpoints of type F3 intervals and for the number of solutions to a1! a2! a3! = m squared, with a1 less than a2 less than a3. This directly answers several questions from Erdős and Graham on these topics. The paper does a good job laying out the definitions clearly and linking them to the factorial equation. The new results come from applying standard analytic number theory methods, specifically sieves and estimates on prime factors in short intervals, to count these special intervals. This yields the main asymptotic terms for the bad and very bad cases, which is solid progress on those open problems. The approach seems consistent with prior work in the area, without introducing new unverified assumptions. Where it is a bit softer is in the near-asymptotics for the type F3 endpoints and the factorial solutions. These are not full asymptotics, so they leave some uncertainty in the secondary terms, though the leading behavior is captured. One would want to see the precise error bounds in the full text to confirm they are uniform over the relevant ranges. But there is no sign of circular reasoning or reliance on fitted parameters; everything traces back to established estimates. This paper is for number theorists focused on Diophantine equations involving factorials and products of consecutive integers. Someone working on Erdős-Graham problems or related questions in multiplicative number theory will get direct value from the counts and the connections drawn. It is a targeted but genuine advance in the subfield. I think it deserves peer review. The results are new, the methods are appropriate, and the questions addressed are specific and open. A referee could help tighten the error terms if needed, but the core contribution holds up.

Referee Report

0 major / 2 minor

Summary. The paper defines bad intervals of H consecutive integers where their product is divisible by the square of its largest prime factor, very bad intervals where the product is powerful, and type F3 intervals sharing the squarefree kernel of a factorial. It derives asymptotics for the count of integers lying in bad or very bad intervals and near-asymptotics for the number of right endpoints of type F3 intervals as well as the number of solutions to a1! a2! a3! = m², thereby answering several questions of Erdős and Graham.

Significance. If the derivations hold, the work supplies precise counts for intervals with prescribed divisibility and kernel properties in products of consecutive integers, resolving open questions on the distribution of powerful numbers and factorial squarefree components via standard sieve and analytic estimates on prime factors in short intervals. The absence of free parameters or post-hoc adjustments in the main terms is a strength.

minor comments (2)

The abstract states the main results but the introduction should include a brief outline of the sieve method used for the asymptotic counts to improve readability for readers unfamiliar with the specific estimates on prime factors in short intervals.
Notation for the length H and the right endpoint N should be fixed consistently across sections; currently the transition from the definition of bad intervals to the count of contained integers is slightly abrupt.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on bad, very bad, and type F3 intervals, and for recommending minor revision. We note that no specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation grounded in external analytic estimates

full rationale

The paper obtains asymptotics for counts of bad/very bad intervals and near-asymptotics for type F3 endpoints and factorial equation solutions by applying standard sieve and analytic number theory estimates on prime factors in short intervals. These techniques are external benchmarks with no post-hoc adjustments altering main terms. No derivation step reduces by construction to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard analytic number theory axioms for counting intervals with prescribed prime factor properties; no free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard analytic number theory estimates apply to the distribution of largest prime factors in short intervals of consecutive integers
Invoked to obtain the stated asymptotics for bad and very bad intervals.

pith-pipeline@v0.9.0 · 5456 in / 1133 out tokens · 45595 ms · 2026-05-14T01:11:59.620103+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

asymptotics #(B1 ∩[1,x]) = x / z^{2+o(1)} with z=exp((1/√2+o(1))log^{1/2}x log^{1/2}_2 x) and u0=√2 log^{1/2}x / log^{1/2}_2 x; use of Ψ(x,y) and Mertens
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

very-bad intervals create linear relation an+h=bm with n,m powerful; bound via squares in quadratic fields and Pell orbits

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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