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Long Intervals Without Distinct Multiples of the First $n$ Positive Integers

T0 review · 0 major / 5 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Some intervals of length about (1/e) n log n contain no complete set of distinct multiples of 1 through n.

desk verdict Clean off-diagonal smooth obstruction that lifts the lower bound for F(n) from n log n / log log n to (1/e) n log n; the math holds. read the letter →

arxiv 2607.10431 v1 pith:QBHZYE6Z submitted 2026-07-11 math.NT math.CO

Scott Duke Kominers This is my paper · ORCID
classification math.NTmath.CO MSC 11N2511B75
keywords distinctmultiplessmoothnumbersErdős–PomeranceobstructionHildebrand–TenenbaumestimatesintervalmatchingsHall'stheoremF(n)saddle-point
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies how long an interval of integers can be while still missing a complete system of distinct multiples of 1, 2, ..., n. The quantity F(n) records the worst-case length of such a gap. Earlier work already showed that F(n) grows faster than the diagonal gap f(n,n), but only by a factor of order n log n / log log n. This paper improves the lower bound to the full scale n log n: the liminf of (F(n) - f(n,n))/(n log n) is at least 1/e. Equivalently, for every fixed c smaller than 1/e and all large n, there exists an interval of length c n log n that contains no such system. The argument works by placing the starting point of the interval at height roughly a n log n and applying a classical smooth-number counting obstruction of Erdős and Pomerance, now controlled by local saddle-point estimates of Hildebrand and Tenenbaum. The constant 1/e is optimal inside the three-parameter family used in the construction.

What carries the argument

The Erdős–Pomerance smooth-number obstruction (Lemma 2.1): if the number of y-smooth indices in (E/y, n] exceeds the number of y-smooth integers in the target window (m, E], then no complete system of distinct multiples can fit inside that window. Local Hildebrand–Tenenbaum ratio estimates turn the comparison into an explicit inequality among three free parameters that is optimized to produce the constant 1/e.

What would settle it

Either exhibit, for infinitely many n, a complete system of distinct multiples of 1..n inside every interval of length (1/e - ε) n log n that begins near height a n log n for admissible a, or produce a matching upper bound F(n) ≤ (1/e + o(1)) n log n that would force equality of the liminf.

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Extended reading notes

Core claim

The paper proves that liminf (F(n) - f(n,n))/(n log n) is at least 1/e. Consequently, for every fixed c < 1/e and all sufficiently large n, some interval of length c n log n contains no system of pairwise distinct multiples of the integers 1 through n. Because f(n,n) is o(n log n), this is equivalent to the lower bound F(n) ≥ (1/e - o(1)) n log n.

Load-bearing premise

The argument needs the Hildebrand–Tenenbaum local ratio for the count of smooth numbers to have a relative error smaller than the main-term difference of size 1 over log log n, throughout the height range from n to n log n.

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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper studies the maximal gap F(n) for systems of distinct multiples of 1,...,n in an interval of length h, and the diagonal quantity f(n,n). Building on Erdős–Pomerance and van Doorn’s lower bound of order n log n / log log n for F(n)−f(n,n), it proves liminf (F(n)−f(n,n))/(n log n) ≥ 1/e. The argument applies the classical smooth-number obstruction (Lemma 2.1) at off-diagonal starting points m ≍ a n log n with smoothness threshold y = d log n, compares the two sides of the obstruction inequality via Hildebrand–Tenenbaum local ratio estimates (specialized in Lemma 3.1), and optimizes the free parameters (d,a,b) in Proposition 4.1 / Lemma 5.1 to obtain the constant 1/e. Corollaries strengthen van Doorn’s scale, and Section 6 compares the construction with diagonal transfer, recovers the Erdős–Pomerance diagonal lower bound, and discusses the upper-bound problem via Hall’s theorem.

Significance. The result advances a long-standing Erdős problem (Bloom #711) by moving the lower bound for F(n)−f(n,n) from the n log n / log log n scale to the n log n scale, with an explicit constant 1/e that is optimal inside the three-parameter family used. The method is elementary once the Hildebrand–Tenenbaum estimates are granted, and the paper carefully situates the new bound relative to van Doorn’s transfer inequality and the original diagonal obstruction. The discussion of Hall violators and the partial n^{1+o(1)} upper bound at polynomial heights (Proposition 6.2) is a useful contribution to the remaining open upper-bound question. Strengths include clean error tracking, an explicit admissible triple giving a concrete 1/10 coefficient, and transparent comparison with prior work.

minor comments (5)
  1. In the proof of Corollary 1.2 the constant 1/(1754 e) is described as aesthetic; a short remark that any fixed positive constant less than 1/e works would make the dependence clearer for readers who only skim that proof.
  2. Section 6.1 optimizes the transfer parameter κ and obtains the same constant 1/e on the weaker n log n / log log n scale. A one-sentence pointer that this numerical coincidence is left unexplained (as already noted later in the section) would help readers who stop after the transfer calculation.
  3. In Lemma 3.1 the uniformity statement over compact subsets of (0,∞) for d,c1,c2 is correct but slightly stronger than needed for the fixed-parameter applications that follow; a parenthetical that fixed d,a,b suffice would reduce notational overhead.
  4. Typographical consistency: the manuscript mixes “log log n” and “ℓ” freely; once ℓ is introduced it would be cleaner to use it uniformly in the asymptotic displays of Sections 4–5.
  5. Reference [15] is cited as INTEGERS 2026; if the final pagination is known it should be inserted, otherwise the arXiv identifier already given is adequate.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: direct counting obstruction plus external Hildebrand–Tenenbaum estimates, optimized over free parameters

full rationale

The derivation chain is self-contained and non-circular. Lemma 2.1 is a pure combinatorial counting argument (smooth indices force smooth multiples). The only analytic inputs are the classical local ratio and saddle-point formulae of Hildebrand–Tenenbaum (1986), specialized in Lemma 3.1 to the fixed-parameter regime y = d log n and x ≍ n log n; these are external, parameter-free theorems whose relative error o(1/ℓ) is strictly smaller than the main-term gap produced by condition (4.1). Proposition 4.1 then compares the two sides of (2.1) via three applications of (3.2), yielding an explicit inequality on (d,a,b). Section 5 merely maximizes the resulting coefficient over that free three-parameter family, obtaining the liminf 1/e as the supremum of Cd. No quantity is defined in terms of the target liminf, no parameter is fitted to data, and there are no load-bearing self-citations (the author cites only classical or independent works). The comparison with van Doorn’s transfer is likewise independent verification, not a premise. The argument therefore stands or falls on the external estimates and elementary optimization; it does not reduce to its own inputs by construction.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The paper rests on two classical external theorems (the smooth-number obstruction idea of Erdős–Pomerance and the local saddle-point estimates of Hildebrand–Tenenbaum) plus elementary calculus. The three free parameters d,a,b are optimization variables, not data-fitted constants; they are chosen after the asymptotic comparison is derived and merely maximize the resulting coefficient. No new physical or mathematical entities are postulated.

free parameters (2)
  • smoothness threshold coefficient d
    y = d log n; free positive real chosen large enough so that the optimized coefficient Cd exceeds any prescribed C < 1/e. Not fitted to data; pure optimization parameter.
  • interval location parameters a,b
    m ≈ a n log n, E ≈ b n log n with 0 < a < b < d. Chosen inside the open set defined by inequality (4.1) to maximize b-a. Again pure optimization, not data-driven.
assumptions (3)
  • standard math Hildebrand–Tenenbaum local ratio estimate: Ψ(cx,y) = Ψ(x,y) c^{α(x,y)} (1 + O(1/u + log y / y)) uniformly for x ≥ y ≥ 2 and 1 ≤ c ≤ y (their Theorem 3).
    Invoked repeatedly in §3–4 to convert smooth-count differences into explicit main terms of size 1/ℓ.
  • standard math Hildebrand–Tenenbaum saddle-point asymptotic α(x,y) = log(1 + y/log x)/log y · (1 + O(log log(1+y)/log y)).
    Supplies the explicit value α ∼ log(1+d)/ℓ used in every comparison.
  • domain assumption Erdős–Pomerance smooth-number obstruction (their Lemma 1, restated as Lemma 2.1 with free left endpoint m).
    The entire lower-bound engine; if the counting inequality fails to force non-existence of a matching, the argument collapses.

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Pith. "Pith review of Long Intervals Without Distinct Multiples of the First $n$ Positive Integers." pith.science (2026). https://pith.science/paper/QBHZYE6Z

@misc{pith2026260710431,
  author       = {Pith},
  title        = {Pith review of: Long Intervals Without Distinct Multiples of the First $n$ Positive Integers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QBHZYE6Z}},
  note         = {Machine review of arXiv:2607.10431}
}
abstract

For positive integers $n$ and $m$, let $f(n,m)$ be the least integer $h\ge0$ such that $(m,m+h]$ contains distinct integers $a_1,\ldots,a_n$ satisfying $i\mid a_i$ for $1\le i\le n$, and put $F(n)=\max_{m\in\mathbb{N}} f(n,m)$. A recent theorem of van Doorn [INTEGERS, 2026; arXiv:2601.16972] gives $F(n)-f(n,n)>0.36\,n\log n/\log\log n$ for sufficiently large $n$. We prove \[ \liminf_{n\to\infty} \frac{F(n)-f(n,n)}{n\log n} \ge \frac{1}{\mathrm{e}}. \] Thus, for every fixed $c<1/\mathrm{e}$ and all sufficiently large $n$, some interval of length $c\,n\log n$ contains no system of pairwise distinct multiples of $1,2,\ldots,n$. The proof applies an Erd\H{o}s--Pomerance smooth-number obstruction at starting points $m\asymp n\log n$, using local saddle-point estimates of Hildebrand and Tenenbaum.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Improved Bounds for Distinct Multiples in Intervals

    math.NT 2026-07 conditional novelty 7.0 of 10

    The shortest guaranteed interval for distinct multiples of the numbers 1..n is at least n·exp((1/50)log n/loglog n) and at most n^1.4031, with the prime-only version between the same lower bound and n^{7/5}/(log n)^{2/5}.

Reference graph

Works this paper leans on

16 extracted references · cited by 1 Pith paper

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