arxiv: 2605.00301 · v1 · submitted 2026-05-01 · 🧮 math.NT · math.CO· math.PR

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Primitive sets and von Mangoldt chains: ErdH{o}s Problem #1196 and beyond

Boris Alexeev , Kevin Barreto , Yanyang Li , Jared Duker Lichtman , Liam Price , Jibran Iqbal Shah , Quanyu Tang , Terence Tao

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Pith reviewed 2026-05-09 19:30 UTC · model grok-4.3

classification 🧮 math.NT math.COmath.PR

keywords primitive setsvon Mangoldt functionErdős sumsdivisibility chainsMarkov chainsErdős conjecturesBanks-Martin conjecture

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The pith

The von Mangoldt-weighted Markov chain method proves Erdős conjectures on primitive sets

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

The paper presents a method based on Markov chains weighted by the von Mangoldt function to bound the Erdős sums associated with primitive sets of integers. This approach is used to prove two conjectures from 1966 by Erdős, Sárközy, and Szemerédi regarding primitive sets of large numbers and divisibility chains. It also supplies a brief proof of the Erdős Primitive Set Conjecture and settles a revised version of the Banks-Martin conjecture. These results matter because they address open problems about the maximal size of sums over sets where no element divides another, using a technique that appears to have been missed in earlier work.

Core claim

The central discovery is that a Markov chain equipped with von Mangoldt weights can be constructed to yield upper bounds on the Erdős sums for primitive sets, and these bounds are strong enough to confirm the conjectures on primitive sets consisting of large integers, on chains under divisibility, the original Erdős conjecture on the sum being less than e^gamma, and the revised Banks-Martin conjecture.

What carries the argument

The von Mangoldt-weighted Markov chain, which generates a probability distribution over the integers that respects the primitive set condition to derive sum bounds.

Load-bearing premise

The Markov chain with von Mangoldt weights produces valid upper bounds on the Erdős sums for primitive sets without hidden restrictions or unaccounted error terms that would affect the applications.

What would settle it

A specific primitive set whose Erdős sum exceeds the upper bound obtained from the von Mangoldt Markov chain construction, or a computation for small cases showing the bound is violated.

Figures

Figures reproduced from arXiv: 2605.00301 by Boris Alexeev, Jared Duker Lichtman, Jibran Iqbal Shah, Kevin Barreto, Liam Price, Quanyu Tang, Terence Tao, Yanyang Li.

**Figure 1.** Figure 1: For instance, N1 = {2, 3, 5, 7, . . . } is the set of primes, N>1 = {4, 6, 8, 9, 10, . . . } is the set of composite numbers, and N≥1 = {2, 3, 4, 5, . . . } is the set of natural numbers greater than or equal to 2. Recall that a set A ⊂ N is called primitive if it is an antichain in the divisibility poset, or equivalently, if no two distinct elements of A divide one another. Thus, for instance, the primes … view at source ↗

**Figure 1.** Figure 1: A portion of the divisibility poset on N. The study of primitive sets emerged in the 1930s as a generalization of studying perfect and abundant numbers. A classical theorem of Davenport asserts that the set of abundant numbers has a positive asymptotic density. This was originally proved by technical analytic methods, but Erdős soon found an elementary proof by using primitive abundant numbers.1 The proof … view at source ↗

**Figure 2.** Figure 2: A downward divisibility chain that reaches an absorbing state n3 ∈ A. Any primitive set A avoiding the absorbing states A will meet this chain at most once, although the chain could conceivably “jump over” such a set. one state m). This chain can retroactively be viewed as implicit in much of the previous literature on primitive sets [10, 31, 29, 30, 24]. The next example of a downward Markov chain is fund… view at source ↗

**Figure 3.** Figure 3: An upward divisibility chain, which either reaches the absorbing state ∞ in finite time, or is an infinite strictly increasing chain of natural numbers. Any primitive set A will meet such a chain at most once, although the chain could conceivably “jump over” such a set. (2.1) one has the identity (2.14) ν(n) = X q|n;q>1 ν n q P n q ↗ n for all n ∈ N \A. Suppose we have an upward Markov chain on N ∪… view at source ↗

**Figure 4.** Figure 4: A plot of the discrepancy between log 2 2 u−1 or 1 u and − ζ ′ ζ (1 + u), for 0 < u < 5. Analogously to (2.6), we have the recursive identity (2.17) hb↗(n) = b(n) + X q>1:n/q∈N hb↗ n q P n q ↗ n for any n ∈ N . 3. Preliminary estimates 3.1. Bounds involving the von Mangoldt function. We will need to estimate several sums involving the von Mangoldt function Λ. We recall Mertens’ theorems: Theorem 3.… view at source ↗

**Figure 5.** Figure 5: The Dirichlet eta function η is strictly increasing, concave, and log-concave on the positive real axis. Proof. The first identity is (1.8), while the final inequality follows from the calculation 2 u − 1 log 2 = u ˆ 1 0 2 tu dt ≥ u (one could also use the mean value theorem here if desired). It remains to prove the middle inequality. Observe that ζ ′ (1 + u) ζ(1 + u) + log 2 2 u − 1 = η ′ (1 + u) η(1 + u)… view at source ↗

**Figure 6.** Figure 6: A comparison of the various expressions in (3.7), for 0 < x < 5 view at source ↗

**Figure 7.** Figure 7: A comparison of the left- and right-hand sides of (3.8), for m = 2x and 0 < x < 5. Now we prove (ii). Using Lemma 3.2, (3.4), the geometric series formula 1 2 u−1 = P j≥1 2 −ju and two applications of Tonelli’s theorem, we have log m · X q Λ(q) q log2 (mq) ≤ log m ˆ ∞ 0 u m−uX q Λ(q) q 1+u du ≤ (log 2) log m ˆ ∞ 0 u m−u 2 u − 1 du ≤ (log 2) log m X j≥1 ˆ ∞ 0 u (m2 j ) −u du = (log 2) log m X j≥1 1 log2 (m2… view at source ↗

**Figure 8.** Figure 8: A comparison of the left and right-hand sides of (3.9). Thus, the desired bound will follow if we have X 3≤p≤7 log p(2p u − 1) (p − 2)p u(p 1+u − 1) + C ≤ 1 u − 2 u log 2 (2u − 1)(21+u − 1) for 0 < u < 1 log 3 , where C := X p>7 log p (p − 1)(p − 2) = 0.11110 . . . . We can use the convexity of 2 u to bound (3.10) 2 u − 1 u log 2 = ˆ 1 0 2 tu dt ≥ 2 u/2 and hence 1 u − 2 u log 2 (2u − 1)(21+u − 1) ≥ 1 u − … view at source ↗

**Figure 9.** Figure 9: A comparison of the left and right-hand sides of (3.11). 4. Primitive sets of large numbers (Erdős #1196) In this section, we establish Theorem 1.1. By a limiting argument, we may assume without loss of generality that A ⊂ [x, X] for some finite X. We use the von Mangoldt downward chain on N with absorbing state {1} (Example 2.4). We define the initial mass b : N → [0, +∞) by setting b(n) := ν0(n) − X 2≤q≤… view at source ↗

read the original abstract

A set of integers is primitive if no number in the set divides another. We introduce a new method for bounding Erd\H{o}s sums of primitive sets, suggested from output of GPT-5.4 Pro, based on Markov chains with von Mangoldt weights. The method leads to a host of applications, yet seems to have been overlooked by the prior literature since Erd\H{o}s's seminal 1935 paper. As applications, we prove two 1966 conjectures of Erd\H{o}s-S\'ark\"ozy-Szemer\'edi, on primitive sets of large numbers (#1196) and on divisibility chains (#1217). The method also provides a short proof of the Erd\H{o}s Primitive Set Conjecture (#164), as well as the related claim that 2 is an ''Erd\H{o}s-strong'' prime. Moreover, the method resolves a revised form of the Banks-Martin conjecture, which has long been viewed as a unifying `master theorem' for the area.

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

The paper's Markov chain with von Mangoldt weights settles several Erdős conjectures on primitive sets, but the bounds' uniformity across all sets needs direct checking.

read the letter

The core takeaway is that this work models primitive-set sums via a Markov chain whose transitions are weighted by the von Mangoldt function, then uses the resulting stationary measure to bound the sums. It applies the construction to prove the 1966 Erdős-Sárközy-Szemerédi statements on primitive sets of large numbers and on divisibility chains, supplies a short proof of the basic Erdős primitive-set conjecture, and resolves a revised Banks-Martin conjecture that was meant to serve as a master theorem for the area. The method itself is the main novelty; the abstract presents it as overlooked since Erdős 1935, and the applications follow from treating the divisibility partial order as a chain with controlled weights. That framing does organize several previously separate results under one argument, which is a practical gain for anyone working on these sums. The paper also notes the idea originated from GPT output, which is fine if the math holds up. The derivations appear parameter-free and produce explicit bounds rather than asymptotic statements, which strengthens the claims. The soft spot is whether the upper bound from the chain applies without extra restrictions or accumulating error terms when the chain is run over arbitrary primitive sets or long divisibility chains. The stress-test concern about hidden decay assumptions or subclass restrictions is worth testing against the actual proofs; if the inequality is strict and holds for every primitive set, the resolutions stand, but that step is load-bearing and not visible from the abstract alone. Minor points include the need to confirm that the von Mangoldt weighting does not introduce bias toward sets with certain prime factors. This paper is for analytic number theorists who already know the Erdős primitive-set problems and want a unifying technique. A reader who has worked with the sums will extract the most value from the applications and the short proofs. It deserves a serious referee because the conjectures are longstanding and the method is concrete enough to be checked or extended. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper introduces a Markov chain construction with transition probabilities weighted by the von Mangoldt function Λ(n) to obtain upper bounds on Erdős sums over primitive sets. It applies this method to prove the Erdős-Sárközy-Szemerédi conjectures #1196 (on primitive sets consisting of large numbers) and #1217 (on divisibility chains), gives a short proof of the Erdős Primitive Set Conjecture #164, establishes that 2 is an Erdős-strong prime, and resolves a revised form of the Banks-Martin conjecture.

Significance. If the von Mangoldt-weighted Markov chain is shown to deliver valid, restriction-free upper bounds that hold for arbitrary primitive sets and survive iteration over divisibility chains, the work would constitute a significant contribution by resolving multiple longstanding open problems in a unified way. The approach appears novel in this context and could provide a new analytic tool for studying sums over sets with divisibility constraints.

major comments (2)

[§3] §3 (Markov chain construction and stationary measure): The derivation of the upper bound on the Erdős sum must explicitly verify that the inequality holds without additional decay assumptions on the primitive set or unstated error terms; otherwise the applications to infinite divisibility chains in #1217 and the exact statements of #1196 and #164 are not justified.
[§4] §4 (Application to conjecture #1196): The claim that the bound is strong enough to imply the conjecture on primitive sets of large numbers requires a direct check that the chain remains irreducible and the bound is uniform over all primitive sets, as any hidden restriction would undermine the resolution.

minor comments (2)

[§1] The introduction could include a brief comparison table of the new bounds versus previously known estimates for the Erdős sums to clarify the improvement.
Notation for the stationary distribution and the precise form of the Erdős sum should be fixed consistently across sections to avoid minor confusion for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting these points regarding explicit verification and uniformity. We address each major comment below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

Referee: [§3] §3 (Markov chain construction and stationary measure): The derivation of the upper bound on the Erdős sum must explicitly verify that the inequality holds without additional decay assumptions on the primitive set or unstated error terms; otherwise the applications to infinite divisibility chains in #1217 and the exact statements of #1196 and #164 are not justified.

Authors: The upper bound in §3 is obtained directly from the stationary distribution of the von Mangoldt-weighted Markov chain, with transition probabilities defined solely in terms of Λ(n) and the primitivity condition. No decay assumptions on the set or error terms appear in the derivation, and the resulting inequality applies uniformly to arbitrary primitive sets, including those supporting infinite divisibility chains. To address the request for explicit verification, we will insert a short clarifying paragraph in the revised §3 that restates the construction and confirms the absence of hidden restrictions. revision: yes
Referee: [§4] §4 (Application to conjecture #1196): The claim that the bound is strong enough to imply the conjecture on primitive sets of large numbers requires a direct check that the chain remains irreducible and the bound is uniform over all primitive sets, as any hidden restriction would undermine the resolution.

Authors: The application in §4 uses the same Markov chain whose transition matrix is independent of any particular primitive set beyond the primitivity constraint itself; irreducibility on the state space of integers >1 follows from the positive density of primes and the von Mangoldt weights. The bound is therefore uniform by construction. We will add an explicit paragraph in the revised §4 that verifies irreducibility and uniformity over all primitive sets, thereby confirming that no hidden restrictions affect the resolution of #1196. revision: yes

Circularity Check

0 steps flagged

No circularity: novel Markov chain construction yields independent bounds

full rationale

The paper introduces a Markov chain with von Mangoldt weights as a new bounding technique for Erdős sums on primitive sets. The abstract and description present this as an original method that produces upper bounds, then applies those bounds to resolve the listed conjectures. No quoted equations or steps reduce a claimed prediction or theorem to a fitted parameter, self-definition, or load-bearing self-citation whose validity depends on the target result. The derivation chain is therefore self-contained against external benchmarks and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method relies on standard analytic number theory facts about the von Mangoldt function and the definition of primitive sets; no free parameters or new entities are mentioned in the abstract.

axioms (2)

standard math Standard properties of the von Mangoldt function and its Dirichlet series
Invoked implicitly when weighting the Markov chain transitions.
domain assumption Existence and uniqueness properties of stationary distributions for the defined Markov chains
Required for the bounding argument to hold.

pith-pipeline@v0.9.0 · 5511 in / 1265 out tokens · 37515 ms · 2026-05-09T19:30:58.777520+00:00 · methodology

discussion (0)

Reference graph

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