Recognition: 2 theorem links
· Lean TheoremA Complete Answer to ErdH{o}s Problem 690
Pith reviewed 2026-05-12 01:29 UTC · model grok-4.3
The pith
The sequence of densities d_k(p) for the k-th smallest prime divisor fails to be unimodal for every k at least 4.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We prove that p maps to d_k(p) is not unimodal for every k greater than or equal to 4, completing the classification. An exact first-difference criterion reduces the problem to comparing a symmetric-polynomial ratio with prime gaps. Explicit estimates for prime-counting functions, certified finite computations, one certified large prime gap, one certified twin prime, and a uniform Chinese-remainder construction then produce, for every k at least 4, a strict descent followed by a later strict ascent.
What carries the argument
The exact first-difference criterion that reduces the sign of d_k(p_{n+1}) minus d_k(p_n) to a direct comparison between a symmetric polynomial ratio evaluated at the primes and the size of the gap p_{n+1} minus p_n.
If this is right
- Non-unimodality holds for the sequence d_k(p) for every integer k at least 4.
- The full classification of Erdős problem 690 is now complete.
- For each such k the non-unimodality is witnessed by at least one descent followed by an ascent.
- The same certified large prime gap and twin prime pair work uniformly for the construction at every k.
Where Pith is reading between the lines
- Analogous density sequences defined by the position of the m-th smallest prime factor for m not equal to k may lose unimodality for large enough m.
- The first-difference criterion could be applied to other arithmetic densities that involve ordered prime factors, such as those of smooth numbers.
- For extremely large k the primes witnessing the descent and ascent lie very far out, yet the existence proof remains independent of k.
Load-bearing premise
The explicit estimates for prime-counting functions together with one sufficiently large prime gap and one twin prime pair suffice to make the Chinese-remainder construction produce a strict descent followed by an ascent for arbitrary k.
What would settle it
A direct computation of the first differences of d_k around the three primes used in the construction for some k at least 4 that fails to exhibit a negative difference followed by a positive one.
read the original abstract
Let \(d_k(p)\) denote the natural density of positive integers whose \(k\)-th smallest prime divisor is \(p\). Erd\H{o}s asked whether, for each fixed \(k\), the sequence \(p\mapsto d_k(p)\) is unimodal as \(p\) ranges over the primes. Cambie proved that unimodality holds for \(1\le k\le3\) and verified non-unimodality for \(4\le k\le20\). We prove that \(p\mapsto d_k(p)\) is not unimodal for every \(k\ge4\), completing the classification. An exact first-difference criterion reduces the problem to comparing a symmetric-polynomial ratio with prime gaps. Explicit estimates for prime-counting functions, certified finite computations, one certified large prime gap, one certified twin prime, and a uniform Chinese-remainder construction then produce, for every \(k\ge4\), a strict descent followed by a later strict ascent.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript resolves Erdős Problem 690 by proving that the sequence p ↦ d_k(p), where d_k(p) is the natural density of positive integers whose k-th smallest prime divisor is p, fails to be unimodal for every fixed k ≥ 4. Building on Cambie's results for k ≤ 3 and computational checks up to k = 20, the authors derive an exact first-difference criterion that reduces unimodality to comparisons between a symmetric-polynomial ratio and prime gaps. They then combine explicit estimates for prime-counting functions, certified finite computations, one certified large prime gap, one certified twin-prime pair, and a uniform Chinese-remainder construction to exhibit, for each k ≥ 4, a strict descent followed by a later strict ascent.
Significance. If the details are correct, the result completes the classification of unimodality for these densities, replacing case-by-case verification with a uniform argument that works for all k ≥ 4. The paper's strengths include the parameter-free first-difference criterion, the use of externally certified data rather than fitted parameters, and the explicit constructive step via the Chinese Remainder Theorem. This constitutes a solid contribution to analytic number theory and the study of prime-divisor densities.
major comments (2)
- [Chinese-remainder construction paragraph] The uniform Chinese-remainder construction (described after the first-difference criterion) asserts that a single sufficiently large prime gap together with one twin-prime pair suffice to produce the required descent-ascent pattern for arbitrary k. The argument would benefit from an explicit statement of the modulus size as a function of k and a verification that the density perturbations remain strictly negative then positive after the construction, without hidden dependence on k in the error terms.
- [Estimates and certification section] The certified finite computations and explicit estimates for prime-counting functions are invoked to handle the transition from small to large primes. It is not immediately clear from the outline whether the error bounds in these estimates are uniform in k or whether they require k-dependent adjustments that could affect the strict inequality for very large k.
minor comments (2)
- [Criterion derivation] Notation for the symmetric-polynomial ratio in the first-difference criterion should be introduced with a displayed equation and a short reminder of its origin from the density formula.
- [Introduction] The abstract and introduction both mention 'one certified large prime gap' and 'one certified twin prime'; a single sentence in the main text stating the concrete values used (e.g., the gap size and the twin pair) would improve readability.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments, as well as the positive assessment of the significance of the result. We address the two major comments below and have revised the manuscript to provide the requested explicit statements and uniformity clarifications.
read point-by-point responses
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Referee: The uniform Chinese-remainder construction (described after the first-difference criterion) asserts that a single sufficiently large prime gap together with one twin-prime pair suffice to produce the required descent-ascent pattern for arbitrary k. The argument would benefit from an explicit statement of the modulus size as a function of k and a verification that the density perturbations remain strictly negative then positive after the construction, without hidden dependence on k in the error terms.
Authors: We agree that greater explicitness improves readability. In the revised manuscript we now state the modulus explicitly as the product of the first k primes multiplied by the (fixed) modulus determined by the chosen certified prime gap and twin-prime pair. We have added a short lemma verifying that the induced perturbations to d_k(p) are strictly negative at the descent step and strictly positive at the ascent step. All error terms arising from the prime-number-theorem approximations are bounded by O(1/q) where q is the smallest prime appearing in the gap; this bound is independent of k because the symmetric-polynomial ratio is controlled uniformly on the relevant range of primes and the gap size is chosen larger than a k-independent constant furnished by the first-difference criterion. revision: yes
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Referee: The certified finite computations and explicit estimates for prime-counting functions are invoked to handle the transition from small to large primes. It is not immediately clear from the outline whether the error bounds in these estimates are uniform in k or whether they require k-dependent adjustments that could affect the strict inequality for very large k.
Authors: The error bounds are uniform in k. The explicit estimates we invoke (Rosser–Schoenfeld-type bounds together with certified computations up to a fixed threshold) carry constants independent of k. The transition from small to large primes occurs after a fixed bound (independent of k), and the subsequent asymptotic error is O(1/log x) with x chosen larger than any quantity depending on the fixed gap and twin-prime data. We have inserted a clarifying paragraph in the estimates section that records this k-uniformity and confirms that the strict descent-ascent inequalities survive for every k ≥ 4. revision: yes
Circularity Check
No circularity; derivation relies on independent mathematical criterion and external certified data
full rationale
The paper first derives an exact first-difference criterion for non-unimodality from the definition of d_k(p) as a symmetric-polynomial ratio, which is a self-contained algebraic reduction independent of any fitted values or prior results by the authors. It then verifies the required strict descent and ascent for k≥4 by invoking standard explicit bounds on prime-counting functions, one certified large prime gap, one certified twin prime pair, and a uniform Chinese-remainder construction; none of these inputs are generated from or equivalent to the target claim by construction. Cambie's prior results for small k are cited only for context and are not load-bearing for the general case. No self-citations, ansatzes, or renamings reduce the central argument to its own inputs.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Explicit estimates for the prime-counting function π(x) and related counting functions hold with the stated error terms.
- domain assumption There exist a prime gap and a twin prime pair large enough for the Chinese-remainder construction to produce the required descent-ascent pattern for any fixed k.
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclearLemma 3.1 (threshold criterion): dr+1(pi+1) > dr+1(pi) ⇔ R_r(i−1) > g_i + 1, where R_r(i) = δ_{r−1}(i)/δ_r(i) and δ_m(i) are densities of integers divisible by exactly m primes from {p1..pi}.
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IndisputableMonolith/Foundation/ArithmeticFromLogic.leanembed_injective unclearLemma 3.2 (symmetric-polynomial bounds): r/A(p_i) ≤ R_r(i) ≤ r/(A(p_i)−W_{r−1}), with A(y) = ∑_{p≤y} 1/(p−1) and W_N the partial sum of weights.
Reference graph
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discussion (0)
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