REVIEW
Surjectivity of Finite Rank-Capped Enots-Wolley Sequences
Number Theory · Submitted 13 Sep 2026 · 8 references
Pith's verdict: Solid proof of a conjectured subfield result; the analytic estimates are correctly handled and the combinatorial core is clean.
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
For each fixed integer K at least 2, the rank-capped Enots-Wolley sequence EW[2..K] is a permutation of the integers with 2 to K prime factors plus 1 and 2.
The reading
What carries the argument
The argument rests on two main pillars. First, the combinatorial structure of the tail after a nonsaturated support T: using the 'near‑height proper replacement' (Lemma 5.1) every full term v is paired with an earlier proper term w(v) > v/p (p = min T), giving the envelope v < p H(N) and the inequality N_full(N) ≤ M_{K,T}(p H(N)). Second, at a proper record H the 'record shadow' R_{K,T}(H;B) of rank‑K integers below H that contain exactly one prime of T and avoid the two‑back support B must already have been selected earlier (Lemma 6.1), yielding R_{K,T}(H;B) ≤ C_T + N_prop(n). Combining with N_prop ≤ N_full gives the fixed bound R(H;B) – M(pH) ≤ C_T. Landau's fixed‑order estimates (Theorem 7.4) then show that R(H;B) ∼ c_{K,T} H (log log H)^{K-2} / log H while M(pH) has strictly smaller logarithmic order (since |T|≥2), so the difference tends to infinity along proper records—a contradiction. Thus every support must be saturated.
Extended reading notes
Core claim
Theorem 1.1: For every fixed integer K ≥ 2, the rank-capped Enots-Wolley sequence EW[2..K]—defined as the lexicographically earliest infinite sequence a_1=1, a_2=2, and for n≥3, a_n is the least unused integer in {m≥1 : 2 ≤ ω(m) ≤ K} satisfying the three local conditions (retain a prime from a_{n-1}, avoid primes of a_{n-2}, and introduce a new prime not in a_{n-1})—is a permutation of the set {1,2} ∪ {m ≥ 1 : 2 ≤ ω(m) ≤ K}. In other words, every integer with at least two and at most K distinct prime factors occurs exactly once.
Load-bearing premise
The proof assumes that Landau’s fixed-order estimates on the number of integers with a prescribed number of prime factors remain valid with uniform error terms when up to K primes are excluded (the two-back support B), a uniformity established in Lemma 7.3 by bounding the loss from each excluded prime using the same Landau asymptotics.
Editorial extensions
If this is right
- For each fixed K, the greedy sequence EW[2..K] is not just a subset but a full permutation of its allowed domain, providing an explicit infinite permutation of an arithmetic set.
- The structural lemmas (FIFO service, full return, episode grammar) are likely generalizable to other lexicographically earliest sequences with a local admissibility condition based on prime support.
- The result separates the rank‑capped cases from the unrestricted Enots‑Wolley sequence, showing that the cap is essential for the counting argument; the unrestricted conjecture remains open.
- The method demonstrates a template for proving surjectivity in greedy sequences that combine combinatorial episode dynamics with analytic density comparisons.
- The asymptotic constants in Theorem 7.4 (involving sums of 1/(t-1)) suggest that the occurrence order of integers depends quantitatively on the specific primes in their support.
Reading between the lines
- The same episode-and-shadow technique may be applicable to other greedy sequences in number theory or combinatorics where a similar 'full return' property holds and a counting inequality can be forced at record values.
- The reliance on the fixed cap K for uniformity (Lemma 7.3) indicates that removing the cap would require a fundamentally different approach, as the moving blocker could then contain an unbounded number of primes, breaking the uniform error estimate.
- The paper’s careful treatment of uniform error terms over a moving set of excluded primes (via Lemma 7.3) provides a useful template for other problems where Landau’s theorem is applied with varying forbidden sets.
Original abstract
For each fixed integer $K\ge2$, we consider the Enots--Wolley sequence in which every term after the initial $1,2$ is required to have between two and $K$ distinct prime divisors. We prove that every integer satisfying this restriction occurs. If an exact prime support $T$ were selected only finitely often, then after a finite cutoff the terms meeting $T$ would form short episodes, and every full term would force an earlier proper term at comparable numerical height. The case $|T|=K$ is then ruled out directly. In the remaining case $|T|<K$, at a record proper value $H$, greediness forces every unblocked rank-$K$ integer below $H$ containing exactly one prime of $T$ to have occurred earlier. Fixed-order Landau estimates show that this proper population has order $H(\log\log H)^{K-2}/\log H$, while the entire possible full population on the same scale has strictly smaller logarithmic order. This contradiction proves surjectivity. The theorem concerns each fixed rank cap and does not settle surjectivity of the unrestricted Enots--Wolley sequence.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
For each fixed integer $K\ge2$, we consider the Enots--Wolley sequence in which every term after the initial $1,2$ is required to have between two and $K$ distinct prime divisors. We prove that every integer satisfying this restriction occurs. If an exact prime support $T$ were selected only finitely often, then after a finite cutoff the terms meeting $T$ would form short episodes, and every full term would force an earlier proper term at comparable numerical height. The case $|T|=K$ is then ruled out directly. In the remaining case $|T|<K$, at a record proper value $H$, greediness forces every unblocked rank-$K$ integer below $H$ containing exactly one prime of $T$ to have occurred earlier. Fixed-order Landau estimates show that this proper population has order $H(\log\log H)^{K-2}/\log H$, while the entire possible full population on the same scale has strictly smaller logarithmic order. This contradiction proves surjectivity. The theorem concerns each fixed rank cap and does not settle surjectivity of the unrestricted Enots--Wolley sequence.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For each fixed integer $K\ge2$, we consider the Enots--Wolley sequence in which every term after the initial $1,2$ is required to have between two and $K$ distinct prime divisors. | background 0.64 | 0.42 | 0.03 | asserted 0.62 |
| s1 | We prove that every integer satisfying this restriction occurs. | result 0.98 | 0.95 | 0.94 | asserted 1.00 |
| s2 | If an exact prime support $T$ were selected only finitely often, then after a finite cutoff the terms meeting $T$ would form short episodes, and every full term would force an earlier proper term at comparable numerical height. | method 0.96 | 0.56 | 0.03 | asserted 0.66 |
| s3 | The case $|T|=K$ is then ruled out directly. | method 0.87 | 0.86 | 0.03 | asserted 1.00 |
| s4 | In the remaining case $|T|<K$, at a record proper value $H$, greediness forces every unblocked rank-$K$ integer below $H$ containing exactly one prime of $T$ to have occurred earlier. | method 0.95 | 0.77 | 0.03 | asserted 1.00 |
| s5 | Fixed-order Landau estimates show that this proper population has order $H(\log\log H)^{K-2}/\log H$, while the entire possible full population on the same scale has strictly smaller logarithmic order. | method 0.75 | 0.69 | 0.05 | asserted 1.00 |
| s6 | This contradiction proves surjectivity. | result 0.63 | 0.90 | 0.68 | asserted 1.00 |
| s7 | The theorem concerns each fixed rank cap and does not settle surjectivity of the unrestricted Enots--Wolley sequence. | limitation 1.00 | 0.68 | 0.06 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
Discussion
Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.