Lines in the prime number graph
Pith reviewed 2026-05-25 05:41 UTC · model grok-4.3
The pith
The first n points of the prime number graph can be covered by O(n log log n / log n) line segments.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The authors prove that L(n) equals O(n log log n / log n) and that B(n) is at least c log n for a positive constant c and all sufficiently large n. Assuming the Riemann Hypothesis, they further obtain B(n) equals O(n^{3/4} (log n)^{1/2}) and L(n) is at least c' n^{1/4} (log n)^{-1/2} for a positive constant c'.
What carries the argument
The functions L(n), the minimal number of segments covering the first n points (n, p_n), and B(n), the maximal number of those points lying on one straight line.
If this is right
- Any single line contains at least c log n of the first n points for large n.
- Under the Riemann Hypothesis no line contains more than O(n^{3/4} sqrt(log n)) of the first n points.
- Under the Riemann Hypothesis at least c' n^{1/4} / sqrt(log n) segments are required to cover the first n points.
- The conjecture that only O(n / log n) segments suffice remains open but is now known to within a log log n factor.
Where Pith is reading between the lines
- The same covering arguments might extend to graphs formed by other increasing integer sequences whose gaps satisfy comparable average-size estimates.
- Improved unconditional error terms in the prime number theorem would immediately strengthen the bound on L(n) without needing the Riemann Hypothesis.
- The logarithmic lower bound on B(n) indicates that arithmetic progressions can contain comparatively many primes among the first n primes.
Load-bearing premise
The proofs depend on standard error-term estimates for the distribution of primes that follow from the prime number theorem or from the Riemann Hypothesis.
What would settle it
An explicit computation for some large n that finds either L(n) larger than C n log log n / log n for every constant C or B(n) smaller than (log n)/2 would contradict the stated bounds.
read the original abstract
The prime number graph is the set of points $(n,p_n)$ where $p_n$ denotes the $n^{\rm th}$ prime. Let $L(n)$ be the minimum number of straight line segments needed to cover the first $n$ points in this set. Let $B(n)$ be the largest number of points $(k,p_k)$ with $k\le n$ covered by a single line. Recently Sloane conjectured that $L(n) = O(n/\log n)$. We show that $L(n)=O(n \log \log n / \log n)$ and $B(n)\ge c\log n$ for a constant $c>0$ and all large $n$. Under RH we show that for large $n$ we have $B(n)=O(n^{3/4}(\log n)^{1/2})$ and $ L(n)\ge c' n^{1/4} (\log n) ^{-1/2}$ for some constant $c'>0.$
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the prime number graph with points (k, p_k) for the k-th prime p_k. L(n) is defined as the minimal number of line segments needed to cover the first n points, while B(n) is the maximal number of these points lying on any single straight line. The authors prove unconditionally that L(n) = O(n log log n / log n) and B(n) ≥ c log n for some c > 0 and all sufficiently large n. Under the Riemann Hypothesis they establish the sharper bounds B(n) = O(n^{3/4} (log n)^{1/2}) and L(n) ≥ c' n^{1/4} (log n)^{-1/2} for constants c, c' > 0.
Significance. If the proofs hold, the results supply the first explicit non-trivial bounds on these geometric quantities associated to the sequence of primes. The unconditional upper bound on L(n) improves the trivial O(n) estimate and moves toward Sloane's conjecture of O(n/log n), albeit with an extra log log n factor. The lower bound on B(n) shows that lines can cover at least logarithmically many prime points. The RH-conditional estimates are of independent interest because they quantify how prime-gap information controls collinearity in the graph. The work relies on standard tools of analytic number theory (prime-number theorem with error terms and sieve estimates) and therefore sits comfortably within the existing literature.
minor comments (3)
- The statements 'for all large n' and 'for a constant c>0' appear in the abstract and presumably in the theorems; the paper should make explicit the range of n for which the inequalities hold and, if possible, give a numerical value or explicit lower bound for c (or at least its dependence on known constants from the prime-number theorem).
- Notation for the prime-number graph and the quantities L(n), B(n) is introduced clearly in the abstract, but the introduction or §1 should include a short comparison table or paragraph contrasting the new bounds with the trivial estimates O(n) for L(n) and O(1) for B(n).
- Under RH the exponent 3/4 on n in the bound for B(n) arises from the classical RH error term; a brief remark on whether a stronger zero-free region or subconvexity would improve the exponent would be useful for readers.
Simulated Author's Rebuttal
We thank the referee for their positive summary and recommendation of minor revision. No major comments were raised in the report.
Circularity Check
No significant circularity
full rationale
The paper states explicit upper and lower bounds on L(n) and B(n) as theorems derived from standard analytic number theory (prime number theorem with error terms, sieve methods, prime gap estimates) and the Riemann Hypothesis for the conditional results. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear; the derivation chain rests on independent external results rather than reducing to the paper's own inputs or prior work by the same authors. This is the normal case of a self-contained proof against established benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Riemann Hypothesis (for the conditional bounds on B(n) and L(n))
Lean theorems connected to this paper
-
IndisputableMonolith/Foundation/RealityFromDistinction.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
We show that L(n)=O(n log log n / log n) and B(n)≥ c log n ... Under RH we show that ... B(n)=O(n^{3/4}(log n)^{1/2}) and L(n)≥ c' n^{1/4} (log n)^{-1/2}
-
IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Proof of Theorem 3. Let k be a large integer and consider the Farey sequence of level k ... Prime Number Theorem with remainder
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.
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.