REVIEW
Prime pairs along rays of prime indices
Number Theory · Submitted 7 Sep 2026 · 17 references
Pith's verdict: A technically serious new density result for prime-index ratios, with the only real caveat being ineffective constants from a Siegel-tail estimate rather than a structural gap.
Why Pith says this
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Referee report
The paper
The paper claims
The sets of prime-index ratios m/n for which p_m + p_n or |p_m − p_n| is a perfect square are both dense in the positive real numbers.
The reading
What carries the argument
The paper extends the circle method of Baier and Zhao to three quadratic prime-value problems: $h^2 + k$, $h^2 - k$, and $k - h^2$, each averaged over square‑free $k$. The key innovation is a weighted $T_1E_2$ estimate and a regrouping trick that controls the non‑principal character contributions, together with control of the singular series tail via Siegel’s theorem.
Extended reading notes
Core claim
The central discovery is that the arithmetic conditions m/n ≈ r and p_m + p_n = □ (or |p_m − p_n| = □) are simultaneously satisfiable with high density. For every fixed open interval I ⊂ ℝ_{>0}, there exists a constant c_I > 0 such that for any L > 0, as N → ∞, all but O(N/(log N)^L) indices n ≤ N admit at least c_I √n/(log(2n))^{3/2} choices of m with m/n ∈ I and p_m + p_n a square; the same holds for the difference and for the roles of numerator and denominator. It follows that the sets R_Σ = {m/n : p_m + p_n is a square} and R_Δ = {m/n : |p_m − p_n| is a square} are both dense in ℝ_{>0}.
Load-bearing premise
The quantitative lower bounds rely on an ineffective constant from Siegel's theorem, meaning the constants $c_I$ cannot be explicitly computed; however, the density statements themselves remain unconditional.
Editorial extensions
If this is right
- For any positive real number $r$, there exist infinitely many pairs $(m,n)$ such that $m/n$ is arbitrarily close to $r$ and $p_m + p_n$ is a perfect square; the same holds for $|p_m - p_n|$.
- For any nonempty open interval $I$, almost every prime index $n$ has at least $c_I\sqrt{n}/(\log n)^{3/2}$ distinct partners $m$ with $m/n \in I$ and $p_m+p_n$ a square (and analogously for the difference).
- The density of $R_\Sigma$ and $R_\Delta$ implies that a conjecture requiring exact rational ratio for all positive rational numbers is at least consistent with an approximate version.
- The quantitative lower bounds show that the number of partners grows like $\sqrt{n}/(\log n)^{3/2}$ for a full-density set of indices, not merely for infinitely many.
Reading between the lines
- The same circle-method approach might be adaptable to other quadratic forms such as $h^2 + ck$ for fixed $c$, provided the corresponding singular series and character sums can be controlled.
- The ineffective constants in the quantitative results mean that constructing explicit partner pairs for a given index would require additional number-theoretic input, such as effective bounds for $L$-functions.
- The density result is unconditional, but the quantitative constants depend on an ineffective Siegel bound; a future effective version would give fully explicit lower bounds.
- The method suggests that similar density phenomena should hold for other arithmetic functions, such as $m/n$ with $p_m p_n$ a square, though the circle method would need substantial modifications.
Original abstract
Let $p_j$ be the prime with index $j$. We prove that the ratios $m/n$ for which $p_m+p_n$ is a square are dense in $\mathbb R_{>0}$. The same is true when $|p_m-p_n|$ is a square. In every nonempty open interval, almost every index can be used as a numerator and as a denominator. We also give a quantitative lower bound for the number of possible partners.
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
Let $p_j$ be the prime with index $j$. We prove that the ratios $m/n$ for which $p_m+p_n$ is a square are dense in $\mathbb R_{>0}$. The same is true when $|p_m-p_n|$ is a square. In every nonempty open interval, almost every index can be used as a numerator and as a denominator. We also give a quantitative lower bound for the number of possible partners.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $p_j$ be the prime with index $j$. | background 1.00 | 0.08 | 0.01 | asserted 0.67 |
| s1 | We prove that the ratios $m/n$ for which $p_m+p_n$ is a square are dense in $\mathbb R_{>0}$. | result 0.98 | 0.96 | 0.81 | asserted 1.00 |
| s2 | The same is true when $|p_m-p_n|$ is a square. | result 0.99 | 0.91 | 0.14 | asserted 1.00 |
| s3 | In every nonempty open interval, almost every index can be used as a numerator and as a denominator. | result 0.97 | 0.89 | 0.23 | asserted 1.00 |
| s4 | We also give a quantitative lower bound for the number of possible partners. | result 0.78 | 0.95 | 0.05 | asserted 1.00 |
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
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