Admissible sets of integers in imaginary quadratic fields of class number 1

Daniel Shiu

REVIEW

Admissible sets of integers in imaginary quadratic fields of class number 1

Daniel Shiu

Number Theory · Submitted 25 Oct 2026

Pith's verdict: Solid computational number theory that delivers new admissible sets and explicit counterexamples, but Theorem 2's disc-based constructions lack the verifiability needed for a formal proof.

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arXiv ID2610.34556 (version 2)

Pith IDYKLEM6ML, Pith's permanent record of this paper and its review

CitationDaniel Shiu. "Admissible sets of integers in imaginary quadratic fields of class number 1." arXiv:2610.34556 (2026). https://pith.science/paper/YKLEM6ML

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Editor's note

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Dan,

This paper does two things well. First, it improves the best known diameters for admissible sets of rational integers in several imaginary quadratic fields, using a straightforward adaptation of the Polymath sieve (avoid inert primes, double up on split primes). The exhaustive search in Section 2 is sound, and the resulting bounds for k=50 (the key input for bounded gaps) are concrete improvements over Vatwani's earlier work. Theorem 1 is legitimate.

Second, Section 4 gives explicit counterexamples to a natural analogue of the second Hardy-Littlewood conjecture for all class number 1 fields except Q(√-11). These are explicit and verifiable—you can check that the listed elements are indeed primes in the right disc. That part is a clean contribution.

The soft spot is Theorem 2 (disc-based admissible sets). The paper claims these sets exist based on 'computations in bold in table 8 and the associated images'. But the actual sets are not listed, and the greedy algorithm is described only qualitatively. The output depends on tie-breaking and candidate ordering, and the claimed diameters are not independently reproducible from the description. The authors acknowledge that exhaustive search is infeasible here, which is fine, but then the burden shifts to providing the constructed sets (or at least the residue classes chosen). Without that, Theorem 2 is not really a theorem—it's an experimental claim. The stress-test note is right: the gap between description and verifiability is critical for this part.

The heuristics in Section 5 are speculative but harmless; they don't affect the main results.

Who is this for? Number theorists working on bounded gaps in number fields, or anyone interested in explicit admissible sets and Hardy-Littlewood type conjectures. The paper is worth engaging with, but the authors should be asked to either supply the disc-based admissible sets (in digital supplementary material) or clearly downgrade Theorem 2 to an experimental observation. I would accept it for peer review with a request for supplementary data.

Referee report

Major objections (2)

[§3 (Theorem 2 and Figure 4)] The proof of Theorem 2 rests on admissible sets constructed by a greedy sieving algorithm. The paper states that 'the computations in bold in table 8 and the associated images provide a proof', but neither the table nor the images specify the actual ring elements or residue classes selected. Because the greedy algorithm’s output depends on tie‑breaking and candidate ordering, a reader cannot independently verify the existence of an admissible set of the required size inside the claimed disc. Without the explicit admissible sets (or at least a detailed specification of the residue choices and candidate list), the argument for Theorem 2 is incomplete and not independently verifiable. The authors should provide the exact admissible sets as supplementary data.
[§3, first paragraph] The relationship between the initial disc radius B and the diameter of the final admissible set is unclear. The text says 'the norm of whose difference is less than 4B', but later the reported diameters (e.g., 218) do not obviously equal 4B. Clarify how the diameter is computed from the candidate set and how the figures in Table 4 (Figure 4) are derived.

Minor objections (7)

[§1, after Theorem 1] 'Vatawani' should be 'Vatwani' (also in the caption of Figure 1).
[§2, Figure 1] The column header 'D:k=50' is ambiguous; replace with 'Diameter (k=50)' for clarity.
[§3 (pp. 6–7)] Several instances of 'lies is a set' should be 'lies in a set' (e.g., 'lies is a set of (Gaussian) diameter 404').
[§5] The heuristic discussion is interesting but its connection to the main computational findings could be made sharper. Consider stating explicitly that this section does not affect the validity of the theorems.
[References] Reference [15] (Song and Yue) is listed as an eprint; if it has been accepted or published, please update the reference.
[§4] The notation π(0, A) is used for the disc prime count; earlier the paper uses π(ξ, A) for a generic centre. To avoid confusion, consider renaming the disc count, e.g., π_disc(0, A).
[§5 (Figure 8)] The 'advantage measure' A_K(x)/log x is presented without explanation of how it relates to the likelihood of finding counterexamples. A brief interpretive sentence would help.

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The paper

The paper claims

Construction of dense admissible sets yields new prime gap records in the nine imaginary quadratic fields of class number 1 and produces explicit counterexamples to a generalized second Hardy-Littlewood conjecture.

The reading

The paper constructs admissible sets of integers in the rings of integers of the nine imaginary quadratic fields of class number 1, including the Gaussian and Eisenstein integers. Using sieving techniques adapted to the splitting behavior of rational primes in these fields, the author produces admissible sets with large size relative to their diameter. For the one-dimensional case (elements differing by rational integers), exhaustive search yields the smallest known diameters, giving new upper bounds on the rational integer difference between large primes in each field (Theorem 1). For the disc-based two-dimensional case, greedy algorithms produce admissible sets with smaller norm differences than previously known; for Gaussian primes the bound is 218 (Theorem 2). Additionally, very small admissible sets are used to supply explicit counterexamples to the natural generalization of the second Hardy-Littlewood conjecture for all fields except Q(√-11) (Theorem 3).

What carries the argument

The central objects are admissible sets: finite subsets of the ring of integers such that for every prime ideal p, the reductions of the set modulo p do not cover all residue classes. For the one‑dimensional case, the Schinzel sieve is adapted to the field’s ramification and splitting: inert primes eliminate no rational candidates, and split primes can be assigned a common residue class to halve the elimination rate, allowing exhaustive optimization. For the disc‑based case, the initial candidate set is taken as the integers of norm less than a bound, and residue classes modulo prime ideals of norm ≤ k are chosen greedily (or by a “greedy composites” variant) to maximize the number of survivors; the set’s diameter is then bounded by the disc diameter. The choice of disc centre (origin vs. (1+α)/2) gives an additional degree of freedom not available in the rational case.

Extended reading notes

Core claim

The paper establishes three theorems. Theorem 1: For each of the nine class‑1 imaginary quadratic fields, there exist admissible sets of size 50 (and 49, 48) contained in short intervals of rational integers, yielding new records on the smallest rational integer difference between large primes in those fields (e.g., ≤136 for Gaussian primes, ≤93 for Eisenstein primes). Theorem 2: When admissible sets are allowed to be drawn from two‑dimensional discs, the smallest norm difference between distinct prime elements in the field is reduced; for Gaussian primes the bound is ≤218, and similar improvements are given for the other eight fields. Theorem 3: The generalization of the second Hardy‑Littlewood conjecture to these fields — that the number of primes in a disc is maximized when the disc is centered at the origin — is false for all except Q(√-11); explicit sets of prime elements are provided as counterexamples.

Load-bearing premise

For the disc‑based admissible sets, the greedy algorithms that choose residue classes are not proven to find the smallest possible diameter; the claimed bounds are the best known from those heuristics, not proven optimal.

Editorial extensions

If this is right

The improved admissible sets can be combined with the bounded‑gaps results of Castillo, Hall, Lemke Oliver, Pollack, Thompson and Vatwani to assert that there are infinitely many pairs of prime elements in each field whose norm difference is bounded by the numbers in Theorems 1 and 2.
The explicit counterexamples to the generalized second Hardy‑Littlewood conjecture show that the natural analog of that conjecture fails for all but one of the class‑1 fields, casting doubt on other uniform distribution conjectures for primes in these rings.
The same sieving strategy can be applied to other number fields with class number 1 (or small discriminant) to obtain analogous prime‑gap records.
The heuristic of Section 5 suggests that for sufficiently large discs, dense admissible sets should exist in every field, implying that counterexamples to the generalized conjecture should exist even for Q(√-11) at larger radii.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The failure to find a counterexample in Q(√-11) is not likely to be fundamental: the small primes split in that field, making the sieving less efficient, but the heuristic indicates that a counterexample should appear for a larger disc if the greedy search is extended or improved.
The greedy algorithms used for disc‑based sets are not proven optimal; the reported diameters are only the best known. A more systematic search (e.g., by optimization over residue class assignments for composite moduli) might yield further reductions.
The paper’s disc‑based approach is methodologically related to the “Gaussian zoo” problem, but with the goal of maximizing survivors rather than achieving connectivity among primes, suggesting a new computational direction for studying the local distribution of primes in imaginary quadratic fields.

Original abstract

Let $\mathcal O_K$ be the ring of integers of a number field $K$. A finite subset $\mathcal H=\{\eta_1,\ldots,\eta_k\}\subset\mathcal O_K$ is said to be \emph{admissible} if for every prime ideal $\mathfrak p$ in $\mathcal O_K$, the reductions of the $\eta_i$ modulo $\mathfrak p$ do not represent every residue class. The diameter of a finite subset $S$ of $\mathcal O_K$ is defined to be \[\mathrm{diam}(S):= \max_{\eta_i,\eta_j\in S:\eta_i\neq\eta_j}|\mathcal N(\eta_i-\eta_j)|\] where $\mathcal N$ denotes the norm down to the rational field. This paper computationally investigates admissible subsets of rings of integers in imaginary quadratic fields of class number 1 (such as the Gaussian integers or the Eisenstein integers) that are large relative to their diameter. The motivation is twofold. Firstly, there is the improvement of results analogous to the question of an upper bound for the least gap between large rational primes, which has been one of the great number theoretic successes of the twenty-first century. A second motivation is that there is the opportunity to disprove a variation of the second Hardy-Littlewood conjecture. The calculations in the paper provide new records on the smallest distance between primes in the fields in question, and explicit counter-examples to a generalisation of the second Hardy-Littlewood conjecture in all bar one of the fields.

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.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-27: 1 request, 6,567 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.32 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.16 · compares with prior results 0.54 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Let $\mathcal O_K$ be the ring of integers of a number field $K$. A finite subset $\mathcal H=\{\eta_1,\ldots,\eta_k\}\subset\mathcal O_K$ is said to be \emph{admissible} if for every prime ideal $\mathfrak p$ in $\mathcal O_K$, the reductions of the $\eta_i$ modulo $\mathfrak p$ do not represent every residue class. The diameter of a finite subset $S$ of $\mathcal O_K$ is defined to be \[\mathrm{diam}(S):= \max_{\eta_i,\eta_j\in S:\eta_i\neq\eta_j}|\mathcal N(\eta_i-\eta_j)|\] where $\mathcal N$ denotes the norm down to the rational field. This paper computationally investigates admissible subsets of rings of integers in imaginary quadratic fields of class number 1 (such as the Gaussian integers or the Eisenstein integers) that are large relative to their diameter. The motivation is twofold. Firstly, there is the improvement of results analogous to the question of an upper bound for the least gap between large rational primes, which has been one of the great number theoretic successes of the twenty-first century. A second motivation is that there is the opportunity to disprove a variation of the second Hardy-Littlewood conjecture. The calculations in the paper provide new records on the smallest distance between primes in the fields in question, and explicit counter-examples to a generalisation of the second Hardy-Littlewood conjecture in all bar one of the fields.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $\mathcal O_K$ be the ring of integers of a number field $K$.	background 1.00	0.04	0.01	asserted 0.78
s1	A finite subset $\mathcal H=\{\eta_1,\ldots,\eta_k\}\subset\mathcal O_K$ is said to be \emph{admissible} if for every prime ideal $\mathfrak p$ in $\mathcal O_K$, the reductions of the $\eta_i$ modulo $\mathfrak p$ do not represent every residue class.	background 1.00	0.05	0.02	asserted 0.94
s2	The diameter of a finite subset $S$ of $\mathcal O_K$ is defined to be \[\mathrm{diam}(S):= \max_{\eta_i,\eta_j\in S:\eta_i\neq\eta_j}\|\mathcal N(\eta_i-\eta_j)\|\] where $\mathcal N$ denotes the norm down to the rational field.	background 0.96	0.13	0.02	asserted 0.98
s3	This paper computationally investigates admissible subsets of rings of integers in imaginary quadratic fields of class number 1 (such as the Gaussian integers or the Eisenstein integers) that are large relative to their diameter.	contribution 0.98	0.93	0.07	asserted 0.97
s4	The motivation is twofold.	background 0.63	0.17	0.01	asserted 0.83
s5	Firstly, there is the improvement of results analogous to the question of an upper bound for the least gap between large rational primes, which has been one of the great number theoretic successes of the twenty-first century.	background 0.96	0.06	0.03	asserted 0.97
s6	A second motivation is that there is the opportunity to disprove a variation of the second Hardy-Littlewood conjecture.	background 0.82	0.14	0.03	hedged possibility 0.74
s7	The calculations in the paper provide new records on the smallest distance between primes in the fields in question, and explicit counter-examples to a generalisation of the second Hardy-Littlewood conjecture in all bar one of the fields.	result 0.99	0.95	0.85	asserted 1.00

What kind of paper the classifier reports this is: simulation study 0.84 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Admissible sets in an interval results 5

**Figure 2.** Zhang and Schinzel approaches One plausible source of the difference between the rational and quadratic field cases is the greater non-uniformity in the representation of residue classes within our chosen region. In the rational case, our region is an interval of the real line of length, say, L, and the number of integer candidates eliminated by any single residue class modulo a prime p is either [L/p] or [L/p] + 1. In the Gaussian case with a disc of area, say A, if we consider the residue class zero modulo some prime ρ, the candidates eliminated 7

**Figure 3.** Polymath, odd greedy, and pure greedy approaches

**Figure 4.** Admissible sets inside discs results The shortest diameters found are highlighted in bold. For these sets we plot the Minkowski embeddings of the corresponding admissible sets below: Note that the computations in bold in table 8 and the associated images provide a proof of Theorem 2. The diversity in regions of bounded diameter makes an exhaustive search of the full system of residues like that of section 2 more challenging. There is considerable scope for improvement of the results of this section and hopefully other researchers will be able to contribute improved sets. 9

**Figure 5.** Winning sets ∆ = −3, −4, −7

**Figure 6.** Winning sets ∆ = −8, −11, −19

**Figure 7.** Winning sets ∆ = −43, −67, −163 10

**Figure 8.** Advantage measure across fields and bounds

Figures reproduced from arXiv:2610.34556 by the authors.

References

Discussion

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