PaperlayerView exact revision on arXiv

Source-linked AI summary

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

Daniel Shiu

arXiv:2610.34556v1math.NTcs.DM

TL;DR

The paper addresses how to construct dense admissible sets in imaginary quadratic fields, where rational-set methods do not transfer well. It computationally searches for such sets and reports new prime-gap records and counterexamples to a Hardy–Littlewood analogue, with one field excepted.

  • Problem

    Methods for rational admissible sets do not transfer well to the two-dimensional rings, while dense sets could improve prime-gap bounds and test a second Hardy–Littlewood analogue.

    Source: §1 · p. 2

  • Method

    The paper computationally constructs admissible sets in imaginary quadratic fields of class number 1, including sets with rational-integer differences and sets contained in discs.

    Source: §1 · p. 1, §2 · p. 4, §3 · p. 6

  • Results

    The constructions yield new records for smallest prime differences and explicit counterexamples to the conjecture analogue in all considered fields except Q(√−11).

    Source: §6 · p. 15

  • Takeaways & Limitations

    Rational-integer-difference sets give records for prime differences, while disc-contained sets give records for prime differences measured by norm.

    Source: §6 · p. 15

    The heuristic model is unreliable: one-dimensional constructions have asymptotic excess factor Ω(√log z), rather than merely a constant.

    Source: §5 · p. 13

Abstract

from arXiv · show

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.

1 Introduction

The paper computationally studies admissible sets large relative to their diameter in imaginary quadratic fields of class number 1. It seeks improved prime-gap results and tests a second Hardy–Littlewood conjecture analogue through dense admissible sets.

Admissible-set bounds

  • The constructions improve admissible-set bounds both when elements differ by rational integers and in the general case.Earlier work used the rational-integer admissible set of the Polymath project for Gaussian integers, without optimizing that choice.Source: §1 · pp. 1–2

Prime-pair results

  • For rational primes represented by a²+b², infinitely many pairs have shift 0 < |h| ≤136; in Q(√2), the corresponding bound is 168.The paper also gives bounds 93, 126, 100, 80, 59, 55, and 50 for d = 1, 2, 3, 5, 11, 17, and 41, respectively.Source: §1 · p. 2

Motivation and scope

  • The search concerns dense admissible sets in bounded-diameter regions such as discs, since restricting differences to rational integers constrains the two-dimensional rings.Source: §1 · pp. 1–2

Prime-pair results

  • Infinitely many pairs of distinct Gaussian primes satisfy N(π1 − π2) ≤218.Source: §1 · p. 2

Hardy–Littlewood analogue

  • The proposed Hardy–Littlewood analogue predicts prime concentration around the origin; an admissible set in a disc with more elements than π(0, A0) could challenge it.The introduction says a suitably small shift might then capture all the set’s primes and provide an explicit counterexample.Source: §1 · p. 3
  • The methods yield explicit counterexamples in most cases, but not in Q(√−11), where the authors could not construct a sufficiently dense admissible set.Source: §1 · p. 4

2 Short line segments containing admissible sets

The section develops sieve-based searches for admissible sets in short rational intervals and reports improved diameter records. Scaling these sets also yields bounds on prime differences that are rational integer multiples of a fixed ring element.

2 Short line segments containing admissible sets

  • The search sieves a candidate interval by assigning residue classes to small primes, then greedily chooses classes for larger primes and makes endpoint and residue-class adjustments.A target-sized admissible set is selected when enough candidates survive the sieve.Source: §2 · p. 4
  • For k=50, the least rational diameter is D=136 (Gaussian diameter 136^2); for k=49 and k=48, the least rational diameters are 133 and 131.The k=50 results prove Theorem 1; the other columns align with anticipated improvements to k(2).Source: §2 · pp. 5–6
  • Multiplying an admissible set by a ring element coprime to the sieving set preserves admissibility and scales the prime-difference norm bound by that element’s norm.If the multiplier is not coprime, its prime divisors can instead be removed from the sieving set.Source: §2 · p. 6

3 Discs containing admissible sets

The section searches for large admissible sets by sieving elements in bounded-diameter discs, comparing several residue-class strategies and disc centers. The shortest-diameter sets highlighted in Table 8 and their plotted embeddings provide the proof of Theorem 2.

3 Discs containing admissible sets

  • The search sieves ring elements inside N(η) < B discs by choosing residue classes modulo prime ideals with norms at most k.Survivors avoid all specified residue classes; more than k(2) survivors guarantee pairs of primes whose difference has norm less than 4B.Source: §3 · p. 6
  • The cited cases give Gaussian diameters 592 at the 489th candidate (norm 153), 298 at the 250th (norm 81), and 244 for odd greedy at the 190th (norm 61), each at the 50th survivor.A candidate of norm 58 lies in a set of Gaussian diameter 218.Source: §3 · pp. 6–7
  • The diagrams suggest that Zhang creates a wide moat, while Schinzel adds central and outer-ring survivors at the cost of a wider moat.Polymath seeks more survivors in the moat, and greedy approaches capture some useful structure without explicitly building it into their choices.Source: §3 · p. 7
  • Disc residue-class counts are less uniform than interval counts, motivating boundary-aware choices and a composite-modulus greedy sieve that matches prime-greedy for Gaussian k=50 but improves other fields.The composite sieve chooses a residue class modulo a product of small prime ideals to minimize the surviving candidate count.Source: §3 · pp. 7–8
  • Centering the starting disc at (1 + α)/2 worsened the Gaussian k=50 result but improved results in some other fields.The section compares this center with the origin-centered disc across four approaches per field.Source: §3 · pp. 8–9
  • The shortest-diameter sets highlighted in Table 8 and their plotted embeddings provide the proof of Theorem 2.The section identifies the winning sets for the fields shown in Figures 4–7.Source: Figure 4 · p. 9, §3 · p. 9, Figure 5 · p. 10
  • The variety of bounded-diameter regions makes exhaustive searches over the full residue system more challenging, and the reported sets leave considerable room for improvement.The authors invite further contributions of improved sets.Source: §3 · p. 9

4 Extending the second Hardy-Littlewood conjecture

The section searches for admissible sets that can contradict an extended second Hardy–Littlewood conjecture, beginning with fields where prime norms have useful gaps. It constructs explicit prime counterexamples in several fields, while the search does not succeed in Q(√−11).

4 Extending the second Hardy-Littlewood conjecture

  • The search proceeds in reverse discriminant order, anticipating easier counterexamples in fields with larger discriminants because their very small prime sets are sparse.The authors relate this expectation to the small values of π(0, A) in those fields.Source: §4 · p. 11
  • In Q(√−163), the disc N(η) ≤ 4 contains five integers, and selecting a residue class for the inert prime 2 yields a translated admissible set whose elements are prime.The resulting five-tuple is explicitly listed.Source: §4 · p. 11
  • In Q(√−19), sieving by the prime factors of 5 requires eliminating at least one element, but the four-element set {−2, −1, 1, 2} remains admissible.Its translate gives an explicit counterexample; the passage also notes that the prime element α19 could enlarge it.Source: §4 · p. 11
  • For Q(√−11), the greedy search finds no suitable set among up to 4,000 candidates, but this does not rule out such a set within them.The authors identify this as a limitation of the search methods, not evidence of nonexistence.Source: §4 · p. 11
  • In Q(√−8), A0=9 gives 23 candidates and requires at least seven survivors, from which an admissible set is readily found.The passage gives one such seven-element set.Source: §4 · p. 11
  • Explicit prime counterexamples are given in Q(√−2), Q(√−7), the Gaussian and Eisenstein integers, and Q(√−163), Q(√−67), Q(√−43), and Q(√−19).The constructions include five primes in Q(√−163), corresponding to a translated admissible set of five elements.Source: §4 · pp. 11–12

5 Lies, damned lies and heuristics

The section asks whether counterexamples to equation (3) persist for arbitrarily large A, and develops heuristics for the size and distribution of admissible sets in discs. Its model predicts that typical discs fall short of the target, while unusually dense discs may exceed the average.

5 Lies, damned lies and heuristics

  • For class-number-1 fields, the candidate set is taken from elements in a disc of norm less than A and considered modulo a lattice of ideals for primes below z.The quotient points coprime to that ideal represent the admissible candidates considered by the heuristic.Source: §5 · pp. 12–13
  • A random disc is expected to yield wA exp(−γ)/log A admissible elements, below the target, though some discs may exceed the average by more than exp(γ).The target size is the lower bound that would, subject to the first generalized Hardy–Littlewood conjecture, give a counterexample to (3).Source: §5 · p. 13
  • Randomly selecting r residue classes gives a hypergeometric count of admissible residues, tending to a Poisson distribution when r is small relative to NP.The proposed disc model uses Poisson parameter λ = wA exp(−γ)/log A.Source: §5 · p. 13
  • exp(A/log A(1 + o(1))) ≈ exp(1.781λ/w) discs are considered, which offers heuristic hope that dense admissible sets exist.The section asks whether counterexamples to equation (3) occur for arbitrarily large A.Source: §5 · pp. 12–13
  • The model is unreliable: one-dimensional constructions have asymptotic excess Ω(√log z), rather than merely a constant.Small counterexamples may also be partly explained by subsidiary terms in the asymptotics; the proposed advantage measure is approximately large negative A_K(x)/log x, alongside small π(0,A).Source: §5 · pp. 13–14

6 Conclusion

The paper develops methods for dense admissible sets in imaginary quadratic fields of class number 1, since rational constructions transfer poorly. These constructions yield prime-difference records and counterexamples to the generalized second Hardy–Littlewood conjecture in all but one field.

6 Conclusion

  • The paper demonstrates methods for dense admissible sets in imaginary quadratic fields of class number 1, while finding that rational-integer methods transfer poorly.The conclusion also reports new prime-difference records and conjecture counterexamples arising from these constructions.Source: §6 · pp. 14–15
  • The constructions set new records for the smallest rational-integer differences and smallest norms of differences between large primes in the fields studied.Rational-integer differences come from admissible sets whose elements differ by rational integers; norm records come from admissible sets inside small-radius discs.Source: §6 · p. 15
  • Very small admissible sets give explicit counterexamples to the natural generalized second Hardy–Littlewood conjecture in all but Q(√−11).The paper also offers heuristics for the expected distribution of admissible-set sizes.Source: §6 · p. 15
Loading 2610.34556v1…