Newton–Gregory kissing number problem in unresolved dimensions

Determine the maximum number of non-overlapping unit spheres that can simultaneously touch a given unit sphere in each Euclidean dimension, thereby resolving the Newton–Gregory kissing number problem in all dimensions.

Background

The paper introduces the general spherical-code problem of determining the largest cardinality of a set of unit vectors in Euclidean space subject to a prescribed angular separation. The special case with angular parameter θ=π/3\theta=\pi/3 is the Newton–Gregory kissing number problem, which asks for the largest number of unit spheres that can touch a central unit sphere without overlapping.

The paper notes that the kissing number is known in several dimensions, including dimensions 1, 2, 3, 4, 8, and 24, but remains unresolved in every dimension. This is the only explicitly unresolved problem identified in the paper; the subsequent results concern an upper bound for newly defined p-adic spherical codes rather than a complete determination of their kissing numbers.

References

The case $\theta=\pi/3$ is known as the famous (Newton-Gregory) kissing number problem. With extensive efforts from many mathematicians, it is still not completely resolved in every dimension (but resolved in dimensions $d=1$ ($n=2$), $d=2$ ($n=6$), $d=3$ ($n=12$), $d=4$ ($n=24$), $d=8$ ($n=240$), $d=24$ ($n=196560$)) .

— p-adic Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender Bound  (2503.05654 - Krishna, 22 Feb 2025) in Section 1, Introduction

In dimensions $d\leq 8$, the best known lower bounds are attained by configurations already known in the nineteenth century or earlier; the classical root systems provide examples of all these sizes. Their persistence does not mean that the corresponding optimality questions have all been settled: the exact kissing numbers in dimensions $5$, $6$, and $7$ remain unknown.

— New lower bounds for kissing numbers in dimensions $25$--$29$ and $31$  (2609.21591 - Takhanov et al., 18 Sep 2026) in Section 1, Introduction