The hexagonal versus the square lattice
| dc.creator | Moree, Pieter | |
| dc.creator | Riele, Herman J. J. te | |
| dc.date | 2002-04-28 | |
| dc.date.accessioned | 2026-07-07T04:48:06Z | |
| dc.date.available | 2026-07-07T04:48:06Z | |
| dc.description | We establish Schmutz Schaller's conjecture that the hexagonal lattice is `better' than the square lattice. Schmutz Schaller (Bulletin of the AMS 35 (1998), p. 201), motivated by considerations from hyperbolic geometry, conjectured that in dimensions 2 to 8 the best known lattice sphere packings have `maximal lengths' and goes on to write: "In dimension 2 the conjecture means in particular that the hexagonal lattice is `better' than the square lattice. More precisely, let 0<h_1<h_2<... be the positive integers, listed in ascending order, which can be written as h_i=x^2+3y^2 for integers x and y. Let 0<q_1<q_2<... be the positive integers, listed in ascending order, which can be written as q_i=x^2+y^2 for integers x and y. Then the conjecture is that q_i<=h_i for i=1,2,3,..." Our proof requires computational prime number theory in combination with methods from a preprint of the first author (to appear in Math. Comp.), arXiv:math.NT/0112100. | |
| dc.description | 24 pages, 6 figures, 2 tables | |
| dc.identifier | https://arxiv.org/abs/math/0204332 | |
| dc.identifier | http://arxiv.org/abs/math/0204332 | |
| dc.identifier | Math. Comp. 73 (2004), 451-473 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63925 | |
| dc.subject | Number Theory | |
| dc.subject | Metric Geometry | |
| dc.subject | 11N13;11Y35;11Y60 | |
| dc.title | The hexagonal versus the square lattice | |
| dc.type | text |