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"reference_text": "K. Sch\u00fctte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96\u2013124, doi:10.1007/BF02054944.3\u2062x3\u22129\u2062x2\u22123\u2062x+1",
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"reference_text": "K. Sch\u00fctte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96\u2013124, doi:10.1007/BF02054944.7\u2062x2+2\u2062x\u22121",
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"reference_text": "K. Sch\u00fctte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96\u2013124, doi:10.1007/BF02054944.3\u2062x\u22121",
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"reference_text": "K. Sch\u00fctte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96\u2013124, doi:10.1007/BF02054944; L. Danzer, Finite point-sets on S2 with minimum distance as large as possible, Discrete Math. 60 (1986), 3\u201366, doi:10.1016/0012-365X(86)90002-6.7\u2062x3\u22124\u2062x2\u22122\u2062x+1",
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"reference_text": "K. Sch\u00fctte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96\u2013124, doi:10.1007/BF02054944; L. Danzer, Finite point-sets on S2 with minimum distance as large as possible, Discrete Math. 60 (1986), 3\u201366, doi:10.1016/0012-365X(86)90002-6.5\u2062x2\u22121",
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"reference_text": "L. Fejes [T\u00f3th], \u00dcber eine Absch\u00e4tzung des k\u00fcrzesten Abstandes zweier Punkte eines auf einer Kugelfl\u00e4che liegenden Punktsystems, Jber. Deutsch. Math.-Verein. 53 (1943), 66\u201368.5\u2062x2\u22121",
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"reference_text": "K. Sch\u00fctte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96\u2013124, doi:10.1007/BF02054944; O. R. Musin and A. S. Tarasov, The strong thirteen spheres problem, Discrete Comput. Geom. 48 (2012), 128\u2013141, doi:10.1007/s00454-011-9392-2.x8\u221224\u2062x7\u221212\u2062x6+8\u2062x5+38\u2062x4+24\u2062x3\u221212\u2062x2\u22128\u2062x+1",
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"reference_text": "K. Sch\u00fctte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96\u2013124, doi:10.1007/BF02054944; O. R. Musin and A. S. Tarasov, The Tammes problem for N=14, Exp. Math. 24 (2015), 460\u2013468, doi:10.1080/10586458.2015.1022842.4\u2062x4\u22122\u2062x3+3\u2062x2\u22121",
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"reference_text": "K. Sch\u00fctte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96\u2013124, doi:10.1007/BF02054944; D. A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158\u2013165, doi:10.1107/S0108767390011370; mt0-svg, The Tammes problem for fifteen points, preprint, 2026, doi:10.5281/zenodo.23450670, https://github.com/mt0-svg/tammes-15.13\u2062x5\u2212x4+6\u2062x3+2\u2062x2\u22123\u2062x\u22121",
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"reference_text": "K. Sch\u00fctte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96\u2013124, doi:10.1007/BF02054944.23\u2062x6+6\u2062x5+5\u2062x4+4\u2062x3\u22123\u2062x2\u22122\u2062x\u22121",
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"reference_text": "T. Tarnai and Zs. G\u00e1sp\u00e1r, Improved packing of equal circles on a sphere and rigidity of its graph, Math. Proc. Cambridge Philos. Soc. 93 (1983), 191\u2013218, doi:10.1017/S0305004100060485.216\u2062x11+108\u2062x10+118\u2062x9+209\u2062x8\u2212100\u2062x7\u2212160\u2062x6+40\u2062x5+46\u2062x4\u221220\u2062x3\u221212\u2062x2+2\u2062x+1",
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"reference_text": "D. E. Lazi\u0107, V. \u0160enk, and I. \u0160e\u0161kar, Arrangements of points on a sphere which maximize the least distance, Bull. Appl. Math. (Technical University Budapest) 47 (1987), 7\u201321.Minimal polynomial listed in plain text data",
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"reference_text": "B. L. van der Waerden, Punkte auf der Kugel. Drei Zus\u00e4tze, Math. Ann. 125 (1952), 213\u2013222, doi:10.1007/BF01343118.21\u2062x3\u22129\u2062x2\u22125\u2062x+1",
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"reference_text": "D. A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158\u2013165, doi:10.1107/S0108767390011370.",
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"reference_text": "D. A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158\u2013165, doi:10.1107/S0108767390011370.Minimal polynomial listed in plain text data",
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"reference_text": "T. Tarnai and Zs. G\u00e1sp\u00e1r, Arrangement of 23 points on a sphere (on a conjecture of R. M. Robinson), Proc. Roy. Soc. London Ser. A 433 (1991), 257\u2013267, doi:10.1098/rspa.1991.0046.",
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"reference_text": "R. M. Robinson, Arrangements of 24 points on a sphere, Math. Ann. 144 (1961), 17\u201348, doi:10.1007/BF01396539.7\u2062x3+x2\u22123\u2062x\u22121",
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"reference_text": "D. A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158\u2013165, doi:10.1107/S0108767390011370.Minimal polynomial listed in plain text data",
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"reference_text": "N. J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994\u20131996.",
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"reference_text": "N. J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994\u20131996.",
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"reference_text": "N. J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994\u20131996; J. Buddenhagen and D. A. Kottwitz, Multiplicity and symmetry breaking in (conjectured) densest packings of congruent circles on a circle, undated manuscript.",
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"reference_text": "N. J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994\u20131996; K. J. Nurmela, Constructing spherical codes by global optimization methods, Research Report A32, Digital Systems Laboratory, Helsinki University of Technology, 1995, https://research.ics.aalto.fi/publications/info/bibdb.HUT-TCS-A32.shtml.",
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