cp's OEIS Frontend

A096315 Dimensions n such that the integer lattice Z^n contains n+1 equidistant points (i.e., the vertices of a regular n-simplex).

%I A096315 #20 Sep 02 2025 14:13:11
%S A096315 1,3,7,8,9,11,15,17,19,23,24,25,27,31,33,35,39,43,47,48,49,51,55,57,
%T A096315 59,63,67,71,73,75,79,80,81,83,87,89,91,95,97,99,103,105,107,111,115,
%U A096315 119,120,121,123,127,129,131,135,139,143,145,147,151,155,159,161,163,167
%N A096315 Dimensions n such that the integer lattice Z^n contains n+1 equidistant points (i.e., the vertices of a regular n-simplex).
%C A096315 Schoenberg proved that a regular n-simplex can be inscribed in Z^n in the following cases and no others: (1) n is even and n+1 is a square; (2) n == 3 (mod 4); (3) n == 1 (mod 4) and n+1 is the sum of two squares.
%H A096315 Munenori Inagaki, Hideki Matsumura, Masanori Sawa, and Yukihiro Uchida, <a href="https://arxiv.org/abs/2508.20733">On a group of normalized solutions of the higher-dimensional Prouhet-Tarry-Escott problem</a>, arXiv:2508.20733 [math.NT], 2025. See p. 6.
%H A096315 Hiroshi Maehara and Horst Martini, <a href="https://doi.org/10.1007/s00010-018-0557-4">Elementary geometry on the integer lattice</a>, Aequationes mathematicae, 92 (2018), 763-800. See Sec. 3.2.
%H A096315 I. J. Schoenberg, <a href="https://doi.org/10.1112/jlms/s1-12.45.48">Regular Simplices and Quadratic Forms</a>, J. London Math. Soc. 12 (1937) 48-55.
%e A096315 There is no equilateral triangle in the plane whose vertices have integer coordinates, so 2 is not on the list. But there is a regular tetrahedron in space whose vertices have integer coordinates, namely (0,0,0), (0,1,1), (1,0,1), (1,1,0), hence 3 is on the list.
%p A096315 select(n->(is(n,even) and issqr(n+1)) or (n mod 4 = 3) or ((n mod 4 = 1) and (numtheory[sum2sqr](n+1)<>[])),[ $1..200]);
%Y A096315 Contains A033996 except 0.
%Y A096315 Cf. A000290, A001481, A016813, A097269.
%K A096315 easy,nonn,changed
%O A096315 1,2
%A A096315 _David Radcliffe_, Aug 01 2004