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

1, 3, 7, 8, 9, 11, 15, 17, 19, 23, 24, 25, 27, 31, 33, 35, 39, 43, 47, 48, 49, 51, 55, 57, 59, 63, 67, 71, 73, 75, 79, 80, 81, 83, 87, 89, 91, 95, 97, 99, 103, 105, 107, 111, 115, 119, 120, 121, 123, 127, 129, 131, 135, 139, 143, 145, 147, 151, 155, 159, 161, 163, 167

Offset: 1

Author

David Radcliffe, Aug 01 2004

Keywords

• easy
• nonn
• changed

Comments

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.

Examples

			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.
		

Links

• Munenori Inagaki, Hideki Matsumura, Masanori Sawa, and Yukihiro Uchida, On a group of normalized solutions of the higher-dimensional Prouhet-Tarry-Escott problem, arXiv:2508.20733 [math.NT], 2025. See p. 6.
• Hiroshi Maehara and Horst Martini, Elementary geometry on the integer lattice, Aequationes mathematicae, 92 (2018), 763-800. See Sec. 3.2.
• I. J. Schoenberg, Regular Simplices and Quadratic Forms, J. London Math. Soc. 12 (1937) 48-55.

Crossrefs

Contains A033996 except 0.

Cf. A000290, A001481, A016813, A097269.

Programs

Maple

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]);