A334736 Dimensions d such that the integer lattice Z^d does not contain the vertices of a regular d-simplex.

2, 4, 5, 6, 10, 12, 13, 14, 16, 18, 20, 21, 22, 26, 28, 29, 30, 32, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 50, 52, 53, 54, 56, 58, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 82, 84, 85, 86, 88, 90, 92, 93, 94, 96, 98, 100, 101, 102, 104, 106, 108

Offset: 1

Author

Harry Richman, May 08 2020

Keywords

• nonn

Comments

List contains d such that (1) d is even and d+1 is not a square, or (2) d == 1 (mod 4) and d+1 is not a sum of two squares; proved by Schoenberg.

Examples

			2 is in the list because there is no equilateral triangle in the plane whose vertices all have integer coordinates.
3 is not in the list because there is a regular tetrahedron in space whose vertices have integer coordinates; e.g. (1,1,0), (1,0,1), (0,1,1), (0,0,0).
		

Links

• 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

Complement of A096315.

Cf. A000290, A001481, A022544, A097269.

Extensions

More terms from Jinyuan Wang, May 09 2020