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A096315
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Dimensions n such that the integer lattice Z^n contains n+1 equidistant points (i.e., the vertices of a regular n-simplex).
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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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]);
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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