Dodecahedral number

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A dodecahedral number is a figurate number that represents a dodecahedron. The nth dodecahedral number is given by the formula

The first such numbers are 0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, 5456, 7140, 9139, 11480, … (sequence A006566 in the OEIS).

Primality[edit]

A dodecahedral number can never be prime--the nth dodecahedral number is always divisible by n. This can be proved very simply:

  • n is part of the numerator. There are no fractions in the numerator alone, so the numerator is divisible by n.
  • Out of or , one of the two must be even. Therefore, the numerator is divisible by 2.
  • Given the above, the numerator must be divisible by 2n.
  • Noting the denominator, . Therefore, the nth dodecahedral number is always divisible by n.

References[edit]

Kim, Hyun Kwang, On Regular Polytope Numbers (PDF), archived from the original (PDF) on 2010-03-07