Subject: Re: Uniqueness of number 30
Date: Thu, 31 Oct 1996 00:00:08
From: Alex Bogomolny

Hello,

Have you thought about the question you asked me? I came up with several properties of 30:

  1. 30 is the max number of edges that may be had by a regular convex polyhedron.
  2. 30 is the largest number such that all smaller numbers prime relative to it are actually prime.
  3. There are exactly 30 ways to color a cube with 6 colors.
  4. 12 is 30 in base 4 - the claim uses the first five digits.
  5. 30 is a primorial number, viz., 30 = primorial(5) = 2*3*5. (primorial(p) is defined for prime p's and equals the product of all primes up to and including p.)
  6. 30 is the smallest integer with three distinct prime divisors.
  7. There are only two Pythagorean triangles with perimeter equal their areas: 5-12-13 and 6-8-10. In the first case the number is 30, in the second 24.

Perhaps you can declare a competition in your math school. Somebody may actually come up with a novel property.

A useful book to look for unusual properties of numbers is

D.Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1986.

Best regards,
Alex

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