By the Numbers - some interesting facts of mathematics

2 - Every even number greater or equal to 4 can be expressed as a sum of exactly two (not necessarily distinct) prime numbers. The first few examples are: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 5 + 5, 12 = 5 + 7, etc. Cf. Goldbach's Conjecture - Alright, so this technically isn't a "fact" yet. But this conjecture is strongly suspected to be true. Certainly, it has been proven to be true for "very large" numbers, and no known counter-example of this has ever been found in over 2 centuries.

4 - Every positive integer can be expressed as a sum of the squares of 4 (or less) integers. Cf. the Four-Square Theorem by Lagrange.

4 - The minimum number of colors needed to paint an arbitrary map so that all adjacent regions are guaranteed to be of different colors. Cf. the Four-Color Theorem.

4, 6, 8, 12, and 20 - The number of sides of the 5 regular solids, usually known as the Platonic Solids. A regular solid is a one whose facets are all congruent to one another and are either a right triangle, a square, or a regular pentagon. Euclid proved over 2000 years ago that only 5 such solids exist. They are tetrahedron (4 sides), cube (6 sides), octahedron (8 sides), dodecahedron (12 sides), and icosahedron (20 sides).

5 - There does not exist a general formula, one consisting solely of elementary arithmetic operations and radicals, that gives the roots of all polynomials of degree 5 or higher. (You all know about the quadratic formula which can solve every second degree polynomial. There are also cubic and quartic formulas solving all polynomials of degree 3 and 4, respectively.) This fact is one consequence of the Galois Theory.

6 - The smallest perfect number. A perfect number is an integer such that the sum of its proper divisors is equal to the integer itself: 6 = 1 + 2 + 3.

1729 - The smallest integer that can be expressed as the sum of the cubes of two other integers in two different ways. 1729 = 9³ + 10³ = 1³ + 12³. (This was the subject of a very famous mathematical anecdote involving Srinivasa Ramanujan and G.H. Hardy, circa 1917. See A Mathematician's Apology by Hardy.

2^6972593 - 1 - The value of the largest known prime number (as of September, 1999). It is an instance of Mersenne primes. It has 2098960 digits!

n - The exact number of roots (not necessarily distinct), real and/or complex, that an nth-degree polynomial with real or complex coefficients will have. Cf. the Fundamental Theorem of Algebra.

n(n + 1)/2 - The sum of first n positive integers, 1, 2, 3, ... , n - 1, n. For example, when n = 100, the sum of 1 + 2 + 3 + ... +100 is (100 * 101)/2 = 5050, as you probably already know - from an anecdote about a young Carl Friedrich Gauss, perhaps?

A couple of things we see in this column will be explored further next time - perfect numbers and their relationship with Mersenne primes. Thus:

Next time: Perfect Numbers