Visual Definition of a Prime Number

Most textbooks define a prime number as a number that can be evenly divided only by itself and the number 1.

I have always been interested in the visual and geometric aspect of mathematics, and prefer to define a prime number in another, more graphic manner.

Take an ordinary set of children's building blocks. Assuming that all the blocks are cubes of the same size, it is possible to build solid symmetrical rectangular blocks of various sizes out of these blocks by stacking them together. Some simple examples would be taking four blocks and putting them together to form a square, or taking eight blocks and building a cube with four on top and four on the bottom, or building a rectangular-shaped block by stacking thirty of them together, two by three by five.

Considering the three simple examples above, we can easily see that they in a sense are graphic representations of non-prime numbers and their factors. Two times two is four, represented by the flat square pattern of four blocks. Two times two times two is eight, represented by the block built by stacking eight blocks evenly, two by two by two. Similarly, the factors of the number 24 are represented by the last example given above, the rectangular block of 2 by 3 by 4.

Non-prime numbers with two prime factors can be represented 2-dimensionally by a 2-dimensional assembling of blocks. Non-primes with three prime factors can be represented by a 3-dimensional stacking of blocks. Numbers with more than three factors require building rectangular shapes of four dimensions or more, hyperspace extensions of various rectangular-like hyperblocks.

For example, the number 24 can be represented by a 4-D rectangle, two by two by two by three. 72 would be a 5-D rectangle, three of the previous example of 24 stacked up into a fifth dimension of hyperspace.

Now take a prime number of blocks and try doing the same. Try it with three. It is impossible to stack three blocks together into a rectangular shape. With three you can only build an "L" shape, or a line of three in a row. A similar problem occurs with five or seven blocks, or any other prime number of blocks. No matter how you assemble a prime number of blocks, and no matter how many dimensions you use, you can never obtain a perfectly solid rectangular shape. There will always be one left over.

The only thing you can do with a prime number of blocks is to build a single column of blocks, one on top of the other.

Therefore, my "visual" definition of a prime number is* any number
in the set of all numbers of blocks which cannot be used to build a regular, symmetrical,
solid rectangular shape, regardless of the number of dimensions used, other than by
placing them together to build a single, one-dimensional column.*

(Note also that this applies to the prime numbers 1 and 2.)

Prime numbers, then, are the set of all "one-dimensional" numbers. They can be arranged only in one dimension. All non-prime numbers are "multi-dimensional", having two or more dimensions. For instance, non-prime numbers with two prime factors are "2-dimensional" numbers, and non-primes with three prime factors are "3-dimensional".

I realize that this is merely saying the same thing as the traditional definition of a prime number, a number which is divisible only by itself and 1. I just think that putting into these terms makes the concept a little more tangible.

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Copyright © 1998 by Bill Price