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Partitions and Prime Numbers

Partitions and Prime Numbers

When you break a number into "chunks" it's called a partition. For example {2,3,5} is a partition of 10 because 10=2+3+5. It's the partition that breaks 10 up into 3 chunks, one of size 2, one of size 3 and one of size 5. In fact 10 has 42 different partitions.

All numbers have partitions. In fact, any given number has lots and lots of partitions. But if you require that all chunks in a partition be the same size then things get very interesting. There are some numbers that cannot be broken into partitions with same size pieces. These numbers are called prime numbers.

It turns out that prime numbers play a fundamental role in mathematics.

So 3 is prime, but it's such a small number that it's prime property is not displayed that well. Let's look at some slightly bigger numbers. Is 10 prime? No, because it can be broken into equal sized pieces, for example you can break it into 2 pieces each of size 5. That's the partition 10 = {5,5}. Or you could do the partition 10 = {2,2,2,2,2}.

OK, then what about 11, is that prime? Yes. Try it. You will never break 11 into equal size pieces.

Are there other prime numbers?
You bet! Here's the list of all prime numbers below 100..
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

So we have the following "sophisticated" definition..

A prime number is one that cannot be broken into equal size pieces.

Of course, this definition is exactly equivalent to the "standard" definition you find in most math books..

A prime number is one with no divisors (other than 1 and itself).

I prefer the "sophisticated" definition, because it's phrased in terms of real world operations i.e. trying to break a group of objects into equal size pieces. This demonstrates the real world implications.

For example, your farm has 827 beautiful acres. Your Will states that when you die the acres will be divided equally amongst your children. Big mistake. No matter how many children you have this is not possible because 827 is a prime number. So it cannot be divided into equal size chunks.

Content written and posted by Ken Abbott