Prime Numbers

The numbers $\,2, 3, 4,\, \ldots\,$ can be expressed as products in a very natural way! Just keep ‘breaking them down’ into smaller and smaller factors until you can't get the ‘pieces’ any smaller.

For example:

OR

OR

No matter how the number is ‘broken down’, you'll always get to the same place, except for possibly different orderings of the factors.

In the example above, you always get: $$360 = 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 5$$ Three factors of $\,2\,,$ two factors of $\,3\,,$ and one factor of $\,5\,.$

These smallest ‘pieces’ (like $\,2\,,$ $\,3\,$ and $\,5\,$ above) are, in a very real way, basic ‘building blocks’ for numbers being represented as products. They're very, very, very important! So, you shouldn't be surprised that these ‘multiplicative building blocks’ are given a special name:

Notes on the Definition

• The first few prime numbers are $\,2\,,$ $\,3\,,$ $\,5\,,$ $\,7\,,$ $\,11\,,$ $\,13\,,$ $\,17\,,$ $\,19\,,$ $\,23\,,$ $\,29\,,$ $\,31\,$ and $\,37\,.$ Want more? Hop up to WolframAlpha and type in (say):

  prime numbers less than 100

• The number $\,1\,$ is not prime. A reason for this is discussed in the next section.
• Recall that there are several different ways to say ‘goes into evenly’. In particular, ‘$\,n\,$ goes into $\,p\,$ evenly’ is equivalent to ‘$\,p\,$ is divisible by $\,n\,$’.
• Every number is divisible by $\,1\,,$ and every number is divisible by itself. If a number is prime, then—that's all there is!

• The divisibility tests are useful for determining if a number is not prime. That is, by looking at just the last digit of the number we can quickly conclude that certain numbers aren't prime.

  For example:

  • If a number (greater than $\,2\,$) ends in the digits $\,0\,,$ $\,2\,,$ $\,4\,,$ $\,6\,,$ or $\,8\,,$ then it is divisible by $\,2\,,$ and hence not prime.
  • If a number (greater than $\,5\,$) ends in the digit $\,5\,,$ then it is divisible by $\,5\,,$ and hence not prime.

  Thus, we've eliminated ending digits of $\,0\,,$ $\,2\,,$ $\,4\,,$ $\,5\,,$ $\,6\,,$ and $\,8\,.$

• It follows that all prime numbers greater than $\,5\,$ must end in one of these digits: $\,1\,,$ $\,3\,,$ $\,7\,,$ $\,9\,$

  That is: If a number greater than $\,5\,$ is prime, then its base ten representation must end in one of these digits: $\,1\,,$ $\,3\,,$ $\,7\,,$ $\,9\,$

  The other direction is NOT true! If a number ends in (say) $\,7\,,$ then it may or may not be prime! For example, $\,17\,$ is prime, but $\,27\,$ is not prime.

• Here's a ‘pile interpretation’ of prime numbers: Suppose you have a bunch of candy (that can't be broken into smaller pieces without being ruined). You want to break the candy into equal piles. If the number of pieces is prime, then there are only two ways to make the piles:

  • one big pile
  • individual piles (piles of size $\,1\,$)

  For example, five pieces of candy can only be piled in two configurations:

  one big pile of five	five piles of one
  XXXXX	X     X     X     X     X

  Compare this with (say) six pieces of candy (which is not prime):

  one big pile of six	two piles of three
  XXXXXX	XXX       XXX
  three piles of two	six piles of one
  XX       XX       XX	X     X     X     X     X     X

• Definitions are conventionally given as ‘ if ’ statements, when they're actually ‘ if and only if ’ statements. (Recall that ‘ if and only if ’ is another way to say ‘ is equivalent to ’.) This is only true for definitions!!

  That is, the definition of prime numbers actually tells us two things for counting numbers greater than $\,1\,$:

  • If the number is prime, then it is divisible only by itself and $\,1\,.$
  • If the number is divisible only by itself and $\,1\,,$ then it is prime.

• Here is a more correct (but less conventional) version of the definition:

  A counting number greater than $\,1\,$ is called prime if and only if the only numbers that go into it evenly are itself and $\,1\,.$

• Here is another more correct (but less conventional) version of the definition:

  Let $\,n\,$ be a counting number greater than $\,1\,.$ Then, $\,n\,$ is prime if and only if the only numbers that go into $\,n\,$ evenly are itself and $\,1\,.$

Practice