Definition: Prime Numbers

Definition 2: A prime number is a natural number that is greater than 1 and is only divisible by itself and 1. //

Notes:

  • By “divisible” I mean “evenly divisible”, meaning that there is no remainder after dividing.  Thus 4 is divisible by 2 but not by 3, and 10 is divisible by 5 but not by 4.
  • 1 is not a prime number, because 1 is not greater than 1!
  • 2, 3, 5, 7, 11, 13, and 17 are prime.  For example, 5 ONLY equals 5\times 1.
  • 4, 6, 8, 9, 10, 12 and 14 are not prime.  For example, 4=4 \times 1 AND 4=2\times 2, which is not allowed for a prime.
  • Every number is either prime or not prime—there is no “in-between” or “both”.  This should be clear, because something is either “divisible by something else” or it isn’t. Period.
  • Whole numbers that are not prime are called composite numbers.

Math is not just about making definitions, it is also about proving stuff with these definitions.  Once we’ve made this definition, which we are totally allowed to do, we can now ask questions about the definition.  How many primes are there?  What is the smallest prime?  How many primes are sums of smaller primes?  What patterns exist amongst the primes?

There are an infinite number of questions that we could ask about prime numbers once we have the definition, and some are harder to answer than others.  Elsewhere on this site we have proved that there are an infinite number of primes, but some simple questions about primes have yet to be answered conclusively.  We’ll explain some of these problems later on, but for now just know that there is immense beauty wrapped up in this seemingly harmless definition!

If you check out this note on the method of “proof by contradiction”, then you’ll be ready to understand the incredibly gorgeous proof that there are indeed infinitely many primes!

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