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Another Look at Prime Numbers

Primes are numeric celebrities: they're used in movies, security codes, puzzles, and are even the subject of forlorn looks from university professors.

But mathematicians delight in finding the first 20 billion primes, rather than giving simple examples of why primes are useful and how they relate to what we know. Somebody else can discover the largest prime -- today let's share intuitive insights about why primes rock:

So what are prime numbers again?

A basic tenet of math is that any number can be written as the multiplication of primes. For example:

And primes are numbers that can't be divided further, like 3, 5, 7, or 23. Even the number 2 is prime, if you think about it. And the number 1?

Well, 1 is special and isn't considered prime, since things get crazy because 1 = 1 * 1 * 1... and so on. Even mathematicians take shortcuts sometimes, and leave 1 out of the discussion.

Rewriting a number into primes is called prime decomposition, math speak for "find the factors". Primes seem simple, right?

Well, not really. It turns out that

God, nature, or the flying spaghetti monster -- whatever determined the primes, it made a whole lot of 'em and distributed them in a quirky way.

Analogy: Prime Numbers and Chemical Formulas

Prime numbers are like atoms. We can rewrite any number into a "chemical formula" that shows its parts. In chemistry, we can say a water molecule is really H20:

And for a number, we can break it into primes

Neat relationship, right? In chemistry the "exponent" happens to go underneath -- I'd really prefer exponents above, but the American Chemical Society hasn't replied to my letters.

Why is this interesting? Well, when chemists arranged their basic elements into the periodic table, new insights emerged:

Not bad for reorganizing existing data, eh? Similarly, we can imagine putting the primes (numerical "elements") into a table. But there's a problem.

Nobody knows what the table looks like! Primes are infinite and although we've tried for centuries to find a pattern, we can't. We have no idea where the gaps are or when the next prime is coming. (That's not quite true -- there's interesting hypotheses and conjectures, but the riddle is not solved).

But we won't cry about it, breaking our pencil and sobbing home. You and I are going to make use of the primes even though we don't know every detail.

Organic Chemistry and Functional Groups

I'm no chemistry expert, but I can see a relationship to the primes. Chemical elements have properties based on their location in the periodic table of the elements:

And in organic chemistry there's an idea of a functional group: several atoms can determine the class of the entire molecule. For example:

Those are the basics, if I didn't mess it up. Now let's see what happens when we treat numbers like chemicals.

First Example: Guessing Evenness

In general, an organic chemical contains carbon (not quite, but it's a good starting point). No matter what elements you mix together, if you never add any carbon then you can't create an organic compound.

"Evenness" works the same way. A number is even if it has a 2 in its prime decomposition -- i.e., 2 was used to make the number. There could be a single 2 or fifty; if you have a single 2, you are even, and that's that. If you don't have a 2, you're odd.

Now, remember those math questions that ask how odd and even numbers multiply?

How would you solve this? Guess? Try a few examples? ("Let's see, 3 times 2 is.. 6, but 3 times 3 is 9... so...").

Here's one way to think about it. Multiplication is combining the "prime formulas" for the numbers. Since even numbers contain a "2" somewhere, we can guess that:

Pretty cool, eh? And since 2 is prime, we know we can't "manufacture" a 2 by combining other numbers together.

Thank you prime chemistry, for giving us another way to think about this problem. Now you can even answer questions like this:

It's even, since we mixed in a 2 at the end.

Another Example: Ending with 0

I've read your mind: you want another chemical example, this time with functional groups.

Suppose a number has a "2*5" functional group -- it has one or more 2s and one or more 5s. For example:

Notice a pattern? If a number has a 2 * 5 "functional group", it ends in 0.

Why? Well, 2 * 5 = 10. So having 2 * 2 * 2 * 5 is really like having (2 * 2) * 10. Any whole number multiplied by 10 ends in 0. In general,

So just by looking at the "prime formula" you can determine that the number ends with a 0. You never had to do the multiplication out.

And Another Example: Sum of Digits

What's that? You want another example with functional groups? If you insist.

Let's think about numbers with the "33" functional group. A number could have 400 threes, but as long as there's at least 2 we're interested. If a number has (33) it means

Here's an example:

Again, this is pretty cool. We know something about the sum of digits just by finding a certain functional group in the prime decomposition of the number.

Primes in the Real World

Primes have properties that come in useful.

1. Large numbers are hard to factor. We essentially resort to trial-and-error when doing prime decomposition: one method is to keep trying to divide it by other numbers, up to its square root. The fact that primes and prime decompositions are "secret" can be a good thing for cryptography -- we'll get into this later.

2. Primes don't play well with other numbers. Prime numbers don't "overlap" with the regular numbers: they intersect at the last possible moment. For example, 4 and 6 "overlap" at 12, which is pretty early. Their first "required" overlap is at 4 * 6 = 24.

Primes, however, intersect at the last possible moment. 5 and 7, for example, only coincide at 35 (5*7). There's no intermediate value where they both show up.

You'd think a lack of rhythm would be a bad thing, but in nature it can be an advantage.

The cicada insect sprouts from the ground every 13 or 17 years. This means it has a smaller chance of "overlapping" with a predator's cycle, which could be at a more common 2 or 4-year cycle.

3. Primes are prime everywhere.

The movie "Contact" used primes as a universally understood sequence. It's a non-trivial sequence (2, 3, 5, 7, 11, 13) that would be hard to generate by accident (1, 0, 1, 0 could be made by a swinging pendulum, for example).

And prime numbers are prime in any number system. "1/3" is only a repeating fraction in base 10 (.33333), and you could even argue that pi (3.14159...) is not irrational in base "pi". But everyone can agree that certain numbers are prime and can't be divided. You can even transmit primes in a unary number system that lacks a decimal point:

II
III
IIIII
IIIIIII

So, primes are an infinite, non-repeating, universally-understood sequence, and a good choice for transmitting a message.

Conclusion

Don't hate the primes because they're different -- see how their properties can be useful. "Not fitting in" is a great if it means you don't overlap with a predator, right? Being hard to factor is great if you're making a secret message, right? For a long time primes were considered a purely theoretical curiosity, but lo and behold, we've found situations where they apply.

And that's a large part of math, in my opinion: seeing how strange properties can be useful or relate to the real-world. Math gives us rules, often for games we don't yet play. Our job is to find situations where we want to follow those rules.

There's much more I'd like to say in upcoming posts. If you want to dive into primes, check out Music of the primes which is a decent introduction to the issue of the primes, and motivated me to think about this topic.

Other Posts In This Series

  1. Techniques for Adding the Numbers 1 to 100
  2. Rethinking Arithmetic: A Visual Guide
  3. Quick Insight: Intuitive Meaning of Division
  4. Quick Insight: Subtracting Negative Numbers
  5. Surprising Patterns in the Square Numbers (1, 4, 9, 16…)
  6. Fun With Modular Arithmetic
  7. Learning How to Count (Avoiding The Fencepost Problem)
  8. A Quirky Introduction To Number Systems
  9. Another Look at Prime Numbers
  10. Intuition For The Golden Ratio
  11. Different Interpretations for the Number Zero