the natural history of abstract objects
-->

Prime Numbers Aren’t

Today, in “lies you’ve been told all your lives:” not all prime numbers are prime!!! Your teachers told you \(5\) was prime. It’s not:

\[5 = (2+i)(2-i)\]

We can’t factor \(5\) using integers—but using complex numbers, we can!!!

\[\begin{align*} (2+i)(2-i) &= 2^2 + 2i - 2i -i^2\\ &= 4 -(-1)\\ &= 4 + 1 \\ &= 5 \end{align*}\]

Are the prime numbers no longer prime? Can we all of a sudden factor every prime number using complex numbers? Now that we’ve gone so deep into our imagination, is anything real anymore?!?

What about non-real complex numbers? Are any of them prime? Or can we just keep factoring them into smaller and smaller factors? For example, here’s an non-real complex number:

\[3+93i\]

This isn’t prime, since we can factor out a \(3\):

\[3 + 93i = (3)\cdot(1+31i)\]

Can we factor it yet further? We know that \(3\) is prime as an integer, but can we factor it further in the complex numbers? What about \(1+31i\)? Can we factor that any further, or is it prime? We can’t factor out from it any normal integers, since \(1\) and \(31\) have no factors in common (\(1\) doesn’t really count). Can we perhaps factor it into complex integers?

We can!

\[1+31i = (3+2i)(5+7i)\]

Without explaining how we would come up with this factorization, we can check that it works:

\[\begin{align*} (3+2i)(5+7i) &= 3\cdot5 + 3\cdot7i + 5\cdot2i + (2i)(7i)\\ &= 15 + 21i + 10i + 14i^2\\ &= 15 + 31i - 14\\ &= (1 + 31i) \end{align*}\]

So, I guess that \(1+31i\) isn’t prime! What about its factors, \(3+2i\) and \(5+7i\)? Are they prime? Can we factor them any more? \(2\), \(3\), \(5\), and \(7\) are all prime among the integers, but apparently that doesn’t mean that \(3+2i\) and \(5+7i\) must also be prime.

So we have:

\[\begin{align*} \underbrace{3+93i}_{\text{not prime}} \,\, &= (3)\cdot(1+31i) \\ &= \underbrace{(3)\cdot(3+2i)\cdot(5+7i)}_{\text{are any of these prime???}} \end{align*}\]

Before we go totally off the deep end, let’s be a bit more clear about our definitions. When we normally talk about factoring numbers into primes, we’re talking about factoring integers. For any (positive) integer, there’s only one way to factor it into a product of prime numbers (ignoring things like negatives, extra \(1\)s, and ordering the factors differently). So, for example, as an integer, \(5\) is prime, because we can’t factor it into smaller integers. \(12\) isn’t prime (it’s composite), because we can factor it into a bunch of primes:

\[\begin{align*} 12 &= 2\cdot 2\cdot3 \\ &= 2^2\cdot 3 \end{align*}\]

What about when we start talking about complex numbers? Let’s define a complex integer as a number of the form \(a+bi\), where \(a\) and \(b\) are both integers:

\[\text{a complex integer} = a + bi, \quad a,b\in \mathbb{Z}\]

So these are like the complex versions of integers (as opposed to the complex versions of real numbers, which is most of what we’ve been talking about). Most people call these the Gaussian integers, and while Gauss was a great guy, I think complex integer is a more evocative name.

So, for example, here are some complex integers:

\[5+2i\]

\[3-4i\]

\[7\quad \text{(it's also a normal integer!)}\]

\[12i\quad \text{(it's also a normal imaginary number!)}\]

Every complex integer is also a normal integer:

VENN DIAGRAM PICTURE

Here are some complex numbers that aren’t complex integers:

\[1.6 + 2.4i\]

\[8 - \pi i\]

\[11.46\]

\[-17.9i\]

Here are two overwhelming questions:

And here are some more specific questions: