Why SixteenThousand?

It's short for 16,256.

Why 16,256?

I use 16256 as a bit of a calling card, so I get the same question semi-regularly: What does 16256 mean?

First off, let's get your first idea out of the way:

No, it's not a power of two

I get it. You see 16, you see 256, if you're into maths and/or computers you've seen those numbers before, so a part of you subconciously thinks it must be part of the same set.

Well, it's not.

...But could it be?

That is to say, are there any powers of two which when concatenated give another power of two?

The answer's still no, but it is slightly interesting to see why (skip to the next section if you just don't care about this). Now, if you concatenate two numbers, let's call them $a$ and $b$, you get something that looks like this:

where $x$ is the number of digits in $b$. So, e.g., if you concatenate 16 and 256, you can express that as

If $a$ and $b$ are powers of two, and we substitute 2 * 5 for 10, then for a 16,256-like number to be a power of two, we would need:

for some $x,y,z,w$.

If this is true, then , since . Dividing both sides by $2^x$ or $2^z$ (whichever is smaller), we get

or

In either case, we have an odd number on one side and an even number on the other⁶, a contradiction.

Ok but why then?

Because 16,256 is abundant. To explain what that means, we need to define a few other things:

A proper factor of a number $n$ is a number that divides $n$ cleanly and isn't $n$. So the proper factors of 10 are 1,2 and 5, and the proper factors of 12 are 1,2,3,4 and 6.

From this, we can calculate the sum of the proper factors of a number, which in our examples is 1+2+5=8 and 1+2+3+4+6=16. We then split numbers into three groups according to how big this sum is:

A number is deficient if the sum of its proper factors is less than the number itself.

So 10 is deficient, as 1+2+5=8<10. "Most"¹ numbers are deficient.

A number is perfect if the sum of its proper factors is equal to the number itself.

This is "rare"². The first few perfect numbers are 6, 28, 496, 8128³.

The last group is our group:

A number is abundant if the sum of its proper factors is greater than the number itself.

So 12 is abundant, because 1+2+3+4+6=16>12. Abundant numbers are...kind of rare, but not really⁴?

Well, I was reading about this when I was about ...uhhh... let's say 14, because that makes me look cooler. Anyway, I was reading this super hard maths that makes me look smart and cool, and I had a thought:

If a number is perfect, then isn't double that number also perfect? After all, if $p$ is perfect, then $2p$ has all the factors of $p$, plus $p$ itself!

Now, there are a lot of unsolved problems related to perfect numbers, so if this was true, it would be quite interesting, so I got a little excited, and set to writing my proof, knowing I was about to make strides in the field, and I immediately realised I was wrong.

I mean, see the examples above: 12 = 2 * 6, and 12 is abundant. But then again, it was still clear that double a perfect number was perfect or abundant⁵, and abundant numbers are kind of interesting, so... it'd do. And it meant that 2 * 8,128 = 16,256 was abundant! That's cool!

Was there any point to this story?

Mostly it was just about writing down the explanation publicly so I don't have to try verbally explaining again. But also it's also having something personal to you be something you put some work into getting, and how that's nice.