There ought to be a law, or, my career as a mathematician

I've been a chemist, formally, for 32 years. Before that I was a biochemist or biophysicist. And one of the frustrating things about chemistry is all the important laws seem to be taken, and also that they all seem to have to be of major scientific importance. So we have The Second Law of Thermodynamics (big) and the Law of Definite Proportions (important, if not always obeyed) and the Ideal Gas Law (ditto). After 32 years in the field, you'd think I could have a law too, but it looks increasingly unlikely. Sad trombone.

But then I found out that in mathematics, apparently there are all sorts of laws lying around to be discovered, that don't seem that, well, important. Like Benford's law. Benford's Law says if you have a big list of random natural numbers, the most common leading digit is 1, followed by 2, followed by 3, etc. I've explained it all in a previous post. The great thing is, once I'd gotten over the astonishment that they would actually call this a law, and once I'd spent the 90 seconds it took, including three sips of coffee, to prove it, I realized there are thousands of laws in mathematics, just as important as Benford's Law, just lying around, waiting to be named.

So (ta-da!) I present Harbison's Law

If you add two random n-digit non-negative integers, the result will be an n digit number more than half the time, and an n + 1 digit number less than half the time.

Of course, you could prove this on the back of an envelope, but I decided if it's going to be a Law, it needs to be given proper respect, so I programmed it up in Mathematica. It was less than 3 weeks work. Actually, to be truthful, it was less than three minutes work.

exponent = 1; count = 0;
For[n = 0 , n < 10^exponent, n++,
For[m = 0, m < 10^exponent, m++,
p = n + m;
If[p >= 10^(exponent), count++];
];
];
Print[count,"/", 10^(2*exponent)];

And the all important data

Exponent = 1, result = 45/100
Exponent = 2, result = 4950/10000
Exponent = 3, result = 499500/1000000
Exponent = 4, result = 49995000/100000000

Whoa! Dude! Do you see that? A trend. I feel like conjecturing! Hell, I'm going to conjecture not once, but twice!

Harbison's first conjecture: If you add two random n-digit non-negative integers, the result will be an n digit number exactly half the time, in the limit of n going to infinity
Harbison's second conjecture: If you add two random n-digit non-negative integers, the result will be less than one half by an amount equal to 5*10^(n-1)/(10^2n)

Needless to say, a man of my vast mathematical abilities could prove both of these, but I'll leave that to some up-and-coming mathematical sprat, who can then put on their CV they proved Harbison's First or Second Conjecture. Should land them an Ivy League job, easy.

And here endeth my career in Mathematics. I'd like to thank Theodore Hill for inspiration, and all the little people out there. I'm going to make myself a wiki page. Tell the Field's Medal committee I might be a little busy, but I'll call them back.