I proved a theorem in number theory last week.
Don’t get me wrong; the theorem is fairly useless. The proof itself is trivial. But the fact that I came up with the proof, in my head, makes me proud…mostly because I’m a physicist, not a mathematician.
Here is the theorem:
Square any odd integer greater than 1, and then subtract 1. The result is evenly divisible by 4.
I’ll leave the proof to you, for fun. Do it in your head!
This is what theoretical physicists daydream about on 7 hour car rides.
Many Worlds Theory 
Pass, but well done nonetheless.
Hey, so it is. And…the square of any even integer (no subtracting 1 in this case) is also divisible by four. And any odd integer plus one all squared is divisible by four. This is fun at 6:00AM.
I feel so smart. I got it in less than 3 minutes. There are more direct ways, but this is how I got it and I think this is the easiest to understand.
1. Lets use m to represent the odd integer greater than 1. I’ll use 7 as an example to explain. Imagine m^2 as being like this, with each x being equal to 1,
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
2. The -1 we remove from one of the corners.
x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
3. Now, we have 6 rows of 7 and one row of 6. I can cut the 6 rows of 7 to become 6 rows of 6 and one column of 6. That gives us 6^2 + 2(6).
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x x x x x
+
x x x x x x
x x x x x x
4. Let’s say that n = m – 1. From what I just listed, it is easy to see that m^2 – 1 is equal to n^2 + 2n. n^2 + 2n factors out to be n(n + 2).
5. n is equal to m – 1, so n will always be divisible by 2, since m is an odd integer. Similarly, n + 2 must also be an even integer, thus divisible by 2. Thus, n(n + 2) must always be divisible by 4.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain…