## Many Worlds Puzzle #5

October 31, 2016 by Matthew Rave

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.

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on October 31, 2016 at 5:50 PM |john zandePass, but well done nonetheless.

on October 31, 2016 at 6:07 PM |rickpsHey, 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.

on November 7, 2016 at 3:59 PM |Daniel He hetiandingI 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.

on November 21, 2016 at 1:26 PM |AnonymousUserI have discovered a truly marvelous proof of this, which this margin is too narrow to contain…