h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

My name is Brendan, i'm a 15 year old high schooler from Australia. I was researching the Syracuse Conjecture for a maths assignment when i noticed that you could post here, and i decided i wanted to finally make this conjecture known. I thought of it in about grade 5 or 6, and have wanted to show people for a long time, but none of maths teachers understood. it's hard to understand, but with an example i'm sure you'll be able to understand. Would any people interested in this please email on [email protected] if you think i'm on to a new theory.

Ok, it is based around square numbers. To work out the next square number in the sequence, you take the square root of the previous square number, take the square of that number, add these two to the number that you want to know the square of and if you understand, then it should all go together.

Example - to find out what 31 squared is, take 900(the square of the previous number - 30), take 30, and take 31, which are the numbers that you have to add as mentioned above. And what does all that equal? 961!

another example, just for a bit more convincement....
take 400(the square of 20), take 20, and take 21, add them together and it is 441, which just happens to be the square of 21, the number after 20.

If you think I've found something, please email me.

I'm not going to e-mail you, Brendan, because that's not the way things happen here on h2g2. I'll put my answer here and you can read when next you come along.

What you have shown is that (n+1)^2 = n^2 + 2n + 1. You've provided a few examples that it is true. You've put it a slightly different way as (n+1)^2 = n^2 + (n+1) + n which is the same thing really. This is not an amazing discovery. It is something that any high school student should be able to show. But since you have discovered this result for yourself, it should be more easy to understand for you.

Good work! Keep searching for other things like this.