Square Numbers

Square numbers are interesting in themselves. But there are patterns hidden in between them that can be more profound.

The difference between consecutive square numbers are the odd natural numbers. 

All square numbers are the sums of two triangle numbers. A triangle number is a number that is the sum of a consecutive list of natural numbers (i.e., 1 + 2 + 3 + 4 + 5 = 15).

This is interesting at face value alone, but this means that the sum of natural numbers up to n added to the sum of all natural numbers up to n + 1 is always a square.

And this is where you get your odd differences. To get to the next square you have to add the next natural number in each consecutive list. Since you are always adding an odd and an even, this is always the case when adding two consecutive naturals, the addend is always odd. And it can be clearly seen that the next addend is always two greater. I leave that to the reader to prove.

Another interesting fact, is that n as we have used it above is always one less then the square root of the square number sum.