 Alright, let's take a look at a couple of different methods of factoring, and one of the most important methods of factoring is based on the difference of squares, and one approach is the following. If I want to factor some large number n, what I can do is I can try and find a perfect square, b squared, where if I add n and b squared, I get another perfect square. And why is that useful? Well, I can rearrange things, and it's going to be the difference of two squares, and the difference of squares factors, and so here is a factorization a minus b times a plus b, whose product is equal to n. And so I want to find a and b, and the efficient way of looking for what those are is I can look at the values of n plus a perfect square, and see if any of those are also perfect squares. For example, suppose I wanted to factor 2,491. So I can look at 2491 plus a perfect square for k equals 1, 2, 3, and so on. So I add 1 squared, I get 2492, and one thing that is nice to do, we don't really have to do this, but it makes our job a little bit easier when we're doing this by hand. In a base 10 system, a perfect square has to end in 0, 1, 4, 5, 6, or 9. So that means any number that ends in 2 can't possibly be a perfect square, so I'll ignore it and move on to the next. 2491 plus 2 squared is 2495, which could be a perfect square, but it isn't. So we'll move on to the next one. 2491 plus 3 squared is 2,500, which is definitely a perfect square. It's 50 squared, and so that gives me the option of writing 2491 as 50 squared minus 3 squared, and then that right-hand side factors, and my number is a product of 4753. Well, there's a couple of nice algebraic simplifications that we can take advantage of. One useful algebraic fact is the k plus 1 squared is k squared plus 2k plus 1, and what this means is I can find the square of any number by looking at the square of the preceding number and adding an odd number to it. And what that translates into into our algorithm is that if I want to obtain n plus k plus 1 squared, I can add 2k plus 1 to n plus k squared. So when I did my computations, once I found 2491 plus 1 squared is 2492, my next number, 2491 plus 2 squared, I could have obtained that by taking my previous number, 2492, and adding the next odd number plus 3. And likewise, if I wanted to get 2493 plus 3 squared, that's the preceding value plus the next odd number. And what this does is it allows us to compute n plus k squared much more easily. For example, let's take a look at 6497. So again, I'm going to find the results of adding the perfect squares to the number. And again, because I know what perfect squares have to end in, I can discard some possibilities right away. Now, again, my next perfect square is the preceding number plus the next odd number. So here I'm adding 1. My next value, I'll add 3 to get 6501, not a perfect square. Next one, I added 3 to get this. I'll add 5 to get the next one. Add 7. Add 9. Add 11. Add 13. Add 15. And it turns out that that's a perfect square, and that says that 6497 can factor as a product 73 times 89. Thank you.