How Many Ways Can 945 Be Written as the Difference of Squares?

How Many Ways Can 945 Be Written as the Difference of Squares? An introduction to the mathematical way of thinking

by Dr. Mark Faucette Department of Mathematics State University of West Georgia

Mathematical research begins, above all else, with curiosity. Mathematicians are people who constantly ask themselves questions. The Nature of Mathematical Research

Most of these questions require a considerable mathematical background, but many do not. As long as you’re inquisitive, you can always find problems to ask. The Nature of Mathematical Research

Questions, Questions

Let’s start with a question anyone can understand: Which numbers can be written as a difference of two squares of numbers? Questions, Questions

Which numbers can be written as a difference of two squares of numbers? Let’s think of some examples: Ponder the Possibilities

Thinking Like The Ancient Greeks

Thinking Like The Greeks The ancient Greeks didn’t have algebra as a tool. When the ancient Greeks talked about squares, they meant geometric squares.

For instance, here is the picture of how Pythagoras reached the theorem which bears his name. Thinking Like The Greeks

Thinking Like The Greeks First, draw a square of side length a and a square of side length b side by side as shown.

Thinking Like The Greeks Next, measure b units from the corner of the first square along the bottom side. Connect that point to the upper left corner of the larger square and the upper right corner of the smaller square.

Thinking Like The Greeks Notice that we now have two congruent right triangles. The sides of the triangles are colored pink and the hypoteni are colored green.

Thinking Like The Greeks Now, detach those two right triangles from the picture.

Thinking Like The Greeks Slide the triangle at the bottom left to the upper right. Slide the triangle at the bottom right to the upper left.

Thinking Like The Greeks Notice these two triangles complete the picture to form a square of side length c, which we have colored green.

Difference of Squares

Difference of Squares Let’s think about our problem the way the ancient Greeks might have. We start with any odd number, say 2k+1 for some natural number k.

Difference of Squares First, draw k dots in a horizontal row.

Difference of Squares Next, draw k dots in a vertical row, one unit to the left and one unit above the horizontal row. This gives 2k dots.

Difference of Squares Put the last of the 2k+1 dots at the corner where the row and column meet. This gives all our 2k+1 dots.

Difference of Squares Now, we have a right angle with k+1 dots on each side.

Difference of Squares Complete this picture to a square by filling in the rest of the dots.

Difference of Squares From this picture, we see that the 2k+1 red dots can be written as the number of dots in the larger square minus the number of dots in the smaller, yellow square.

Difference of Squares By this argument, the ancient Greeks would conclude that any odd number (greater than one) can be written as the difference of two squares. (Then again, 1=12-02.)

In modern terms, we have shown using diagrams of dots the equation at right: So, we see that any odd number can be written as the difference of two squares. Difference of Squares

Can 2 be written as the difference of two squares? Difference of Squares

Suppose this is true for some whole numbers n and m. Then we can factor the left side as the difference of two squares. Difference of Squares

Since n and m are both whole numbers and we must have n>m, we see that n+m and n-m are both natural numbers. Difference of Squares

Since 2 is prime, it follows that n+m=2 and n-m=1. Adding these two equations, we get 2n=3, which means n is not a whole number. This contradiction shows n and m don’t exist. Difference of Squares

Difference of Squares So, 2 can’t be written as the difference of squares.

What Have We Learned?

What Have We Learned? Well, so far, we’ve learned that every odd number can be written as the difference of two squares, but 2 cannot.

Our result has led us to a number of new questions: Can some even number be written as a difference of squares? If so, which ones can? Questions, Questions

We already know the answer to the first question: The answer is given in our examples. Questions, Questions

So, let’s ask the second question: Which even numbers can be written as the difference of squares? Difference of Squares

Let’s suppose that an even number, 2k, can be written as the difference of squares of whole numbers n and m: Difference of Squares

Let’s try factoring the left side again and see what that tells us: Difference of Squares

Since the right side is even, the left side must also be even. By the Fundamental Theorem of Arithmetic, either n+m or n-m is even. Difference of Squares

Suppose n+m is even. Then n+m = 2j for some whole number j. Difference of Squares

Then the following computation shows that if n+m is even, then n-m must also be even. Difference of Squares

Looking back at our original assumption, since both n+m and n-m are even, the even number on the right must actually be divisible by 4. Difference of Squares

What Have We Learned?

We’ve learned that every odd number can be written as a difference of squares. We’ve learned that if an even number can be written as the difference of squares, it must be divisible by 4. What Have We Learned?

Now we can refine our last question to this: Can every natural number divisible by 4 be written as a difference of squares? Questions, Questions

Once again, let’s take an arbitrary natural number which is divisible by 4 and suppose it can be written as a difference of squares: Difference of Squares

Let’s try factoring the left side again and see what that tells us: Difference of Squares

Notice that the right side of this equation is divisible by 4. So the left side of this equation must also be divisible by 4. Difference of Squares

How Many Ways Can 945 Be Written as the Difference of Squares?