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

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

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

An introduction to the mathematical way of thinking

Mathematical research begins, above all else, with curiosity.

Mathematicians are people who constantly ask themselves questions.

The Nature of Mathematical ResearchMost 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 ResearchLet’s start with a question anyone can understand:

Which numbers can be written as a difference of two squares of numbers?

Questions, QuestionsWhich numbers can be written as a difference of two squares of numbers?

Let’s think of some examples:

Ponder the PossibilitiesThinking 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

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 SquaresSuppose 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 SquaresSince 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 SquaresDifference of Squares

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

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

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, QuestionsOnce 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

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

By an argument similar to what we did for 2, if n-m is even, then n+m must also be even.Difference of Squares

Since the right side is divisible by 4, we may choose two factors, s and t, of 4k so that both s and t are even.Difference of Squares

Then, we have these equations:

Comparing these, we see that we can set s=n+m and t=n-m and solve for n and m.

Difference of SquaresThe solution is given by the equations at right.

Notice that n and m are integers since both s and t are even.

Difference of SquaresWe’ve learned that an even number can be written as the difference of squares if and only if it is a multiple of 4. What Have We Learned?

Now we can ask one last question:

How many ways can numbers be written as differences of squares?

Questions, QuestionsLet’s answer this question first for an odd number 2k+1.

We already know it can be written as the difference of two squares of numbers n and m.

Difference of SquaresChoose any factors s and t of 2k+1 so that s ≥ t and st=2k+1.

If either s or t were even, then the product st=2k+1 would be even, so it follows that s and t are both odd.

Difference of SquaresSo, if we set s=n+m and t=n-m and solve the resulting system for n and m, we get the following solution:Difference of Squares

So, for any pair of factors s and t with s ≥ t and st=2k+1, we get a pair of whole numbers n and m so that 2k+1 is the difference n2-m2.Difference of Squares

Conversely, for any pair of whole numbers n and m so that 2k+1 is the difference n2-m2, then we get factors s and t with s ≥ t and st=2k+1.Difference of Squares

First, we list all the factors of 945 paired so that the product of each pair is 945:Difference of Squares

Setting n=(s+t)/2 and m=(s-t)/2, we get eight ways to write 945 as the difference of squares:

And these are all the ways in which 945 can be written as the difference of two squares.

Difference of SquaresNow, I’ll leave you with one last question: How many different ways can an even number be written as the difference of two squares?Questions, Questions

Personally, I care because it’s fun to think about these things. I consider it a kind of mental gymnastics. You know, it’s sort of like calisthentics for the mind.Why Do We Care?

If you don’t like that answer, let me offer you a question which is equally easy to state which has a real reason to solve:Why Do We Care?

Why Do We Care?

Question: Can every even number greater than 2 be written as the sum of two prime numbers?

Before you think too hard about this one, this question is a famous one in number theory and is known as the (Modern) Goldbach Conjecture.Why Do We Care?

It was originally posed in a letter from Christian Goldbach to Leonhard Euler in 1742.Why Do We Care?

The Goldbach Conjecture has been investigated for all even numbers up to 4 times 1011.

So far, no counterexamples have been found.

Why Do We Care?Now, 252 years after it was first posed, The Goldbach Conjecture is still unsolved.Why Do We Care?

However, if you ask why anyone would care about this problem, there is a one million dollar prize for a correct mathematical solution of this conjecture.Why Do We Care?

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