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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.

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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

by Dr. Mark Faucette

Department of Mathematics

State University of West Georgia

the nature of mathematical research
Mathematical research begins, above all else, with curiosity.

Mathematicians are people who constantly ask themselves questions.

The Nature of Mathematical Research
the nature of mathematical research4
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 questions6
Let’s start with a question anyone can understand:

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

Questions, Questions
thinking like the 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.

thinking like the greeks11
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 greeks12
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 greeks13
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 greeks14
Thinking Like The Greeks

Now, detach those two right triangles from the picture.

thinking like the greeks15
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 greeks16
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 squares18
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 squares19
Difference of Squares

First, draw k dots in a horizontal row.

difference of squares20
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 squares21
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 squares22
Difference of Squares

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

difference of squares23
Difference of Squares

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

difference of squares24
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 squares25
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.)

difference of squares26
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
difference of squares28
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
difference of squares30
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 squares31
Difference of Squares

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

what have we learned33
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.

questions questions35
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
difference of squares37
So, let’s ask the second question:

Which even numbers can be written as the difference of squares?

Difference of Squares
difference of squares40
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
difference of squares41
Suppose n+m is even. Then

n+m = 2j

for some whole number j.

Difference of Squares
difference of squares43
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 learned45
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?
questions questions47
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
difference of squares48
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
difference of squares50
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
difference of squares52
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
difference of squares53
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 Squares
difference of squares55
The solution is given by the equations at right.

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

Difference of Squares
what have we learned57
We’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?
questions questions59
Now we can ask one last question:

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

Questions, Questions
difference of squares60
Let’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 Squares
difference of squares61
Choose 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 Squares
difference of squares64
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
difference of squares65
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
difference of squares69
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 Squares
questions questions71
Now, 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
why do we care73
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?
why do we care74
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 care75
Why Do We Care?

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

why do we care76
Why Do We Care?

Examples: 4 = 2 + 2

6 = 3 + 3

8 = 3 + 5

154 = 151 + 3

1062 = 1051 + 11

why do we care77
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?
why do we care79
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?
why do we care81
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?