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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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An introduction to the mathematical way of thinking
Department of Mathematics
State University of West Georgia
Mathematicians are people who constantly ask themselves questions.The Nature of Mathematical Research
As long as you’re inquisitive, you can always find problems to ask.The Nature of Mathematical Research
Which numbers can be written as a difference of two squares of numbers?Questions, Questions
Let’s think of some examples:Ponder the Possibilities
The ancient Greeks didn’t have algebra as a tool. When the ancient Greeks talked about squares, they meant geometric squares.
First, draw a square of side length a and a square of side length b side by side as shown.
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.
Notice that we now have two congruent right triangles.
The sides of the triangles are colored pink and the hypoteni are colored green.
Now, detach those two right triangles from the picture.
Slide the triangle at the bottom left to the upper right.
Slide the triangle at the bottom right to the upper left.
Notice these two triangles complete the picture to form a square of side length c, which we have colored green.
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.
First, draw k dots in a horizontal row.
Next, draw k dots in a vertical row, one unit to the left and one unit above the horizontal row.
This gives 2k dots.
Put the last of the 2k+1 dots at the corner where the row and column meet.
This gives all our 2k+1 dots.
Now, we have a right angle with k+1 dots on each side.
Complete this picture to a square by filling in the rest of the dots.
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.
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.)
So, we see that any odd number can be written as the difference of two squares.Difference of Squares
Then we can factor the left side as the difference of two squares.Difference of Squares
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
So, 2 can’t be written as the difference of squares.
Well, so far, we’ve learned that every odd number can be written as the difference of two squares, but 2 cannot.
Can some even number be written as a difference of squares?
If so, which ones can?Questions, Questions
By the Fundamental Theorem of Arithmetic, either n+m or n-m is even.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?
We already know it can be written as the difference of two squares of numbers n and m.Difference of Squares
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
And these are all the ways in which 945 can be written as the difference of two squares.Difference of Squares
Question: Can every even number greater than 2 be written as the sum of two prime numbers?
Examples: 4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
154 = 151 + 3
1062 = 1051 + 11
So far, no counterexamples have been found.Why Do We Care?