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

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

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

4. 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.

5. Questions, Questions

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

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

8. Thinking Like The Ancient Greeks

9. 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.

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

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

12. 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.

13. 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.

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

15. 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.

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

17. Difference of Squares

18. 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.

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

20. 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.

21. 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.

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

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

24. 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.

25. 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.)

26. Difference of Squares
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.

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

28. 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.

29. 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.

30. 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.

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

32. What Have We Learned?

33. 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.

34. Questions, Questions

35. Questions, Questions
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?

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

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

38. 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:

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

40. 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.

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

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

43. 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.

44. What Have We Learned?

45. 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.

46. Questions, Questions

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

48. Difference of Squares
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:

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

50. 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.

51. 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.

52. 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.

53. 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.

54. Difference of Squares
So, we have this system of equations and we’re looking for integer solutions:

55. Difference of Squares The solution is given by the equations at right.
Notice that n and m are integers since both s and t are even.

56. What Have We Learned?

57. What Have We Learned?
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.

58. Questions, Questions

59. Questions, Questions Now we can ask one last question:
How many ways can numbers be written as differences of squares?

60. Difference of Squares
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.

61. Difference of Squares
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.

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

63. Difference of Squares
Since s and t are both odd, both n and m are whole numbers.

64. 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.

65. 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.

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

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

68. Difference of Squares
These are all the possible pairs s and t so that st=945.

69. 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.

70. Questions, Questions

71. Questions, Questions
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?

72. Why Do We Care?

73. Why Do We Care?
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.

74. 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:

75. Why Do We Care?
Question: Can every even number greater than 2 be written as the sum of two prime numbers?

76. Why Do We Care? Examples: 4 = 2 + 2 6 = 3 + 3 8 = 3 + 5 154 = 151 + 3
1062 =

77. Why Do We Care?
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.

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

79. 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.

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

81. 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.

82. I Care!
Now there are a million reasons to major in mathematics!