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

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by Dr. Mark Faucette

• Department of Mathematics
• University of West Georgia

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The Nature of Mathematical Research

• Mathematical research begins, above all else,
  with curiosity.
• Mathematicians are people who constantly ask
  themselves questions.

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The Nature of Mathematical Research

• Most of these questions require a considerable
  mathematical background, but many do not.
• As long as youre inquisitive, you can always
  find problems to ask.

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Questions, Questions
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Questions, Questions

• Lets start with a question anyone can
  understand
• Which numbers can be written as a difference of
  two squares of numbers?

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Ponder the Possibilities

• Which numbers can be written as a difference of
  two squares of numbers?
• Lets think of some examples

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Thinking Like The Ancient Greeks
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Thinking Like The Greeks

• The ancient Greeks didnt have algebra as a tool.
  When the ancient Greeks talked about squares,
  they meant geometric squares.

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Thinking Like The Greeks

• For instance, here is the picture of how
  Pythagoras reached the theorem which bears his
  name.

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Thinking Like The Greeks

• First, draw a square of side length a and a
  square of side length b side by side as shown.

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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.

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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.

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Thinking Like The Greeks

• Now, detach those two right triangles from the
  picture.

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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.

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Thinking Like The Greeks

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

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Difference of Squares
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Difference of Squares

• Lets think about our problem the way the ancient
  Greeks might have.
• We start with any odd number, say 2k1 for some
  natural number k.

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Difference of Squares

• First, draw k dots in a horizontal row.

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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.

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Difference of Squares

• Put the last of the 2k1 dots at the corner where
  the row and column meet.
• This gives all our 2k1 dots.

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Difference of Squares

• Now, we have a right angle with k1 dots on each
  side.

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Difference of Squares

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

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Difference of Squares

• From this picture, we see that the 2k1 red dots
  can be written as the number of dots in the
  larger square minus the number of dots in the
  smaller, yellow square.

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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, 112-02.)

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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.

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Difference of Squares

• Can 2 be written as the difference of two squares?

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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.

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Difference of Squares

• Since n and m are both whole numbers and we must
  have ngtm, we see that nm and n-m are both
  natural numbers.

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Difference of Squares

• Since 2 is prime, it follows that nm2 and
  n-m1.
• Adding these two equations, we get 2n3, which
  means n is not a whole number.
• This contradiction shows n and m dont exist.

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Difference of Squares

• So, 2 cant be written as the difference of
  squares.

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What Have We Learned?
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What Have We Learned?

• Well, so far, weve learned that every odd number
  can be written as the difference of two squares,
  but 2 cannot.

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Questions, Questions
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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?

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Questions, Questions

• We already know the answer to the first question
  The answer is given in our examples.

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Difference of Squares

• So, lets ask the second question
• Which even numbers can be written as the
  difference of squares?

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Difference of Squares

• Lets suppose that an even number, 2k, can be
  written as the difference of squares of whole
  numbers n and m

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Difference of Squares

• Lets try factoring the left side again and see
  what that tells us

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Difference of Squares

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

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Difference of Squares

• Suppose nm is even. Then
• nm 2j
• for some whole number j.

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Difference of Squares

• Then the following computation shows that if nm
  is even, then n-m must also be even.

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Difference of Squares

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

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What Have We Learned?
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What Have We Learned?

• Weve learned that every odd number can be
  written as a difference of squares.
• Weve learned that if an even number can be
  written as the difference of squares, it must be
  divisible by 4.

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Questions, Questions
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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?

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Difference of Squares

• Once again, lets take an arbitrary natural
  number which is divisible by 4 and suppose it can
  be written as a difference of squares

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Difference of Squares

• Lets try factoring the left side again and see
  what that tells us

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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.

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Difference of Squares

• By an argument similar to what we did for 2, if
  n-m is even, then nm must also be even.

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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.

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Difference of Squares

• Then, we have these equations
• Comparing these, we see that we can set snm and
  tn-m and solve for n and m.

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Difference of Squares

• So, we have this system of equations and were
  looking for integer solutions

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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.

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What Have We Learned?
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What Have We Learned?

• Weve learned that an even number can be written
  as the difference of squares if and only if it is
  a multiple of 4.

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Questions, Questions
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Questions, Questions

• Now we can ask one last question
• How many ways can numbers be written as
  differences of squares?

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Difference of Squares

• Lets answer this question first for an odd
  number 2k1.
• We already know it can be written as the
  difference of two squares of numbers n and m.

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Difference of Squares

• Choose any factors s and t of 2k1 so that s t
  and st2k1.
• If either s or t were even, then the product
  st2k1 would be even, so it follows that s and t
  are both odd.

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Difference of Squares

• So, if we set snm and tn-m and solve the
  resulting system for n and m, we get the
  following solution

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Difference of Squares

• Since s and t are both odd, both n and m are
  whole numbers.

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Difference of Squares

• So, for any pair of factors s and t with s t
  and st2k1, we get a pair of whole numbers n and
  m so that 2k1 is the difference n2-m2.

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Difference of Squares

• Conversely, for any pair of whole numbers n and m
  so that 2k1 is the difference n2-m2, then we
  get factors s and t with s t and st2k1.

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

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

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Difference of Squares

• These are all the possible pairs s and t so that
  st945.

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Difference of Squares

• Setting n(st)/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.

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Questions, Questions
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Questions, Questions

• Now, Ill leave you with one last question How
  many different ways can an even number be written
  as the difference of two squares?

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Why Do We Care?
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Why Do We Care?

• Personally, I care because its fun to think
  about these things. I consider it a kind of
  mental gymnastics. You know, its sort of like
  calisthentics for the mind.

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Why Do We Care?

• If you dont like that answer, let me offer you
  a question which is equally easy to state which
  has a real reason to solve

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Why Do We Care?

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

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Why Do We Care?

• Examples 4 2 2
• 6 3 3
• 8 3 5
• 154 151 3
• 1062 1051 11

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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.

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Why Do We Care?

• It was originally posed in a letter from
  Christian Goldbach to Leonhard Euler in 1742.

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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.

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Why Do We Care?

• Now, 252 years after it was first posed, The
  Goldbach Conjecture is still unsolved.

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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.

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I Care!

• Now there are a million reasons to major in
  mathematics!