Transcript 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
Department of Mathematics
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
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.
Questions, Questions
Questions, Questions
Let’s start with a
question anyone
can understand:
Which numbers can
be written as a
difference of two
squares of
numbers?
Ponder the Possibilities
Which numbers can
be written as a
difference of two
squares of
numbers?
Let’s think of some
examples:
2 2 -12 = 3
5 2 - 32 = 16
32 -12 = 8
52 - 4 2 = 9
32 - 2 2 = 5
6 2 - 32 = 27
4 2 - 32 = 7
6 2 - 4 2 = 20
Thinking Like The
Ancient 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 Greeks
For instance, here is
the picture of how
Pythagoras reached
the theorem which
bears his name.
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
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.)
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.
(k +1)2 - k 2 = 2k +1
Difference of Squares
Can 2 be written as
the difference of two
squares?
n 2 - m2 = 2?
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.
n -m =2
2
2
( n + m )( n - m ) = 2
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.
n -m =2
2
2
( n + m )( n - m ) = 2
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.
n -m =2
2
2
( n + m )( n - m ) = 2
Difference of Squares
So, 2 can’t be written
as the difference of
squares.
What Have We Learned?
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, Questions
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?
Questions, Questions
We already know the
answer to the first
question: The
answer is given in
our examples.
32 -12 = 8
5 - 3 = 16
2
2
6 2 - 4 2 = 20
Difference of Squares
So, let’s ask the
second question:
Which even numbers
can be written as
the difference of
squares?
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:
n 2 - m2 = 2k
Difference of Squares
Let’s try factoring the
left side again and
see what that tells
us:
n 2 - m 2 = 2k
( n + m)( n - m) = 2k
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.
n 2 - m 2 = 2k
( n + m)( n - m) = 2k
Difference of Squares
Suppose n+m is even.
Then
n+m = 2j
for some whole
number j.
n - m = 2k
2
2
( n + m)( n - m) = 2k
(2 j )( n - m) = 2k
Difference of Squares
Then the following
computation shows
that if n+m is even,
then n-m must also
be even.
n - m = n + m - 2m
= ( n + m) - 2m
= 2 j - 2m
= 2( j - m)
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.
n 2 - m 2 = 2k
( n + m)( n - m) = 2k
What Have We Learned?
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.
Questions, Questions
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?
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:
n 2 - m2 = 4k
Difference of Squares
Let’s try factoring the
left side again and
see what that tells
us:
n - m = 4k
2
2
( n - m)( n + m) = 4k
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.
n - m = 4k
2
2
( n - m)( n + m) = 4k
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.
n - m = 4k
2
2
( n - m)( n + m) = 4k
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.
n - m = 4k
2
2
( n - m)( n + m) = 4k
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.
n - m = 4k
2
2
( n - m)( n + m) = 4k
st = 4k
Difference of Squares
So, we have this
system of equations
and we’re looking
for integer
solutions:
ìs = n + m
í
ît = n - m
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.
s+ t
s-t
n=
and m =
2
2
What Have We Learned?
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.
Questions, Questions
Questions, Questions
Now we can ask one
last question:
How many ways can
numbers be written
as differences of
squares?
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.
n - m = 2k + 1
2
2
( n - m)( n + m) = 2k + 1
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.
n - m = 2k + 1
2
2
( n + m)( n - m) = 2k + 1
st = 2k + 1
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:
s+ t
n=
2
s- t
m=
2
Difference of Squares
Since s and t are both
odd, both n and m
are whole numbers.
s+ t
n=
2
s- t
m=
2
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.
s+ t
n=
2
s- t
m=
2
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.
s=n+m
t =n-m
How Many Ways Can 945 Be
Written as the Difference of
Squares?
Difference of Squares
First, we list all the
factors of 945
paired so that the
product of each pair
is 945:
1
945
3
315
5
7
9
189
135
105
15
21
27
63
45
35
Difference of Squares
These are all the
possible pairs s and
t so that st=945.
1
945
3
315
5
7
9
189
135
105
15
21
27
63
45
35
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.
4732 - 472 2 = 945
159 2 -156 2 = 945
97 2 - 92 2 = 945
712 - 64 2 = 945
57 - 48 = 945
2
2
39 2 - 24 2 = 945
33 -12 = 945
2
2
312 - 4 2 = 945
Questions, Questions
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?
Why Do We Care?
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.
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?
Question: Can every even number
greater than 2 be written as the sum of
two prime numbers?
Why Do We Care?
Examples:
4=2+2
6=3+3
8=3+5
154 = 151 + 3
1062 = 1051 + 11
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.
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.
I Care!
Now there are a
million reasons to
major in
mathematics!