Download Week 1: First Examples

Week 1: First Examples
Mathematics is the art of explanation.
Paul Lockhart
Squares and Primes
There’s something a little strange about the “square” numbers (1, 4, 9, 16, 25, 36, etc.): the
only time a square number comes right after a prime number is when the square number is
4, which follows the prime number 3. Test this out with a few other square numbers, and
you may begin to suspect, like I did, that a square number will rarely, if ever, come right
after a prime number.
What I am claiming, though, is that no square number aside from 4 will ever come
immediately after a prime number. It can never happen. But what a broad generalization
this is! How can I be so sure of this statement? After all, we’re talking about a set of
numbers that go on forever (the square numbers) – and don’t the prime numbers also go
on forever? Let me explain how I can make such a bold claim about these infinite families of
numbers with so much confidence.
Let’s say you pick some square number. Let’s call the number “𝑛,” just for the sake of
having a name for it. What does it mean when I say 𝑛 is “square?” It means 𝑛 is the square
of some other number, like 1, 2, 3, 4, etc. In other words, let’s say 𝑛 is the square of some
positive whole number, which we might refer to as “𝑚” (or whatever letter you’d like to call
it). So, in short, 𝑛 = 𝑚2 . Now, what number comes right before 𝑛? That would be the
number 𝑛 − 1, or in other words, 𝑚2 − 1.
Why on Earth couldn’t 𝑚2 − 1 be a prime number? Think about what a prime number is.
Can a number like 𝑚2 − 1 be written as the product of two other positive whole numbers?
Remember that we can “factor” 𝑚2 − 1, like this: 𝑚2 − 1 = (𝑚 − 1)(𝑚 + 1). Now, 𝑚 − 1
and 𝑚 + 1 are both whole numbers, so no matter what 𝑚 is, 𝑚2 − 1 can always be factored
into two whole numbers (specifically, whatever the numbers 𝑚 − 1 and 𝑚 + 1 are).
Since the number 𝑚2 − 1 can always be written as the product of the numbers 𝑚 − 1 and
𝑚 + 1, does that mean 𝑚2 − 1 is never prime? Almost; the one and only way 𝑚2 − 1 could
be prime is if one of those two numbers (either 𝑚 − 1 or 𝑚 + 1) turns out to be 1. Since 𝑚
is a positive whole number, 𝑚 + 1 has to be at least 2. But can 𝑚 − 1 be equal to 1? Sure it
can! But only if 𝑚 is 2. In short, the only way 𝑚2 − 1 can be prime is if m is 2. In other
words, the only way a prime number could come right before a square number is if that
square number is 4.
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The previous three paragraphs constitute an informal proof, in conversational English, of
the statement I made earlier: Aside from 4, no other square number can immediately
follow a prime number. Isn’t it astounding and beautiful to realize that human beings can
know something so infinite in scope with such absolute conviction? This kind of infinite
certainty can only come from a mathematical proof. In mathematics, a proof is an airtight
explanation of why a statement must be true, leaving absolutely no room for doubt.
Exercise 1: Aside from 4, no other square number can immediately follow a prime number.
Can a similar statement be made about the cubic numbers (1, 8, 27, 64, etc.)? Write the
statement and prove it.
Exercise 2: Can a similar statement be made about the fourth-power numbers (1, 16, 81,
256, etc.)? Write the statement and prove it.
More Concise Proofs
A mathematical statement that we intend to prove is often called a proposition. Let’s take
another look at the following proposition and its proof: Aside from 4, no other square
number can immediately follow a prime number.
You will find that when you want to prove a proposition, it will be easiest to compose the
proof if the proposition can be rewritten in the form “If
, then
.” For
example, the proposition above can be rewritten as follows: If a square number
immediately follows a prime number, then the square number must be 4.
A more concise version of the proposition is this: If 𝑚2 − 1 is prime (𝑚 being a positive
whole number), then 𝑚 = 2.
Note that the three versions of the proposition given above are all stating exactly the same
mathematical fact. Most mathematical propositions can be written in the form “If
,
2
then
.” The two blanks here represent simple propositions like “𝑚 − 1 is prime (𝑚
being a positive whole number)” and “𝑚 = 2.” For convenience, we can refer to these two
propositions as “𝑃1 ” and “𝑃2 .”
The most direct way to try to prove a proposition in the form “If 𝑃1 , then 𝑃2 ” is to suppose
that the proposition 𝑃1 is true, and then explain why the proposition 𝑃2 must be true. This
is what is done in the proof on the previous page, in a very conversational style. Below is a
more concise style of writing the same proof.
Proposition: If 𝑚2 − 1 is prime (𝑚 being a positive whole number), then 𝑚 = 2.
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Proof: Suppose 𝑚2 − 1 is prime and 𝑚 is a positive whole number. Since 𝑚2 − 1 =
(𝑚 − 1)(𝑚 + 1), and 𝑚 + 1 cannot be 1, we know that 𝑚 − 1 = 1, and therefore 𝑚 = 2,
which completes the proof.
The first mathematical proofs, written by the ancient Greeks, ended with the phrase “quod
erat demonstrandum” which can be roughly translated as “which completes the proof.”
Until recently, all mathematical proofs (even in English) would end with the abbreviation
“Q.E.D.” If you read a proof in a more modern high school or college mathematics textbook,
you may see some special marking to visually indicate the end of the proof. The symbol
often has a square shape, which is sometimes referred to as a “tombstone” mark. In my
writing, I will use the “∎” symbol1 to indicate that a proof has concluded.
Even Squares
Think about the square numbers again. As it turns out, if a square number is even, then its
“root” must also be even. In other words, if the square of a positive whole number 𝑛 is even,
then 𝑛 must be even. This proposition has the form “If 𝑃1 , then 𝑃2 ,” so you might feel the
urge to prove the proposition in roughly the same way as the previous one: suppose the
square of a positive whole number 𝑛 is even, and then explain somehow why 𝑛 itself must
be even.
What would it mean to say that 𝑛2 is even? To say that 𝑛2 is “even” means that 𝑛2 is a
multiple of 2. That is, 𝑛2 = 2𝑚 for some positive whole number 𝑚. How would we use this
to explain that 𝑛 itself must be a multiple of 2? The answer is that there is no direct way to
explain this. We will take a different approach. Notice that the following two statements
mean exactly the same thing:
1. If the square of a positive whole number 𝑛 is even, then 𝑛 must be even.
2. If a positive whole number 𝑛 is not even, then 𝑛2 cannot be even.
Can you see that these two sentences are just two different ways of saying the same thing
about positive whole numbers? The sentences mean the same thing for the same reason
that the following non-mathematical sentences do:


If it is raining, then the Smiths stay indoors.
If the Smiths don’t stay indoors, then it isn’t raining.
The fact that the two mathematical propositions above mean the same thing does not
necessarily guarantee that either statement is true. But as it turns out, the second
proposition is not terribly hard to prove, and if we prove it, the first proposition must also
be true.
1
The “∎” symbol can be created in Microsoft Word by typing “220e” and then immediately pressing ALT+X.
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Now, thinking about proving proposition #2 above raises a new question – what does it
mean for a positive whole number to be “not even”? Most people would say that a positive
whole number that is not even is “odd,” but that is not very helpful. “Odd” is just a synonym
for “not even.” Here is a more convenient definition: a number that is 1 less than an even
number is called “odd.”
Proposition: If the square of a positive whole number 𝑛 is even, then 𝑛 must be even.
Proof: We will prove the equivalent proposition that if a positive whole number 𝑛 is odd,
then 𝑛2 must be odd. Suppose a positive whole number 𝑛 is odd. This means 𝑛 = 2𝑚 − 1
for some positive whole number 𝑚, and so 𝑛2 = (2𝑚 − 1)2 = 4𝑚2 − 4𝑚 + 1. We can
rewrite this as 2(2𝑚2 − 2𝑚 + 1) − 1, which is 1 less than an even number. This proves that
if a positive whole number 𝑛 is odd, then 𝑛2 must be odd. ∎
Exercise 3: Explain why the sentence “If it isn’t raining, then the Smiths don’t stay indoors”
does not mean the same thing as the other two sentences about the Smiths given before.
Exercise 4: Prove that if 𝑎 is even and 𝑏 is odd, then 𝑎 + 𝑏 is odd.
Exercise 5: Prove this proposition: If a positive whole number 𝑛 is even, then 𝑛2 must be
even.
Exercise 6: Give a statement about odd numbers that means exactly the same thing as the
proposition in Exercise 5.
The First Known Irrational Number
We now take a side trip away from whole numbers (on which we will focus for a large part
of the course) and briefly into the realm of rational numbers and beyond. The reason for
this detour is to show off one of the most beautiful proofs in basic mathematics: the proof
that √2 is irrational – that is, it is not a rational number. Remember that √2 is some
number whose square is 2. To say that a positive number is “rational” means that the
number can be written as a ratio of two positive whole numbers – in other words, the
number can be written as a “reduced fraction” like 2/5 or 11/3 (as opposed to 21/24,
which is a fraction, but is not reduced, because 21 and 24 have a common divisor).
The proof that √2 is irrational is not only beautiful, but vital to this course, because it will
be our first illustration of an important proof technique. To prove that √2 is irrational, we
will imagine that it is rational, and see what would happen if this were true. It is best for
now to write this proof in a somewhat conversational style.
Proposition: √2 is irrational.
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Proof: Imagine that √2 were actually a rational number. This would mean that √2 could be
written as a reduced fraction 𝑎/𝑏. That is, 𝑎 and 𝑏 are positive whole numbers with no
common divisor. Since √2 = 𝑎/𝑏, we know that (𝑎/𝑏)2 = 2, so 𝑎2 = 2𝑏 2 .
Since we now know that 𝑎2 is even, 𝑎 must be even. (We proved this property of even
square numbers earlier.) This means that 𝑎 is 2 times some positive whole number “𝑚.” So
𝑎2 = (2𝑚)2 = 4𝑚2 . Since 𝑎2 = 2𝑏 2 , this means that 4𝑚2 = 2𝑏 2 , so 𝑏 2 = 2𝑚2 . Since this
tells us that 𝑏 2 is even, we now know that 𝑏 must also be even.
Now, 𝑎/𝑏 is a reduced fraction, but 𝑎 and 𝑏 are both even! This cannot possibly be true, and
therefore √2 cannot be a rational number. ∎
Exercise 7: Prove that the six blank spaces in the 4 × 4 square
shown here cannot be filled with positive whole numbers in such a
way that the four rows, four columns, and two diagonals each sum
up to the same number.
Exercise 8: We have quietly assumed in some previous proofs that
an odd number (a number that is 1 less than an even number)
cannot be even, and an even number cannot be odd, but why? Prove
that no positive whole number can be both even and odd.
1
2
3
4
5
7
6
8
9
10
Counterexamples
A mathematical statement that appears to be true but has not been proved is usually called
a “conjecture.” The history of mathematics is full of very convincing conjectures that turned
out to be false, along with those that turned out to be true (and many that have yet to be
proved or disproved). Following are some examples, just to illustrate.
“Goldbach’s Conjecture” claims that every even number greater than 2 can be written as
the sum of two primes (as an example, 54 = 37 + 17). Using computers, mathematicians
have verified that this can be done with all the even numbers from 4 to
400,000,000,000,000,000. However, no one has been able to prove that it is possible for
every even number greater than 2, which is what the conjecture claims. If someone could
find a single even number greater than 2 that cannot be written as the sum of two primes,
this number would disprove Goldbach’s Conjecture. That number, if it were to be found,
would be called a “counterexample” to Golbach’s conjecture.
For hundreds of years, “Fermat’s Last Theorem” was the conjecture that if 𝑛 is a whole
number greater than 2, then it is impossible to find positive whole numbers 𝑎, 𝑏, and 𝑐 so
that 𝑎𝑛 + 𝑏 𝑛 = 𝑐 𝑛 . This relatively simple claim about positive whole numbers is now
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properly known as a theorem because it was eventually proved in 1994. A faulty
counterexample to Fermat’s conjecture was briefly visible in an old episode of The
Simpsons. The supposed counterexample (which is meant as a very obscure joke) is that
178212 + 184112 = 192212 . This “fact” can be “checked” by using a calculator to find
12
√178212 + 184112 , which appears to be 1922, because of the limitations of calculators.
The “Pólya Conjecture” is another statement about all positive whole numbers. The actual
details are slightly too complicated to explain here, but as it turns out, the statement is not
true for the number 906,150,257. This surprisingly large number is the smallest
counterexample to the Pólya Conjecture, but it is enough to disprove the statement.
The “Chinese Hypothesis” is a conjecture claiming that if you find a positive whole number
𝑛 so that 2𝑛 − 2 is a multiple of 𝑛, then 𝑛 must be prime. For example, 223 − 2 is a multiple
of 23, which is prime. However, there are counterexamples to the Chinese Hypothesis. The
smallest counterexample is 𝑛 = 341, because 2341 − 2 is a multiple of 341, but 341 is not
prime.
Exercise 9: Find a counterexample to disprove the claim that if 𝑘 is any positive whole
number, then 𝑘 2 − 𝑘 + 41 is a prime number. (Hint: You shouldn’t need to use a computer,
or search very long, if you think carefully.)
Exercise 10: Without using a calculator, explain why we can tell pretty quickly that the
supposed equation 178212 + 184112 = 192212 could not possibly be true. (Hint: Is 178212
even or odd? What about 184112 ? And what do we know about the sum of an even number
and an odd number?)
A Special Technique
There is still an important gap in our reasoning about even and odd numbers. You proved
in an earlier exercise that no positive whole number can be both even and odd, but can we
really be certain that every positive whole number must be one or the other? The logic of
some of our previous proofs would fall apart if we did not know this for certain. So let’s see
some proof!
Proposition: Every positive whole number is either even or odd.
Proof: We will write this proof in a different way than any of the other proofs so far. We
will prove two facts –
1. The first of all positive whole numbers is either even or odd.
2. Whenever a positive whole number 𝑛 is either even or odd, the next whole number
(𝑛 + 1) must also be either even or odd.
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If both of these facts are true, then every single positive whole number must be either even
or odd.
First of all, the first positive whole number is 1, which is odd because 1 = 2 − 1 (i.e. 1 is 1
less than the even number 2).
Secondly, we need to prove that if a whole number 𝑛 is either even or odd, then the next
whole number, 𝑛 + 1, must also be either even or odd. Suppose 𝑛 is either even or odd. If 𝑛
is even, this means that 𝑛 = 2𝑚 for some positive whole number 𝑚. So 𝑛 + 1 = 2𝑚 + 1 =
2(𝑚 + 1) − 1, which is odd. On the other hand, if 𝑛 is odd, this means that 𝑛 = 2ℓ − 1 for
some positive whole number ℓ, so 𝑛 + 1 = 2ℓ, which is even. In either case (whether 𝑛 is
even or odd), we find that 𝑛 + 1 must also be either even or odd.
The two facts we have just proved about positive whole numbers, taken together, prove
that every positive whole number is either even or odd. ∎
The proof above illustrates a line of thinking that can often be used to prove a statement
that is supposed to apply to every positive whole number, starting at 1 (or in some cases
starting at a different whole number). The idea is to prove two things:
1. That the statement applies to the first positive whole number.
2. That whenever the statement applies to a positive whole number, the statement
must also apply to the next whole number.
In the proof above, we established these two facts. Because of this, since we showed that
the statement applies to the number 1, it must also apply to the number 2. And since it
applies to the number 2, it must also apply to 3. Since it applies to the number 3, it must
also apply to 4. And so on, indefinitely.
This proof technique is very specialized, and even when the technique can be applied to
prove a certain proposition, there may be a more straightforward way to write the proof.
However, there are certain types of mathematical facts that are well-suited to this kind of
proof.
As another example, there is a remarkable fact about what happens when consecutive odd
numbers are added up. For instance, 1 + 3 + 5 = 9, which is a square number. Similarly,
1 + 3 + 5 + 7 + 9 + 11 = 36, which is also a square number. After some experimentation,
you may notice a general pattern that can be summarized as follows.
Proposition: The sum of the first 𝑛 odd numbers is always 𝑛2 .
Proof: We will prove two separate facts –
1. The sum of the first 2 odd numbers is 22 .
2. Whenever the sum of the first 𝑛 odd numbers turns out to be 𝑛2 , the sum of the first
𝑛 + 1 odd numbers must be (𝑛 + 1)2 .
It is easy to see that the first fact is true, since 1 + 3 = 4.
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Now, suppose we find that the sum of the first 𝑛 odd numbers turns out to be 𝑛2 . In other
words, say the sum of the odd numbers from 1 up to 2𝑛 − 1 is 𝑛2 . That is,
1 + ⋯ + [2𝑛 − 1] = 𝑛2 .
After 2𝑛 − 1, the next odd number is 2(𝑛 + 1) − 1. So the sum of the first 𝑛 + 1 odd
numbers is
1 + ⋯ + [2𝑛 − 1] + [2(𝑛 + 1) − 1].
Since 1 + ⋯ + [2𝑛 − 1] = 𝑛2 , this means that the sum of the first 𝑛 + 1 odd numbers must
be
𝑛2 + [2(𝑛 + 1) − 1] = 𝑛2 + 2𝑛 + 1 = (𝑛 + 1)2 .
Taken together, the two facts we have proved show that the sum of the first 𝑛 odd numbers
is always 𝑛2 . ∎
Exercise 11: What is the sum of the first 𝑛 even numbers? Prove that this is always true.
Exercise 12: At some point, you and I have proved both of the following facts, although
they were each worded differently at the time.
(a) If a positive whole number is not even, then it is odd.
(b) If a positive whole number is even, then it is not odd.
Where was statement (a) proved? Where was statement (b) proved?