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CHAPTER 1: FERMAT’S LAST THEOREM
Around 1637, Pierre de Fermat, a lawyer and
amateur mathematician, conjectured that if n is
a natural number greater than 2, the equation
xn(cid:1)yn(cid:2)zn has no solutions where x, y, and
z are non-zero integers. (Solutions where some
of the integers are zero are possible but not
interesting, and are known as trivial solutions.)
Fermat wrote the statement in the margins of
his Latin edition of a book called Arithmetica,
written by the Greek mathematician Diophantus
in the third century A.D. Fermat claimed, “I have
discovered a truly marvellous proof of this,
which, however, this margin is too small to
contain.” The result has come to be known as
Fermat’s Last Theorem because it was the last of his conjectures to remain unre-
solved after his papers were published. It became famous among mathematicians
because for hundreds of years many great mathematicians attempted to prove it
and achieved only partial results.
Investigate
Fermat’s Last Theorem is closely related to the Pythagorean Theorem, and it is
known that x2(cid:1)y2(cid:2)z2 has integer solutions. For example, 32(cid:1)42(cid:2)52.
The numbers (3, 4, 5) are called a Pythagorean triple. There are an infinite number
of Pythagorean triples. Just pick two numbers a and b with a(cid:3)b and set
x(cid:2)2ab, y(cid:2)a2 (cid:4)b2, and z(cid:2)a2(cid:1)b2. Then x2(cid:1)y2(cid:2)z2. Try it. Then prove that
it works for any a and b.
Similar to Fermat’s Last Theorem is Leonhard Euler’s conjecture that there are no
non-trivial solutions to x4(cid:1)y4(cid:1)z4(cid:2)w4. This question remained unresolved for
over 200 years until, in 1988, Naom Elkies found that
2 682 4404(cid:1)15 365 6394(cid:1)18 796 7604(cid:2)20 615 6734
Since x2(cid:1)y2(cid:2)z2 has infinitely many solutions and x4(cid:1)y4(cid:1)z4(cid:2)w4 has at
least one solution, it is hard to believe that xn(cid:1)yn(cid:2)zn has no solution.
DISCUSSION QUESTIONS
1. Does x3(cid:1)y3(cid:2)z3 have a non-trivial solution?
2. Why is there no need to consider negative integer solutions to xn(cid:1)yn(cid:2)zn?
That is, if we knew there were no solutions among the positive integers, how
could we be sure there were no solutions among the negative integers?
3. When Fermat’s Last Theorem was finally proven, its proof made headlines in
newspapers around the world. Do you think the attention was justified? ●
2 CHAPTER 1
t echnology
Section 1.1 — What Is Proof?
APPENDIX P.485
The concept of proof lies at the very heart of mathematics. When we construct
a proof, we use careful and convincing reasoning to demonstrate the truth of a
mathematical statement. In this chapter, we will learn how proofs are constructed
and how convincing mathematical arguments can be presented.
Early mathematicians in Egypt “proved” their theories by considering a number of
specific cases. For example, if we want to show that an isosceles triangle has two
equal angles, we can construct a triangle such as A A
the one shown and fold vertex B over onto
vertex C. In this example, ∠B(cid:2)∠C; but that is
only for this triangle. What if BC is lengthened
or shortened? Even if we construct hundreds of
triangles, can we conclude that ∠B(cid:2)∠C for
every isosceles triangle imaginable? B fold C C, B
Consider the following example.
One day in class Sunil was multiplying some numbers and made the following
observation:
12(cid:2)1
112(cid:2)121
1112(cid:2)12321
11112(cid:2)1234321
111112(cid:2)123454321
He concluded that he had found a very simple number pattern for squaring a num-
ber consisting only of 1s. The class immediately jumped in to verify these calcu-
lations and was astonished when Jennifer said, “This pattern breaks down.”The
class checked and found that she was right. How many 1s did Jennifer use?
From this example, we can see that some patterns that appear to be true for a few
terms are not necessarily true when extended. The Greek mathematicians who
first endeavoured to establish proofs applying to all situations took a giant step
forward in the development of mathematics. We follow their lead in establishing
the concept of proof.
In the example just considered, the pattern breaks down quickly. Other examples,
however, are much less obvious. Consider the statement, The expression
1(cid:1)1141n2, where n is a positive integer, never generates a perfect square. Is this
statement true for all values of n? Does this expression ever generate a perfect
square? We start by trying small values of n.
1.1 WHAT IS PROOF? 3
1141(1)2(cid:1)1(cid:2)1142, which is not a square (use your calculator to verify this)
1141(2)2(cid:1)1(cid:2)4565, which is not a square
1141(3)2(cid:1)1(cid:2)10270, which is not a square
1141(4)2(cid:1)1(cid:2)18257, which is not a square
Can we conclude that this expression never generates a perfect square? It turns
out that the expression is not a perfect square for integers from 1 through to
30 693 385 322 765 657 197 397 207. It is a perfect square for the next integer,
which illustrates that we must be careful about drawing conclusions based on
calculations alone. It takes only one case where the conclusion is incorrect
(a counter example) to prove that a statement is wrong.
We can use calculations or collected data to draw general conclusions. In 1854,
John Snow, a medical doctor in London, England, was trying to establish the
source of a cholera epidemic that killed large numbers of people. By examining
the location of infection and analyzing the data collected, he concluded that the
source of the epidemic was contaminated water. The water was obtained from the
Thames River, downstream from sewage outlets. By shutting off the contaminated
water, the epidemic was controlled. This type of reasoning, in which we draw
general conclusions from collected evidence or data, is called inductive reason-
ing. Inductive reasoning rarely leads to statements of absolute certainty. (We will
consider a very powerful form of proof called inductive proof later in this book.)
After we collect and analyze data, the best we can normally say is that there is
evidence either to support or deny the hypothesis posed. Our conclusion depends
on the quality of the data we collect and the tests we use to test our hypothesis.
In mathematics, there is no dependence upon collected data, although collected
evidence can lead us to statements we can prove. Mathematics depends on being
able to draw conclusions based on rules of logic and a minimal number of
assumptions that we agree are true at the outset. Frequently, we also rely on
definitions and other ideas that have already been proven to be true. In other
words, we develop a chain of unshakeable facts in which the proof of any state-
ment can be used in proving subsequent statements. In writing a proof, it is
important to explain our reasoning and to make sure that assumptions and
definitions are clearly indicated to the person who is reading the proof. When a
proof is completed and there is agreement that a particular statement can be
useful, the statement is called a theorem.
A theorem is a proven statement that can be added to our problem-solving
arsenal for use in proving subsequent statements. Theorems can be used to help
prove other ideas and to draw conclusions about specific situations. Theorems are
derived using deductive reasoning. Deductive reasoning allows us to prove a
4 CHAPTER 1
statement to be true. Inductive reasoning can give us a hypothesis, which might
then be proved using deductive reasoning.
As an example of inductive reasoning, note that if we write triples of consecutive
integers, say (11, 12, 13), exactly one of the three is divisible by 3. If a number of
such triples are written (say two or three by everyone in the class), we can
observe that every triple has exactly one number that is a multiple of 3. This
provides strong evidence for us to conclude inductively that every such triple
contains exactly one multiple of 3, but it is not proof. We will consider deductive
proof in the other sections of this chapter.
Deductive reasoning is a method of reasoning that allows for a progression
from the general to the particular.
Inductive reasoning is a method of reasoning in which specific examples
lead to a general conclusion.
Exercise 1.1
Part A
In each of the following exercises, you are given a mathematical statement. Using
inductive reasoning (that is, testing specific cases), determine whether or not the
claim made is likely to be true. For those that appear to be true, try to develop a
deductive proof to support the claim.
1. All integers ending in 5 create a number that when squared ends in 25. Test
for the first ten positive integers ending in 5.
Knowledge/ 2. The expression f(n)(cid:2)(cid:4)n3(cid:1)5n2(cid:1)5n(cid:1)6, where n is a positive integer,
Understanding gives a composite number for all values of n. Test for n(cid:2)1, 2, 3, 4, 5, 6.
Thinking/Inquiry/ 3. In every set of four positive integers such that the second, third, and fourth are
Problem Solving each greater by 5 than the one preceding, there is always one divisible by 4.
(One such set is 1, 6, 11, 16). Test using 5, 6, 7, 8, 11, 14, and 17 as first
numbers in the set.
Part B
4. a. The expression n2(cid:1)n(cid:1)5 generates a prime number for every positive
integer value of n.
b. The expression n2(cid:1)n(cid:1)11 generates a prime number for every positive
integer value of n.
c. The expression n2(cid:1)n(cid:1)41 generates a prime number for every positive
integer value of n.
1.1 WHAT IS PROOF? 5
Application 5. Straight lines are drawn in a plane such that no two are parallel and no three
meet in a common point. It is claimed that the nth line creates n new regions.
For example, the first line divides the plane into two regions, creating one
region in addition to the original one. Test this claim for n(cid:2)1, 2, 3, 4, 5.
6. If 9 is subtracted from the square of an even integer n greater than 2, the
result is a number that is composite (has factors other than 1). Test this claim
for n(cid:2)4, 6, 8, 10, 12.
7. If 9 is subtracted from the square of an odd integer n greater than 3, the result
is a number that is divisible by 8. Test this claim for n(cid:2)5, 7, 9, 13, 15.
8. If 3 is subtracted from the square of a positive integer n greater than 4, the
result is a composite number. Test this claim for n(cid:2)5, 6, 7, 8, 9.
Communication 9. John Snow, mentioned earlier, is often called the father of statistics. Discover
more about his interesting work in the great cholera epidemic by searching on
the London Cholera Epidemic on the Internet.
Part C
10. It is claimed that the expression n2 – 79n(cid:1)1601 generates a prime number
t echnology
for all integer values of n from 1 to 200. Write a computer program to deter-
mine whether or not a given number is prime and use it to test the claim.
6 CHAPTER 1
Section 1.2 — An Introduction to Deductive Proof
In Section 1.1, we said that when we are proving something to be true we must
clearly explain the steps in our reasoning and state any assumptions that we are
making. The next example demonstrates these principles and provides a commen-
tary to justify each step.
EXAMPLE 1 Prove that in any set of three consecutive positive integers exactly one of them is
divisible by 3.
Proof
If a is any positive integer, then a(cid:1)1 and a(cid:1)2 are the next two integers, so the
triple (a, a(cid:1)1, a(cid:1)2) represents any set of three consecutive positive integers.
If a is divided by 3, there must be a remainder of 0, 1, or 2.
If a leaves a remainder of 0, then it is divisible by 3. Further, a(cid:1)1 and a(cid:1)2
leave remainders of 1 and 2 on division by 3. Exactly one of the numbers is divis-
ible by 3.
Next, suppose that a leaves a remainder of 1 when divided by 3. If a is the small-
est of the three consecutive integers, then the next integer, a (cid:1) 1, will leave a
remainder of 2. The third integer, a (cid:1) 2, will leave a remainder of 0, so a (cid:1) 2 is
divisible by 3. Again, exactly one of the numbers is divisible by 3.
Finally, suppose that the integer a when divided by 3 leaves a remainder of 2.
Using the same argument as before, the next largest integer, a(cid:1)1, will leave
a remainder of 0 when divided by 3, and the third integer, a(cid:1)2, will leave
a remainder of 1, so a(cid:1)1 is divisible by 3. Exactly one of the numbers is
divisible by 3.
We have considered all possibilities, and each one convinces us that when we take
three consecutive positive integers there is always exactly one of them divisible by
3. We conclude that for any set of three consecutive positive integers exactly one
of them will be divisible by 3.
There are a number of things to observe about this proof. First, the result is com-
pletely general. For any set of three consecutive positive integers, no matter how
large or small, we are guaranteed that exactly one of them is divisible by 3.
Second, we stated clearly the fact on which the proof was developed: when a
positive integer is divided by 3, it leaves a remainder of 0, 1, or 2. Third, we con-
sidered all possibilities, leaving nothing to chance. Finally, and very importantly,
the argument was presented completely in the proof. A good proof is complete,
but we always strive to be brief in its presentation.
1.2 AN INTRODUCTION TO DEDUCTIVE PROOF 7
Mathematical proof enables us to construct a chain of proven facts. In the next
example, we use the result of Example 1 to prove a new fact.
EXAMPLE 2 Prove that if n is an integer, n(cid:5)2, then f(n)(cid:2)n3(cid:4)n is always divisible by at
least 6.
Solution
Factoring the expression, we obtain
f(n)(cid:2)n3 – n(cid:2)n(n2 – 1)
(cid:2)n(n – 1)(n(cid:1)1)
If n is an integer and is 2 or greater, then n – 1, n, n(cid:1)1 is a set of three consecu-
tive integers. Hence, one of these numbers is divisible by 3. Further, if we have
three consecutive integers, at least one of them is even (divisible by 2) by an
argument similar to that of Example 1. Then the product of the three numbers is
divisible by 2(cid:6)3 or 6.
Note that the fact proven in Example 1 makes our task in Example 2 simple once
we factor f(n).
EXAMPLE 3 If a, b, and c form an arithmetic sequence, prove that the equation
(b – c)x2(cid:1)(c(cid:4)a)x(cid:1)(a – b)(cid:2)0 has equal roots.
Proof
Two facts will help. First, if a, b, c form an arithmetic sequence, then there is a
constant difference between consecutive terms. Second, for a quadratic equation
having equal roots, the discriminant is 0.
Since a, b, c form an arithmetic sequence, they have a constant difference, say d.
Then b(cid:2)a(cid:1)d and c(cid:2)b(cid:1)d or c(cid:2)a(cid:1)2d.
D(cid:2)(c – a)2 – 4(b – c)(a – b)
(cid:2)[(a(cid:1)2d) – a]2 – 4[(a(cid:1)d) – (a(cid:1)2d)][a – (a(cid:1)d)]
(cid:2)(2d)2 – 4((cid:4)d)((cid:4)d)
(cid:2)4d2 – 4d2
(cid:2)0
Since D(cid:2)0, the roots of the equation are equal.
In these examples, we have constructed proofs that are entirely general in nature.
We used no specific numerical examples in our proof, and no inductive reasoning.
Note, however, that the numerical examples we considered in Section 1.1 did give
us some assurance that the result is true.
8 CHAPTER 1
Our next example uses the idea of parity. When two integers have the same pari-
ty, either they are both even or they are both odd. If two numbers have different
parity, one of the numbers is even and the other is odd. We can say, for example,
that the numbers 3 and 11 or 2 and 6 have the same parity, while 3 and 8 have
different parity. Problems involving parity usually are posed for positive numbers
only.
EXAMPLE 4 Prove that there is exactly one triple of numbers (n, n(cid:1)2, n(cid:1)4) in which all
three are prime numbers.
Proof
If n is an even number, then n(cid:1)2 and n(cid:1)4 are even also. But the only even
prime is 2, so if n is an even prime, then n(cid:2)2. Now, n(cid:1)2 is 4, which is not a
prime. It is not possible for n to be even.
If n is odd, then n(cid:1)2 and n(cid:1)4 are odd also. Now, one of n, n(cid:1)2, and n(cid:1)4 is
always divisible by 3, because one of n, n(cid:1)1, and n(cid:1)2 is divisible by 3.
If n(cid:1)1 is, then n(cid:1)4(cid:2)(n(cid:1)1)(cid:1)3 is also.
If one of the numbers n, n(cid:1)2, n(cid:1)4 is divisible by 3, that number is not
a prime unless it is 3. Hence, n(cid:2)3 because that is the smallest odd prime, so
there is one triple (3, 5, 7).
Any other triple (n, n(cid:1)2, n(cid:1)4) with n odd must contain a number that is a
multiple of 3, and hence is a composite number, so there is only one such triple.
When writing a proof, explain your steps and always state the facts on which
you base the proof. Remember, the purpose is to convince a reader that you
have constructed a clear, airtight argument.
Exercise 1.2
Part A
1. Prove that every positive integer, ending in 5 creates a number that when
squared ends in 25.
2. Prove that if n is an odd positive integer, then one of the numbers n(cid:1)5 or
n(cid:1)7 is divisible by 4.
Knowledge/ 3. Prove that if nis an even positive integer,then n3– 4nis always divisible by 48.
Understanding
1.2 AN INTRODUCTION TO DEDUCTIVE PROOF 9
Application 4. Prove that the square of an odd integer is always of the form 8k(cid:1)1, where k
is an integer.
Part B
Communication 5. Observe that the last two digits of 72 are 49, the last two digits of 73 are 43,
the last two digits of 74 are 01, and the last two digits of 75 are 07. Prove that
the last two digits of 7201 are 07.
6. Prove that there are no integer solutions to the equation 2x(cid:1)4y(cid:2)5.
Part C
Thinking/Inquiry/ 7. Prove that n5 – 5n3(cid:1)4n is divisible by 120 for all positive integers n(cid:5)3.
Problem Solving
8. Given that p and q are two consecutive odd prime integers, prove that their
sum has three or more prime divisors (not necessarily distinct).
9. Let a , a , a , a , and a be any distinct positive integers. Show that there
1 2 3 4 5
exists at least one subset of three of these integers whose sum is divisible
by 3. (Use the fact that every integer can be written in one of the forms
3k, 3k(cid:1)1, 3k(cid:1)2, where k is an integer.)
10. Prove that if 4 is subtracted from the square of an integer greater than 3, the
result is a composite number.
11. Prove that if 25 is subtracted from the square of an odd integer greater than 5,
the resulting number is always divisible by 8.
10 CHAPTER 1
