Page 170 - HOW TO PROVE IT: A Structured Approach, Second Edition

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156 Proofs
b
n
Since b < n, we can conclude that x = 2 − 1 < 2 − 1. Also, since
1
b
ab = n > a, it follows that b > 1. Therefore, x = 2 − 1 > 2 − 1 = 1, so
n
n
y < xy = 2 − 1. Thus, we have shown that 2 − 1 can be written as the prod-
n
uct of two positive integers x and y, both of which are smaller than 2 − 1, so
n
2 − 1 is not prime.
Commentary. We are given that n is not prime, and we must prove that
n
2 − 1 is not prime. Both of these are negative statements, but fortunately
it is easy to reexpress them as positive statements. To say that an integer
larger than 1 is not prime means that it can be written as a product of two
smaller positive integers. Thus, the hypothesis that n is not prime means
+
∃a ∈ Z ∃b ∈ Z (ab = n ∧ a < n ∧ b < n), and what we must prove is that
+
n + + n n
2 − 1 is not prime, which means ∃x ∈ Z ∃y ∈ Z (xy = 2 − 1 ∧ x < 2 −
n
1 ∧ y < 2 − 1). In the second sentence of the proof we apply existential in-
stantiation to the hypothesis that n is not prime, and the rest of the proof is
devoted to exhibiting numbers x and y with the properties required to prove
n
that 2 − 1 is not prime.
As usual in proofs of existential statements, the proof doesn’t explain how
the values of x and y were chosen, it simply demonstrates that these values
work. After the values of x and y have been given, the goal remaining to be
n
n
n
proven is xy = 2 − 1 ∧ x < 2 − 1 ∧ y < 2 − 1. Of course, this is treated
as three separate goals, which are proven one at a time. The proofs of these
three goals involve only elementary algebra.
One of the attractive features of this proof is the calculation used to show
n
that xy = 2 − 1. The formulas for x and y are somewhat complicated, and at
first their product looks even more complicated. It is a pleasant surprise when
n
most of the terms in this product cancel and, as if by magic, the answer 2 − 1
appears. Of course, we can see with hindsight that it was this calculation that
motivated the choice of x and y. There is, however, one aspect of this calculation
that may bother you. The use of “···” in the formulas indicates that the proof
depends on a pattern in the calculation that is not being spelled out. We’ll give
n
a more rigorous proof that xy = 2 − 1 in Chapter 6, after we have introduced
the method of proof by mathematical induction.

Theorem 3.7.2. There are infinitely many prime numbers.
Proof. Suppose there are only finitely many prime numbers. Let p 1 , p 2 ,..., p n
be a list of all prime numbers. Let m = p 1 p 2 ··· p n + 1. Note that m is not
divisible by p 1 , since dividing m by p 1 gives a quotient of p 2 p 3 ··· p n and a
remainder of 1. Similarly, m is not divisible by any of p 2 , p 3 ..., p n .