Why Read This Book? - Index of

Strong Induction
3.5 Variations of the PMI 89
There is another way to build the set S on p. 80. Consider the following. Suppose
S is a subset of the positive integers that has the following properties.
(K1) 1 ∈ S
(K2) If n ≥ 2 and 1, 2,...,n− 1 ∈ S, then n ∈ S.
Then by yet another mimicking of the proof of Theorem 3.4.6, we could show that
S = N. This is called the strong principle of mathematical induction, or SPMI.
Theorem 3.5.17 (SPMI). Suppose S is a subset of the positive integers that has
properties K1–K2. Then S = N.
EXERCISE 3.5.18 This exercise addresses why Theorem 3.5.17 is called strong
induction. To do so, we alter property I 2 slightly to read: If n ≥ 2 and n − 1 ∈ S,
then n ∈ S. For each of the following pairs of statements, state which is stronger.
(a) n ≥ 2 and n − 1 ∈ S n ≥ 2 and 1, 2,...,n− 1 ∈ S
(b) Property I 2 Property K2
(c) Properties I 1–I 2 Properties K1–K2
(d) If I 1 and I 2, then S = N. If K1 and K2, then S = N.
This is why the SPMI can be more powerful than the PMI. If we are required
to prove a result is true for all n ≥ 1 and induction seems to be the way to go,
we might find that regular induction does not provide us with a strong enough
assumption to make the inductive leap. With either form of induction, we would
still need to show that 1 ∈ S. But to make the inductive step, regular induction
would only allow us to assume n ∈ S and require us to conclude n + 1 ∈ S solely
from this. On the other hand, strong induction allows us to assume that all of
1, 2,...,n− 1 ∈ S, and requires us to conclude n ∈ S from this more extensive set
of assumptions. 19
In Section 2.5 we defined a positive integer to be prime provided it has exactly
two distinct positive integer factors. An integer n ≥ 2 that is not prime is called
composite, and such a number can be written in the form n = ab, where a and
b are positive integers strictly less than n. The following theorem addresses the
factorization of positive integers into a product of primes. We use a slight variation
of the SPMI by rooting it at n = 2.
Theorem 3.5.19 (Fundamental Theorem of Arithmetic). Every integer n≥
2 can be written as the product of primes, and this factorization is unique, except
perhaps for the order in which the factors are written.
19 Actually, strong induction is no stronger than regular induction. They are logically equivalent.