epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Special Kinds of Theorems 53
As both d and m are positive numbers larger than or equal to 2 and
smaller than or equal to k, by inductive hypothesis they are either
primes or product of prime numbers.
If they are both prime numbers, the statement is proved.
If at least one of them is not prime, we can replace it with its prime
factors. So fc + 1 will be a product of prime factors in any case.
Therefore, by the second principle of mathematical induction, the statement
is true for all natural numbers n > 1. •
The adjectives "strong" and "weak" attached to the two statements of
the principle of mathematical induction are not always used in the same
way by different authors and they are misleading. When reading the
two statements, it is easy to see that the strong principle of mathematical
induction implies the principle of mathematical induction. Indeed, the
inductive hypothesis of the principle of mathematical induction assumes
that the given statement is true just for an arbitrary number n. The inductive
hypothesis of the strong principle of mathematical induction assumes that
the given statement is true for all the numbers between the one covered
by the base case and an arbitrary number n.
In reahty, these two principles are equivalent. The proof of this claim is not
easy; it consists of proving that both principles are equivalent to a third
principle, the well-ordering principle (see the material in the front of the
book). So, in turn, the strong principle of mathematical induction and the
principle of mathematical induction are equivalent to each other.
Then which one of the two principles should be used in a proof by
induction? The first answer is that really it does not make any difference, as
one could always use the strong principle of mathematical induction, in
general, when proving equalities, the principle of mathematical induction is
sufficient. For some mathematicians, it is a matter of elegance and beauty to
use as httle machinery as possible when constructing a proof; therefore, they
prefer to use the principle of mathematical induction whenever possible.
We will consider another example that requires the use of the strong
principle of mathematical induction.
EXAMPLE 4. A polynomial of degree n > 1 with real coefficients has at
most n real zeroes, not all necessarily distinct.
Proof:
1. Base case. The statement is true for the smallest number we can
consider, which is 1. Indeed a polynomial of degree 1 is of the form
P{x) = aiX-\-ao with a\ ^ 0. Its zero is the number x = -ao/ai. So, a
polynomial of degree 1 with real coefficients has one real zero.