Page 313 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Strong Induction 299
n
+
(b) Let g : Z → R be defined by the formula g(n) = 2 . Suppose a 0 ,
+
a 1 , a 2 ,. . . , a k are real numbers, and define f : Z → R by the
k
2
formula f (n) = a 0 + a 1 n + a 2 n + ··· + a k n . (Such a function is
called a polynomial.) Prove that f ∈ O(g). (Hint: Use exercise 14 and
part (a).)
+
16. Suppose a and b are positive integers. Let S ={x ∈ Z |∃s ∈ Z∃t ∈
Z(x = as + bt)}. Note that S = ∅ since, for example, a = a · 1 + b · 0,
and therefore a ∈ S. Thus, by the well-ordering principle, we can let d be
the smallest element of S.
(a) Prove that d | a and d | b. (Hint: Use the division algorithm to choose
integers q and r such that a = dq + r and 0 ≤ r < d. Now show that
r = 0).
(b) Prove that if c is any integer such that c | a and c | b, then c | d.
(Note that it follows that c ≤ d,so d is the greatest common divisor
of a and b).
17. (a) Suppose a, b, and p are positive integers and p is prime. Prove that if
p | ab then either p | a or p | b. (Hint: Let d be the greatest common
divisor of a and p. By exercise 16, d = as + pt for some integers s
and t. Since p is prime, there are not many possibilities for the value
of d. What are they?)
(b) Suppose a 1 , a 2 ,..., a n is a sequence of positive integers and p is
a prime number. Prove that if p | (a 1 a 2 ... a n ), then p | a i for some
i, 1 ≤ i ≤ n. (Hint: Use part (a) and induction.)
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18. Suppose p 1 , p 2 ,..., p j and q 1 , q 2 ,..., q k are two sequences of prime
numbers and p 1 p 2 ... p j = q 1 q 2 ... q k . Suppose also that both sequences
are nondecreasing; that is, p 1 ≤ p 2 ≤ ... ≤ p j and q 1 ≤ q 2 ≤ ... ≤ q k .
Prove that the two sequences must be the same. In other words, j = k and
p i = q i for all i, 1 ≤ i ≤ j. (Hint: Apply exercise 17 and use induction
on either j or k. Note that this shows that the factorization of an integer
n > 1 into primes in Theorem 6.4.2 is unique.)
19. A sequence a 0 , a 1 , a 2 , ... is defined recursively as follows:

a 0 = 1;
n

for every n ∈ N, a n+1 = 1 + a i .
i=0
Find a formula for a n and prove that your formula is correct.
20. A sequence a 0 , a 1 , a 2 , ... is defined recursively as follows:
∗
a 0 = 1;
1
for every n ∈ N, a n+1 = 1 + .
a n