Chapter 1 What is a Proof?26
Problem 1.17.
FornD 40 , the value of polynomialp.n/WWDn^2 CnC 41 is not prime, as noted
in Section 1.1. But we could have predicted based on general principles that no
nonconstant polynomial can generate only prime numbers.
In particular, letq.n/be a polynomial with integer coefficients, and letcWWDq.0/
be the constant term ofq.
(a)Verify thatq.cm/is a multiple ofcfor allm 2 Z.(b)Show that ifqis nonconstant andc > 1, then asnranges over the nonnegative
integers,N, there are infinitely manyq.n/ 2 Zthat are not primes.
Hint: You may assume the familiar fact that the magnitude of any nonconstant
polynomial,q.n/, grows unboundedly asngrows.
(c)Conclude that for every nonconstant polynomial,q, there must be ann 2 N
such thatq.n/is not prime.Hint:Only one easy case remains.
Exam Problems
Problem 1.18.
Prove that log 912 is irrational.
Problem 1.19.
Prove that log 1218 is irrational.