Mathematics for Computer Science
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“mcs” — 2017/3/3 — 11:21 — page 304 — #312<br />
304<br />
Chapter 9<br />
Number Theory<br />
9.4 The Fundamental Theorem of Arithmetic<br />
There is an important fact about primes that you probably already know: every<br />
positive integer number has a unique prime factorization. So every positive integer<br />
can be built up from primes in exactly one way. These quirky prime numbers are<br />
the building blocks <strong>for</strong> the integers.<br />
Since the value of a product of numbers is the same if the numbers appear in a<br />
different order, there usually isn’t a unique way to express a number as a product<br />
of primes. For example, there are three ways to write 12 as a product of primes:<br />
12 D 2 2 3 D 2 3 2 D 3 2 2:<br />
What’s unique about the prime factorization of 12 is that any product of primes<br />
equal to 12 will have exactly one 3 and two 2’s. This means that if we sort the<br />
primes by size, then the product really will be unique.<br />
Let’s state this more carefully. A sequence of numbers is weakly decreasing<br />
when each number in the sequence is at least as big as the numbers after it. Note<br />
that a sequence of just one number as well as a sequence of no numbers—the empty<br />
sequence—is weakly decreasing by this definition.<br />
Theorem 9.4.1. [Fundamental Theorem of Arithmetic] Every positive integer is a<br />
product of a unique weakly decreasing sequence of primes.<br />
For example, 75237393 is the product of the weakly decreasing sequence of<br />
primes<br />
23; 17; 17; 11; 7; 7; 7; 3;<br />
and no other weakly decreasing sequence of primes will give 75237393. 8<br />
Notice that the theorem would be false if 1 were considered a prime; <strong>for</strong> example,<br />
15 could be written as 5 3, or 5 3 1, or 5 3 1 1, . . . .<br />
There is a certain wonder in unique factorization, especially in view of the prime<br />
number mysteries we’ve already mentioned. It’s a mistake to take it <strong>for</strong> granted,<br />
even if you’ve known it since you were in a crib. In fact, unique factorization<br />
actually fails <strong>for</strong> many integer-like sets of numbers, such as the complex numbers<br />
of the <strong>for</strong>m n C m p 5 <strong>for</strong> m; n 2 Z (see Problem 9.25).<br />
The Fundamental Theorem is also called the Unique Factorization Theorem,<br />
which is a more descriptive and less pretentious, name—but we really want to get<br />
your attention to the importance and non-obviousness of unique factorization.<br />
8 The “product” of just one number is defined to be that number, and the product of no numbers is<br />
by convention defined to be 1. So each prime p is uniquely the product of the primes in the lengthone<br />
sequence consisting solely of p, and 1, which you will remember is not a prime, is uniquely the<br />
product of the empty sequence.