uniqueness of the fundamental theorem of arithmetic, The

Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, Sep 2001 by Sprows, David J

ABSTRACT: This paper describes a short classroom presentation that is designed to give students some awareness of the critical role played by unique factorization in the Fundamental Theorem of Arithmetic and to illustrate the comparative rarity of this property. It is suitable for any level.

KEYWORDS: Primes, unique factorization, rationals.

INTRODUCTION

There is a certain core of mathematical knowledge that we expect that students will bring with them from high school. For the last several years I have been canvassing my freshman classes concerning some of this core. For example, the vast majority of freshman can state the quadratic formula. On the other hand, I have yet to find a single student who can correctly state the Fundamental Theorem of Arithmetic. A few students knew that it has to do with the factorization of a natural number into primes, but none had an appreciation of the critical role played by unique factorization in the statement of this theorem. As an attempt to make students more aware of why the Fundamental Theorem of Arithmetic is so "fundamental" to the structure of the natural numbers and how different mathematics would be without it, I have developed the following unit. I have found that the following can be covered in less than half a class and is of value not only for incoming students, but (unfortunately) often for outgoing seniors.

UNIQUE FACTORIZATION

We begin by considering a family of number systems that are all closed under multiplication. Let I^sub n^ denote the set of all natural numbers that are divisible by n. For example, I^sub 3^ is the set of all multiples of 3 and I^sub 1^ is the set of natural numbers. For n greater than 1, define a "prime" to be a number in I^sub n^ that is not the product of two smaller numbers in I^sub n^. The definition for I^sub n^ is essentially the same, except that in this case there is a technical problem with the unit value 1. In I^sub n^ we define a "prime" to be a non-unit that can not be written as the product of two smaller non-units.

Note that the "primes" in I^sub n^ include any number of the form k times n where the greatest common divisor of k and n is 1 ((k, n) = 1). A direct consequence of the definition of a "prime" is that any number in I^sub n^ can be written as the product of primes. However for n greater than 1 this factorization may not be unique. For example, in I^sub 3^ the number 90 can be factored into "primes" as either (3)(30) or (6)(15). In fact, if we let A = (nk)(nj) where (n, k) = 1, (n, j) = 1, and (k, j) = 1, then A can be factored into "primes" in I^sub n^ as either (n)(nkj) or (nk)(nj).

After showing that unique factorization is a property that does not hold in many number systems, we then give the statement of the Fundamental Theorem of Arithmetic, i.e., Every natural number greater than 1 can be expressed as a product of primes in exactly one way (up to a rearrangement of the factors). Rather than give a proof of this result, I have found it more effective to have the student investigate the rareness of the unique factorization property by considering other number systems that are closed under multiplication such as the set of all numbers that leave a remainder of 1 when divided by (for example) 4. In each of these systems a "prime" is a non-unit in the system that can not be expressed as the product of two smaller non-units in that number system. In the vast majority of such systems it is determined that unique factorization fails to hold.

Finally we look at how not having the Fundamental Theorem of Arithmetic would change not only the arithmetic of the natural numbers, but also aspects of other number systems such as the rationale. Starting with In, define a "rational" number as any expression of the form AIB where A and B are numbers in I,,. Note that every rational number in the usual sense can be expressed as a "rational" in I,. For example, 3/7 can be expressed as 9/21 in I^sub 3^. However, there are subtle changes in the properties of the "rationals" in systems without unique factorization. As an example consider the following theorem which holds for rationale defined in the usual manner using natural numbers.

THEOREM. If a rational is of the form P/Q where P and Q are distinct primes, then the square root of this rational is irrational.

This theorem can be proved by contradiction using the Fundamental Theorem of Arithmetic. For if we assume that the square root of P/Q is of the form A/B where A and B are natural numbers, then it follows that P times B^sup 2^ equals Q times A^sup 2^. In the factorization of A^sup 2^ and of B^sup 2^, the prime P (or Q) must appear an even number of times (this even number may be zero). As a consequence, P times B^sup 2^ contains an odd number of P's in its factorization while Q times A^sup 2^ contains an even number of P's. This is impossible in any number system where unique factorization into primes holds. Note that essentially the same proof can be used to show that the above theorem holds when square roots are replaced by nth roots. The only change required in the proof is to replace even by multiple of n and odd by non-multiple of n.