1. Math 409/409G History of Mathematics Books VII – IX of the Elements Part 3: Prime Numbers

2. Prime and Composite Numbers An integer p > 1 is prime if its only positive divisors are 1 and p. Ex. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … An integer greater than 1 that is not prime is called composite. Ex. 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, … Note: 1 is neither prime nor composite.

3. Well before Euclid, mathematicians made lists of prime numbers in the hope that they could discover a pattern that would generate all prime numbers. To this day, no such pattern has been discovered. The first thirty prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113

4. To begin our brief investigation of prime numbers, let’s first revisit Euclid’s Lemma: If aІbc and gcd(a,b)  1, then aІc. This says that a number divides one of the factors in a product if it is relatively prime to the other factor in the product. What happens when that number is prime? That is, if p is prime and pІbc what conditions are needed to guarantee that p divides one of the factors b or c?

5. Let’s try an example. We know that 3│36 and that 36 = 1·36 = 2·18 = 3·12 = 4·9 = 6·6. Can you see what condition or conditions are needed to guarantee that 3 divides one of the factors of 36?

6. I hope you saw that no condition was necessary. We know that 3│36 and that 36 = 1·36 = 2·18 = 3·12 = 4·9 = 6·6. This leads us to suspect that since 3 is prime, if it divides a product, it will always divide one of the factors in that product.

7. Let’s look at another example: 7│84 and the factors of 84 are 1·84, 2·42, 3·28, 4·21, 6·14, 7·12. Does 7 divide at least one factor?

8. Of course it does. Since 7│84, it also divides one of the factors of 84. 1·84, 2·42, 3·28, 4·21, 6·14, 7·12. So we are lead to suspect that a prime divisor of a product of two numbers is also a divisor of at least one of the factors in the product. Can we prove this?

9. Th 4: If p is prime and pІab, then pІa or pІb. Proof: If p│a then we are done. So assume that p  a. Since the only positive divisors of p are 1 and p itself, p  a  gcd(p,a) = 1. So by Euclid’s Lemma, it follows that p│b. Euclid’s Lemma: If aІbc and gcd(a,b)  1, then aІc.

10. And as a consequence of this theorem we have the following corollary which can be proved by induction. Corollary 4: If p is prime and p│a 1 a 2 ··· a n, then p│a k for some k where 1  k  n.

11. Theorem 4 and its corollary leads us to the cornerstone of the development of prime numbers in particular, and of arithmetic in general.

12. Fundamental Theorem of Arithmetic: Every positive integer greater than 1 is either prime or can be uniquely expressed as a product of primes. The uniqueness of this expression does not take into consideration the order in which the primes occur in the factorization. Ex: 2·3·5 = 3·2·5 = 5·3·2, and so on.

13. What is now called the Fundamental Theorem of Arithmetic is Proposition 14 in Book IX of the Elements. The proof of this theorem is referred to as a “proof by exhaustion,” and it uses an axiom which was added well after Euclid. That axiom is: The Well Ordering Principle (WOP): Any subset of the positive integers has a least element.

14. The proof of the Fundamental Theorem is in your text, but I will illustrate how the proof goes by using an example which shows how to find the prime factorization of 420.

15. Divide 420 by the smallest prime until you can no longer evenly divide by that prime. 420 = 2·2·105 Divide by the next highest prime until you can no longer evenly divide by that prime. 420 = 2·2·3·35 Divide by the next highest prime until you can no longer evenly divide by that prime. 420 = 2·2·3·5·7

16. Since there are only a finite number of numbers, and thus primes, less than the number you are factoring into primes, the WOP says that this process must terminate.

17. WHY SHOULD YOU CARE ABOUT THE FUNDAMENTAL THEOREM OF ARITHMETIC! Well it’s the most important theorem you have when it comes to reducing fractions and finding the least common denominator when adding fractions. The following example says it all.

19. This ends the lesson on Books VII – IX of the Elements Part 3: Prime Numbers