Download Euclid`s proof of the infinitude of primes (with reasons for every step)

Assume that you know about the integers and the four basic operations (addition, subtraction, multiplication, and division), the ordinary rules of “high school algebra,” as well as the following definitions, theorem, and corollary labeled 1—6 below. Definition 1. An integer b is said to divide an integer a if the quotient a/b is an integer. In this case, b is called a divisor of a or a factor of a. Definition 2. An integer p greater than 1 is called prime if the only divisors of p are 1 and p. Definition 3. An integer p is a prime divisor of an integer n if p is prime and p divides n. Theorem 4. (The Fundamental Theorem of Arithmetic) Every integer greater than 1 can be factored into primes in exactly one way (disregarding re-­‐orderings of the factors). Corollary 5. Every integer greater than 1 has at least one prime divisor. Lemma 6. If d divides both a and b, then d divides the difference a — b . Now, we shall provide a proof of Euclid’s theorem by contradiction: Euclid’s theorem. There are infinitely many prime numbers A step-­‐by-­‐step proof is given below. Your job is to provide the reason that justifies each step. For example, your reasons to justify a step might be of the form “by steps (4) and (6)” or “by applying Definition 3 to step (5)” or “by applying elementary algebra to step (7).” Step Statements (1) Suppose there are only finitely many prime numbers (2) There is some unknown number N that is the total number of prime numbers. (3) Let p1, p2, . . . , pN represent the list of all prime numbers. (4) Let Q = 1 + p1p2 . . . pN (5) Q > 1 (6) Q is not prime. (7) There is some prime p such that p divides Q. Reasons This is the opposite of what we intend to prove. Any finite set has a certain number of elements; we will call this unknown number N for convenience. By step 2, and any list of N objects can be so enumerated. Arbitrary definition for convenience. Q is larger than the product of all of the primes. Q is larger than the product of all of the primes, and thus is larger than any particular prime, so it cannot be prime itself. Corollary 5. (8) The prime p that divides Q must be one of p1, p2, . . . , pN. (9) The prime p = pi for some i in {1,2,3,…,N}. (10) The prime pi divides p1p2 . . . pN. (11) Q – p1p2 . . . pN = 1. (12) The prime pi divides 1. (13) Thus pi = 1. (14) We have a contradiction. (15) There are infinitely many prime numbers. Steps 7 and 3. Step 9. By Definition 1, since p1p2 . . . pN / pi = p1p2 . . . pi-­‐1 pi+1pi+2 . . . pN is an integer. Elementary algebra on Step 4. Lemma 6. The only positive divisor of 1 is 1. Step 13 contradicts definition 2; any prime must be greater than 1. Assuming the truth of statement 1 leads to a contradiction, thus statement 1 must be false.