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Unformatted text preview: THE PRIMES: INFINITUDE, DENSITY AND SUBSTANCE PETE L. CLARK 1. There are infinitely many primes The title of this section is surely, along with the uniqueness of factorization, the most basic and important fact in number theory. The first recorded proof was by Euclid, and we gave it at the beginning of the course. There have since been (very!) many other proofs, many of which have their own merits and drawbacks. It is entirely natural to look for further proofs: in terms of the arithmetical function π ( n ) which counts the number of primes p ≤ n , Euclid’s proof gives that lim n →∞ π ( n ) = ∞ . After the previous section we well know that one can ask for more, namely for the asymptotic behavior (if any) of π ( n ). The asymptotic behavior is known – the celebrated Prime Number Theorem , coming up soon – but it admits no proof simple enough to be included in this course. So it is of interest to see what kind of bounds (if any!) we get from some of the proofs of the infinitude of primes we shall discuss. 1.1. Euclid’s proof. We recall Euclid’s proof. There is at least one prime, namely p 1 = 2, and if p 1 ,...,p n are any n primes, then consider N n = p 1 ··· p n + 1 . This number N n may or may not be prime, but being at least 3 it is divisible by some prime number q , and we cannot have q = p i for any i : if so p i | p 1 ··· p n and p i | N n implies p i | 1. Thus q is a new prime, which means that given any set of n distinct primes we can always find a new prime not in our set: therefore there are infinitely many primes. Comments: (i) Euclid’s proof is often said to be “indirect” or “by contradiction”, but this is unwarranted: given any finite set of primes p 1 ,...,p n , it gives a perfectly definite procedure for constructing a new prime. (ii) Indeed, if we define E 1 = 2, and having defined E 1 ,...,E n , we define E n +1 to be the smallest prime divisor of E 1 ··· E n + 1, we get a sequence of distinct prime numbers, nowadays called the Euclid sequence (of course we could get a different sequence by taking p 1 to be a prime different from 2). The Euclid sequence begins 2 , 3 , 7 , 43 , 13 , 53 , 5 ,... Many more terms can be found on the online handbook of integer sequences . The obvious question – does every prime occur eventually in the Euclid sequence with p 1 = 2 (or in any Euclid sequence?) remains unanswered. c Pete L. Clark, 2007. 1 2 PETE L. CLARK (iii) It is certainly a “classic” proof, but it is not “aesthetically perfect” (what- ever that may mean). Namely, there is a moment when the reader wonders – hey, why are we multiplying together the known primes and adding one? One can ad- dress this by pointing out in advance the key fact that gcd( n,n + 1) = 1 for all n ....
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- Spring '11
- Number Theory, Prime number, primes, pete l. clark