Here is another characterization of prime numbers.

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1 Here is another characterization of prime numbers. Theorem p is prime it has no divisors d that satisfy < d p. Proof [ ] If p is prime then it has no divisors d that satisfy < d < p, so clearly no divisor of p satisfies < d p. [ ] If the number p has no divisors d that satisfy < d p, then its only possible divisors, other than, are greater than p. So if p were composite, we could find divisors a and b for which p = ab > p p = p, which is absurd. Therefore, p is prime. // This provides us with a technique for determining whether a number p is prime: perform trial division of p by every number from 2 up to p. If a divisor is not found, then p must be prime. Indeed, only prime numbers need to be tested as potential factors, since any composite divisor will be reached only after all its prime factors have passed the same test. This is the basis for construction of the Sieve of Eratosthenes, an algorithm for listing the primes. From a list of the integers from 2 to n, we note that the first number must be prime. Then strike from the list all its multiples (4, 6, 8, ). The smallest number not struck (3) must then be prime. Next, strike from the list all the multiples of 3 that have not already been struck (9, 5, 2, ). The smallest number not struck

2 (5) must then be prime. Continue in this fashion until you have found the primes up to n. All the remaining numbers between n and n which have not been struck must all be primes. (Why?) Will we ever run out of primes? Well,...

3 Theorem There are infinitely many primes. Proof [Euclid] Suppose there were only finitely many primes. List them as p, p 2,, p n. Then the number N = p p 2 Lp n +, which is clearly larger than all the primes, must be composite. So it must have a prime divisor, but none of the primes can divide it, since by the DA, division of N by any of the p s leaves remainder. So there must be infinitely many primes. // Another Proof! [Euler] Since p < for every prime p, we can use the geometric series formula to write k = p k 0 p. For example, = L = L = L and so on. Multiplying these (convergent) series together, we get

4 2 3 = L = L and so on. In fact, because of the FTA, it follows that L = L which is more compactly written as = all primes p p n n The series on the right side is the (famous) harmonic series, which is well-known to be a divergent series. It follows that the product on the left cannot have only finitely many factors. // Number theorists have long studied the prime number function π(x) = #(primes p x) (e.g., π(00) = 25) and the nth prime function p(n) = (the nth prime number)

5 (e.g., p(00) = 499) with the hope of discovering a pattern to the growth of the prime numbers. (The functions are related directly by the fact that π(p(n)) = n.) These patterns are elusive and subtle. For instance, it is known that p(n) is not representable as a polynomial function, nor is π(x). Here is some data on π(x): x π(x) π(x) / x x /π(x) ln x Note first that the percentage of primes within the range of numbers from to x decreases steadily as x increases; that is, primes become rarer as they get larger. The last two columns indicate that the quantity x /π(x) seems to grow logarithmically. In fact, Gauss conjectured precisely this in the 790s. It took 00

6 years of concerted effort to prove it, and the final discovery required sophisticated methods from complex analysis to be realized! The Prime Number Theorem [Hadamard & de la Vallee Poussin, 896] lim π(x) x x ln x =. // Before we proceed, let s consider a number of curious facts about the set of primes. A pair of prime numbers whose difference is 2 is called a twin prime pair (e.g., 3 and 5, 5 and 7, 4 and 43, 997 and 999, ). A famous unsolved problem in mathematics is the Twin Prime Conjecture, which states that there are infinitely many twin prime pairs; there is no known proof of this, despite strong evidence that it is true. For instance, in 99 Viggo Brun showed that the series twin primes p p converges! Also, it is conjectured that if T(x) = #(primes p x so that p + 2 is also prime), then lim x T(x) x (ln x) 2 =

7 Another famous outstanding unsolved problem is the The Goldbach Conjecture Every even number greater than 2 is the sum of two primes. // It has been exceedingly difficult to make headway towards proving this result. But we will cite two partial results, both of which are quite hard to prove: Theorem [Vinogradov, 937] Every sufficiently large odd number is the sum of three primes. // Theorem [Chen, 966] Every sufficiently large even number is the sum of two numbers one of which is prime and the other of which is the product of at most two primes. //