Chapter 34 Prime and perfect numbers 34.1 Infinitude of prime numbers 34.1.1 Euclid’s proof If there were only finitely many primes: 2, 3, 5, 7, ...,P, and no more, consider the number Q = (2 3 5 P ) + 1. · · ··· Clearly it is divisible by any of the primes 2, 3,..., P , it must be itself a prime, or be divisible by some prime not in the list. This contradicts the assumption that all primes are among 2, 3, 5,..., P . 34.1.2 Fermat numbers 2n The Fermat numbers are Fn := 2 + 1. Note that 2n 2n−1 2n−1 Fn 2 = 2 1= 2 + 1 2 1 = Fn 1(Fn 1 2). − − − − − − By induction, Fn = Fn 1Fn 2 F1 F0 + 2, n 1. − − ··· · ≥ From this, we see that Fn does not contain any factor of F0, F1, ..., Fn 1. Hence, the Fermat numbers are pairwise relatively prime. From this,− it follows that there are infinitely primes. 1 1 It is well known that Fermat’s conjecture of the primality of Fn is wrong. While F0 = 3, F1 = 5, 32 F2 = 17, F3 = 257, and F4 = 65537 are primes, Euler found that F5 = 2 + 1 = 4294967297 = 641 6700417. × 1202 Prime and perfect numbers 34.2 The prime numbers below 20000 The first 2262 prime numbers: 10 20 30 40 50 60 70 80 90 100 bbbbbb bb bbb b bb bb b bb b bb bb bb b b b b bb b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b b bb b b b b bb b b b b b bb b b b b b b b b b b b b b b b b b b bb b b b b b b b b bb b b b bb b b b b b b b b b b b b bb b b b b b b b b b bb b b b b b b b b b b b b b b b b b b bb b b bb b b b b bb b b b b b b b b b b b b b 100 b b bb b bb b bb b b b b bb b b b b b bb b b b b b b b bb b bb b b b b b bb b b b bb b bb b b b b b b b b bb b b b b b b b bb b b b b b b bb b b b b b b bb b bb b b bb b b b b b b b bb b b b b b bb b b b b b b b bb b b b bb b b b bb b bb b b b 200 b b bb b b b b b b b b b b b b b b b b b bb b b b b bb b b b b bb b b b b b bb b b b b b b b b b b b b bb b b b b b b b b b bb b b bb b b b b b b bb b b b b b b b b bb b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b 300 bb b b b b b b b b b b b b b bb b b b b bb b b b b b b b bb b b b b b b b b b bb b b b b bb b b b b b b b b bb b b bb b b bb b b b b b b b b b b bb b bb b b b b b bb b b bb b b b b b bb b b b b b b b b bb b b b b b bb b b b bb b 400 b bb b b b b b b bb b b b b b b b b b b b b b b bb b b bb b b b b b b bb b b b bb b bb b b b b b b b b b b b b bb b b b b b b b b b bb b b b b b bb b b b b b b bb b b bb b bb b b bb b b b bb b bb b b b b b b b b b b 500 bb b b b b b b b b b b b b b b b bb b b b b b b b b bb b b bb b b bb b b b bb b b b b b bb b b bb b b b b b bb b b b b b b b b b bb b b bb b b b b bb b b b b bb b b b b b b b b b b b b b b b b b b bb b b bb b b b 600 b b bb b b b b b b b b b b b b bb b b b b b b bb b b b bb b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b bb b bb b b bb b b b b b b bb b b b b b b b b bb b b b b b b b bb b b b b b b b b b b b b b b b b b 700 b b bb b b b bb b b bb b b bb b b b b b b bb b b b b b b b b bb b b bb b b bb b b b bb b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b bb b b b bb b b b b b b b b b b b b b bb b b bb b bb b b 800 bb b b b b b b b b bb b b b b b b b b b b b b b b b bb b bb b b bb b b b b b b b bb b b b b b b b b b b b b b b b b b b b b b bb b bb b b b b b b b b bb b b b b b b b b b b b b b b bb b b bb b b b bb b b b b 900 b b b b b b b b b b b b b b b b b b b b bb b b b b b bb b b b b b bb b b b b b b bb b b b b b b b b b b b b b bb b b b b b b bb b b b b bb b b b b b b b b b b b b b bb b b b b b b bb b b b b b b b b b bb b b bb b 1000 b b bb b b b b b b b b b b b bb b b b b b bb b bb b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b bb b b b b b b bb b b b b b b b bb b b b b b bb b b b bb b b b b b b b b bb b b b b b b 1100 b b b b bb b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b bb b b bb b bb b b b b b b b b b b bb b b bb b b b b b b b bb b b b b 1200 bb b b b b b b b b b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b bb b b b bb b bb b b b bb b b b b b b b b b b b b b b b b bb b b b b b b b bb b b b b b bb b b bb b b b bb b b b b 1300 b b b b b b b b bb b b bb b b b b b b b b b b bb b b b b b b b b bb b b b b b b b b b b b b b b b b b b b bb b b b b b bb b b b bb b b b b b b bb b b b b b b bb b b bb b b b b b b b b b b b bb b b 1400 b bb b b b b b b b bb b b b b b b b b b b b b b b b b b bb b b b b b b b b bb b b b b b b b bb b b b b b b b b b b b bb b b b b b b b b b b b b b b b bb b b b b bb b b b b b b b b b b b b b b b 1500 b b bb b b b b b bb b bb b b b bb b b b b b bb b b b bb b b b b b b b b b b bb b b b b b b b b b b b b b b b b bb b b b b b b b b bb b b b bb b b b b b b b b b b bb b bb b b b b b b b b b b b b bb b b b b b b 1600 b b b b b b b b b bb b b b b b b b b bb b bb b b bb b b b b b b b bb b b b b b bb b b b b b b bb b b b bb b b b b b b b b b bb b b b b b b b b bb b b b b bb b b bb b b b bb b b b b 1700 bb b b b b bb b b b b b b b bb b b bb b b b b bb b b b b b b b b b b bb b b b b b b b b b bb b b b b b b bb b b b b b b bb b b bb b b bb b b b b b bb b b b b b b b bb b b b b bb b b b b b b b 1800 bb b b b b b b b b b bb b b b b b b b b b bb b b b b b b b b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b bb b b 1900 b b b b b b b b bb b b b b b b b b b b b b bb b b b b b bb b b b bb b b b b b b b b b bb b b b bb b b b b b bb b b b b b b bb b b b b b b b bb b b b b b b b b b b bb b b b b b 34.3 Primes in arithmetic progression 1203 34.3 Primes in arithmetic progression B.