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Prime and Perfect Numbers 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.
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  • 1 Mersenne Primes and Perfect Numbers
    1 Mersenne Primes and Perfect Numbers Basic idea: try to construct primes of the form an − 1; a, n ≥ 1. e.g., 21 − 1 = 3 but 24 − 1=3· 5 23 − 1=7 25 − 1=31 26 − 1=63=32 · 7 27 − 1 = 127 211 − 1 = 2047 = (23)(89) 213 − 1 = 8191 Lemma: xn − 1=(x − 1)(xn−1 + xn−2 + ···+ x +1) Corollary:(x − 1)|(xn − 1) So for an − 1tobeprime,weneeda =2. Moreover, if n = md, we can apply the lemma with x = ad.Then (ad − 1)|(an − 1) So we get the following Lemma If an − 1 is a prime, then a =2andn is prime. Definition:AMersenne prime is a prime of the form q =2p − 1,pprime. Question: are they infinitely many Mersenne primes? Best known: The 37th Mersenne prime q is associated to p = 3021377, and this was done in 1998. One expects that p = 6972593 will give the next Mersenne prime; this is close to being proved, but not all the details have been checked. Definition: A positive integer n is perfect iff it equals the sum of all its (positive) divisors <n. Definition: σ(n)= d|n d (divisor function) So u is perfect if n = σ(u) − n, i.e. if σ(u)=2n. Well known example: n =6=1+2+3 Properties of σ: 1. σ(1) = 1 1 2. n is a prime iff σ(n)=n +1 p σ pj p ··· pj pj+1−1 3. If is a prime, ( )=1+ + + = p−1 4. (Exercise) If (n1,n2)=1thenσ(n1)σ(n2)=σ(n1n2) “multiplicativity”.
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  • +P.+1 P? '" PT Hence
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