7. Prime Numbers Part VI of PJE
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1 7. Prime Numbers Part VI of PJE 7.1 Definition (p.277) A positive integer n is prime when n > 1 and the only divisors are ±1 and +n. That is D (n) = { n 1 1 n}. Otherwise n > 1 is said to be composite. So n > 1 is composite if and only if there exist integers a > 1 and b > 1 such that n = ab. Example The first few prime numbers are Note that it is an extremely hard problem to prove that a given large integer (say with 150 digits) is prime which is exactly what is required to break present day codes. But some very large primes are known such as with digits found by Edson Smith George Woltman Scott Kurowski et al. in 2008 Question What is the connection with the earlier definition of coprime? Lemma If p is prime and p a then p and a are coprime. Recall Two integers m and n are coprime if gcd (m n) = 1. Here gcd (m n) = max D (m n) = max D (m) D (n). Proof By the recollection gcd (p a) = max D (a) D (p). Because p is prime we have D (p) = { p 1 1 p} and so D (a) D (p) { p 1 1 p}. But p a and so p / D (a) and also p / D (a). Thus D (a) D (p) { 1 1}. Yet 1 1 D (n) for any integer n. Hence D (a) D (p) = { 1 1}. Finally therefore gcd (p a) = max D (a) D (p) = 1. 1
2 Theorem Every integer n > 1 is a product of primes. (With the convention that a product could be of just one prime!) Proof p.278 is by strong induction Theorem (Euclid) If p ab then either p a or p b. Proof p.279. The proof makes use of an earlier result that if n ab and gcd (n a) = 1 then n b. Example yet 12 4 and This proves that 12 is not prime! Euclid s Theorem motivates the definition of a prime in more advanced work. Thus p is prime iff for all integers a b p ab implies either p a or p b. Corollary If p a 1 a 2...a n then p a i for some 1 i n. Proof p a 1 a 2...a n p a 1 (a 2...a n ) which by the Theorem implies that either p a 1 or p a 2...a n. If p a 1 we are finished. If p a 2...a n repeat the process. Theorem Fundamental Theorem of Arithmetic. Every positive integer greater than 1 can be written as a product of primes unique up to ordering i.e. for all n 2 n = p 1 p 2...p r where each p i (i = 1... r) is a prime. Proof p.283 is by contradiction Note the similarities between this and the result that permutations in S n can be written as a product of disjoint cycles unique up to ordering. Aside I It must be stressed that it is a very hard problem to find the prime decomposition of large numbers. What do I mean by large? A 232 digit was factored on December : =
3 See numbers. Question How to find primes? Theorem If m is composite then m has a prime divisor not exceeding m i.e. m composite prime p s.t. p m and p m. Proof p.280 is by contradiction. An application of this result is to finding primes. In this method we disgard composite numbers leaving only primes. And how do you know a number m is composite? It is a multiple of an integer (in fact a prime - but that is not important) m. And if we are disgarding composites m less that N then it will be a multiple of an integer m N. Thus we discard the multiples of integers N. Application Sieve of Eratosthenes Write out the list of natural numbers up to N. Strike out all multiples of 2 except for 2. Strike out all multiples of the next remaining number except that number itself (this will be 3). Continue at each step striking out all multiples of the next remaining number except that number itself. Stop when the next remaining number is > N. You will now have a list of primes up to N. So for example if we look for primes up to 100 we need only check for divisibility by primes up to 100 = 10 i.e and 7. See appendix 7-i where this is carried out for N = 100. Question How many primes are there? Theorem There are infinitely many primes. Proof (due to Euler) p.285 is by contradiction. Note The first few examples of p 1 p 2..p r + 1 where p 1 < p 2 <... < p r are the first r primes are = = = =
4 are all primes. But the next one is not: = = Interesting problems concerning primes. The following are conjectures and are all examples of problems that can be simply stated yet for which the answers are as yet unknown. 1) Goldbach s Conjecture Is every even integer n 4 the sum of two primes? 4 = = = = = = = = = Has been checked for all even numbers up to by Oliveira e Silva (July ). Goldbach s Conjecture is a difficult problem because primes are multiplicative objects defined in terms of divisibility and yet this is an additive question. 2) Do there exist infinitely many Twin Primes i.e. pairs of primes p and p such that p p = 2? The first few examples are (3 5) (5 7) (11 13) (17 19) (29 31) (41 43) (59 61) (71 73) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ).... A large prime pair is ± 1 with digits discovered by Kaiser1 & Klahn in ) Is n is prime infinitely often? = = = = = ) For all m 1 does there exists a prime p : m 2 p (m + 1) 2? What is known is Theorem Bertrand s postulate: p : N p 2N. Proof not given in course. For all N 1 there exists a prime 4
5 Binomial Theorem for prime powers Recall that the Binomial Coefficient ( n r) can be expressed as ( ) n = r n! r! (n r)!. Theorem If p is prime then p divides ( p r) i.e. ( ) p p r for 1 r p 1. Proof Since p! = p (p 1)! we have that p p!. If p r! then since p is prime the Corollary above implies that p m for some m in the product r! which means m r p 1. But p m means p m and thus p m p 1 which is a contradiction. Hence p r!. Similarly since p r p 1 we get p (p r)!. Hence p p! yet p r! (p r)! and so p p! r! (p r)! = ( ) p. r Theorem For all a b Z and all primes p we have Proof By the Binomial Theorem (a + b) p = (a + b) p a p + b p mod p. p ( ) p a p r b r r r=0 p 1 ( ) p = a p + a p r b r + b p r r=1 a p + b p mod p using the previous Theorem on all terms of the sum. 5
6 Corollary (Fermat s Little Theorem) For all n 1 and all primes p If gcd (p n) = 1 then Proof by induction. n p n mod p. n p 1 1 mod p. Base case n = 1. 1 p = 1 and so the result holds with equality. Assume the result holds for n = k so k p k mod p. Consider now m = k + 1. Then by the previous Theorem m p = (k + 1) p k p + 1 mod p k + 1 mod p by the inductive hypothesis = m mod p. Thus the result holds for m = k + 1. Hence by induction the result holds for all m 1 Finding and using inverses. From Fermat s Little Theorem we see that if p is prime and p a then a p 1 1 mod p or equivalently a p 2 a 1 mod p. This means that a p 2 is the inverse of a modulo p or in the language of congruence classes [a] 1 p = [ a p 2] p in Z p. An interesting question is with which method is it quickest to calculate the modular inverse. Is it by Euclid s Algorithm or by calculating a p 2 using for example the method of successive squaring? In fact both methods have running time proportional to the number of decimal digits of a. For Euclid s algorithm this result on the running time is the content of Lame s Theorem PJE p.226. A use of inverses is to solve linear congruences. Example Find the integer solutions of 5x 6 mod 19. Solution 19 is prime so Fermat s Little theorem implies mod 19. So mod 19 i.e is the inverse of 5 mod 19. Thus multiplying both sides of the equation by 5 17 gives 5 18 x mod 19 i.e. x mod 19. 6
7 Though this is an answer to the question we normally give x as the least positive residue. Successive squaring gives Thus ( 2) 2 4 and mod 19. x mod 19 which agrees with the answer found in Chapter 3 but which should still be checked by substitution. Calculating powers. With p = 13 and a = 4 we see that mod 13. Thus = ( 4 12) 8 ( 4 2 ) ( 3) 2 9 mod 13 as we have seen before in Chapter 3 using the method of successive squaring. Note Fermat s Little Theorem may appear wonderful in that it helps us solve congruences and simplifies substantially the calculation of large powers modulo p. But the result has two weaknesses. First it only applies to prime modulus. You can often overcome this by using the Chinese Remainder Theorem. The second weekeness is that you need to know that the modulus is prime. It is a very difficult problem showing that a large number is prime. The largest known prime (as of September 2008) is (See for further details). 7
8 Permutations on Z n. Recall from the previous chapter that Z n is the set of invertible classes in Z n and that it can be written as {[r] n : 1 r n gcd (r n) = 1} which because we can take different labels for the classes is the same as {[r] n : 0 r n 1 gcd (r n) = 1}. Definition* Euler s phi-function is for all n 1 given by φ (n) = Z n = {1 r n : gcd (r n) = 1} = {0 r n 1 : gcd (r n) = 1} Examples Simply by checking we see that and φ (5) = { } = 4 φ (7) = { } = 6. In general φ (p) = p 1 for all primes p since all integers strictly less than p are coprime to p. Also by checking φ (8) = 4 and φ (16) = 8. In general φ (2 n ) = 2 n 1 for all n 1 since in these cases we are counting the integers coprime to 2 n i.e. the odd integers. And half of the integers up to 2 n are odd. And you can easily check by hand that φ (6) = 2 φ (9) = 6 and φ (10) = 4. Example What is φ (100)? Solution Start from φ (100) = {0 n 99 : gcd (n 100) = 1}. Then every 0 n 99 can be written uniquely as Note that n = r + s10 with 0 r 9 and 0 s 9. gcd (n 100) = 1 2 n and 5 n 2 (r + s10) and 5 (r + s10) 2 r and 5 r gcd (r 10) = 1. 8
9 Hence φ (100) = {0 r 9 0 s 9 : gcd (r 10) = 1} = 10 {0 r 9 : gcd (r 10) = 1} since there are 10 choices for s = 10 φ (10) = 40. Question for students. What is φ (1000)? Theorem If gcd (a n) = 1 then the function is a permutation. ρ a : Z n Z n [r] n [ar] n Proof The idea for the proof can be seen on p.259 and p.290. Is ρ a well-defined? Does ρ a map into Z n as claimed in the definition of ρ a? If [r] n Z n then gcd (r n) = 1 by definition of Z n. By the assumption in the theorem we have gcd (a n) = 1. Recall that two integers w v are coprime i.e. gcd (w v) = 1 if and only if there exist integers s and t such that 1 = sw + tv. Thus gcd (r n) = 1 s t Z such that 1 = sr + tn gcd (a n) = 1 k l Z such that 1 = ka + ln. Combine these two equalities as 1 = sr 1 + tn = sr (ka + ln) + tn = (sk) ra + (srl + t) n which gives gcd (ar n) = 1. Thus [ar] n Z n and so for every [r] n Z n the image ρ a ([r] n ) is in Z n and the function ρ a is well-defined. Is ρ a a permutation? So is ρ a a bijection from Z n to Z n? The easiest way to check this is to find the inverse map. Since gcd (a n) = 1 there exists an element a 1 such that a 1 a 1 aa 1 mod n. Then we claim that ρ a 1 is the inverse function. To see this note that for [r] n Z n we have (ρ a 1 ρ a ) [r] n = ρ a 1 (ρ a [r] n ) = ρ a 1 [ar] n = [ a 1 ar ] n = [r] n. 9
10 Similarly (ρ a ρ a 1) [r] n = ρ a (ρ a 1 [r] n ) = ρ a [ a 1 r ] n = [ aa 1 r ] n = [r] n. And thus ρ a 1 is the inverse function ρ a a bijection from Z n to Z n and hence ρ a a permutation. Example On Z 8 = { } consider ρ 3. ρ 3 ([1] 8 ) = [3] 8 ρ 3 ([3] 8 ) = [9] 8 = [1] 8 ρ 3 ([5] 3 ) = [15] 8 = [7] 8 ρ 3 ([7] 8 ) = [21] 8 = [5] 8. Dropping the [...] 8 we can represent this as ( ) ρ 3 = = (1 3) (5 7) The other three permutations are ( ) ρ 1 = = Z 8 ( ) ρ 5 = = (1 5) (3 7) ( ) ρ 7 = = (1 7) (3 5) Theorem Euler s Theorem If gcd (a n) = 1 then a φ(n) 1 mod n. Proof is not in PJE s book but the idea is to be found on p.290. Since ρ a is a permutation of Z n then the image of ρ a is simply Z n in some different order. Since order is immaterial in sets we have Z n = Im ρ a i.e. {[r] n : 1 r n 1 gcd (r n) = 1} = {[ar] n : 1 r n 1 gcd (r n) = 1}. 10
11 Taking products of everything in each set we get [r] n = [ar] n = 1 r n 1 gcd(rn)=1 1 r n 1 gcd(rn)=1 1 r n 1 gcd(rn)=1 ([a] n [r] n ) by definition of multiplication in Z n = [a] φ(n) n [r] n 1 r n 1 gcd(rn)=1 since there are φ (n) terms in the product. Now cancel the product from both sides to get [1] n = [a] φ(n) n = [ a φ(n)] n again by the definition of the multiplication of classes. But this final result merely means a φ(n) 1 mod n as required. Example of the method of proof of Euler s Theorem. Z 8 = {[1] 8 [3] 8 [5] 8 [7] 8 } and so φ (8) = 4. Take a = 5 in which case the map ρ 5 is Then [1] 8 [5 1] 8 = [5] 8 [3] 8 [5 3] 8 = [7] 8 [5] 8 [5 5] 8 = [1] 8 [7] 8 [5 7] 8 = [3] 8. [1] 8 [3] 8 [5] 8 [7] 8 = [5 1] 8 [5 3] 8 [5 5] 8 [5 7] 8 = [5] 4 8 [1] 8 [3] 8 [5] 8 [7] 8 Canceling [1] 8 [3] 8 [5] 8 [7] 8 from both sides leaves [5] 4 8 = [1] 8. Thus mod 8 as expected. Taking n = p prime in the Theorem we again get Theorem Fermat s Little Theorem If p is prime and a an integer with p a then a p 1 1 mod p. (1) 11
12 Proof p.290 Corollary If p is prime and a an integer then a p a mod p. Proof Either p a or p a. In the first case Fermat s Little Theorem implies a p 1 1 mod p. Multiply by a to get a p a mod p. in the second case p a and so a 0 mod p. But then a p 1 0 mod p and so a p 1 0 a mod p. In both cases we have a p a mod p. Example Find the last two digits in the decimal expansion of Solution We have to calculate mod 100. Euler s Theorem tells us that 13 φ(100) = mod 10. Thus Now use successive squaring Hence = ( 13 40) mod (2) mod = mod 100. So the last two digits of are 77 as already found in Chapter 3. Note you may now think that Euler s Theorem has none of the problems of Fermat s Little Theorem as we do not have to know that the modulus m is prime Euler s Theorem holds for all integer m. But unfortunately it has another problem how to calculate φ (m)? In general this is very difficult for large m. It can be shown that if m = pq the product of different primes then φ (m) = φ (p) φ (q) = (p 1) (q 1). But to be able to do this we need to factorise m and that is the difficulty. In general the calculating of φ (m) is as hard as factorising m. 12
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