Discrete Math Class 5 ( )

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1 Discrete Math Class 5 ( Istructor: László Babai Notes tae by Jacob Burroughs Revised by istructor 5.1 Fermat s little Theorem Theorem 5.1 (Fermat s little Theorem. If p is prime ad gcd(a, p = 1, the a p 1 1 mod p. Equivaletly, if p is prime the ( a(a p a mod p DO 5.2. Prove that the two versios of FlT are ideed equivalet. Example 5.3. We ca show this simply for p = 2, 3, 5, otig that the product of cosecutive itegers is always divisible by. a 2 a (mod 2 2 a 2 a = a(a 1 a 3 a (mod 3 2 a 3 a = (a 1a(a + 1 a 5 a (mod 5 5 a 5 a = (a 1a(a + 1(a (a 1a(a + 1(a 2 4 = (a 1a(a + 1(a + 2(a 2 HW 5.4. Prove: if a = 3 4 ad b = the gcd(a, b = 1 or 29. (7 poits Your proof sould tae o more tha two lies. 5.2 Ifiitude of primes; primes i arithmetic progressios Theorem 5.5 (Euclid: Elemets. There are ifiitely may primes. Proof. Let us prove this by cotradictios. Suppose P = m i=1 p i where p 1,..., p m are all the primes. Let q be a prime divisor of P + 1, so q P + 1. Sice q is a prime, ( j(q = p j. Therefore q P, but we also have q P + 1, so q 1, a cotradictio. XC 5.6. Prove: there exist ifiitely may primes p 1 (mod 4 (6 poits XC 5.7. If p is prime ad p 4a the p 1 (mod 4. (Hit: Fermat s Little Theorem (5 poits XC 5.8. Use the precedig exercise to prove that there exist ifiitely may primes p 1 (mod 4. (5 poits 1

2 Theorem 5.9 (Dirichlet. ( a, b 1(if gcd(a, b = 1 the ifiitely may primes p b (mod a The proof uses the theory of complex fuctios. DO If p is prime, p 5, the p ±1 (mod 6 DO There exist ifiitely may primes p 1 (mod 6 CH There exist ifiitely may primes p 1 (mod Asymptotic equality; the Prime Number Theorem: Stirlig s formula f(x Defiitio We say f g if lim x = 1. For sequeces {a g(x } ad {b } we say a a b if lim b = 1. Notatio Let π(x deote the umber of primes x. Example π(10 = 4 π(100 = 25 π(2 = 1 π(π = 2 π( 15 = 0 Theorem 5.16 (Prime Number Theorem, Hadamard ad de la Vallée Poussi, π(x π(x Probability that a radom umber from 1 to x is prime: 1. We ote that this x l x goes to zero fairly slowly, so primes are relatively frequet. Hilarious readig (ot relevat to the course by George Mies: How to be Alie. x l x Theorem 5.17 (Stirlig s Formula. (! 2π e We deote the set{1,..., } by []. Now! is the umber of permutatios of [] Defiitio A permutatio of a set A is a bijectio A A 2

3 5.4 Coutig Notatio Σ deotes the set of strigs of legth over the fiite alphabet Σ. Example Example: let Σ = {A, B, C}; the ABBCA Σ 5 ad {A, B, C} = 3. Defiitio The set P(A is the powerset of A: the set of all subsets of A. Defiitio Give B A, the idicator fuctio of B i A: f B : A {0, 1} is defied as follows: f B (x = 1 if x B ad f B (x = 0 if x A \ B. (This fuctio idicates membership i B. It is also be called the characteristic fuctio of B. Notatio Give A, B sets: DO A B = A B. A B = {f : B A fuctios} Hit. A B couts the strigs of legth B over the alphabet A. DO The fuctio B f B is a bijectio from P(A to {0, 1} A. This proves that P(A = 2 A. HW 5.26 (Due Tuesday, Cout the (0, 1 strigs of legth without cosecutive 1s. Express this i closed form (o summatio or product through a sequece we have already ecoutered. Prove your aswer. (7 poits Notatio We deote the umber of -subsets of a -set by the symbol ( ( choose. Theorem (a For 0 we have ( =!. (b If > the (!(! = 0. Statemet (b is obvious. We prove a geeralizatio of statemet (a. Theorem 5.29 (Permutatios with repeated etries. Let X be a strig of letters over the alphabet Σ = {A 1,..., A m }. Let i deote the multiplicity of of A i (umber of occurreces of A i i X. (So i 0 ad m i=1! i =. The the umber of permutatios of X is (i!. Proof. For otatioal coveiece we describe the proof for the case m = 3 ad alphabet Σ = {A, B, C}. The geeral case wors the same way. Let us label the occurreces of A as A 1,..., A 1, the occurreces of B as B 1,..., B 2, etc. Now all letters are distict, so there are! permutatios. Let us ow drop the labels; let us say that two labeled strigs are equivalet if their ulabeled versios are equal. So for istace, A 2 A 3 B 2 A 1 B 1 is equivalet to A 1 A 3 B 1 A 2 B 2 sice whe we ulabel them, both will become the strig AABAB. This is a equivalece relatio of labeled strigs; what we eed to cout is the equivalece classes. Each equivalece class cosists of i! permutatios, so the umber of equivalece classes! is (i!. 3

4 Remar The istructor calls the method used Kig Matthias s shepherd s method. (The shepherd couted his sheep by coutig their legs ad dividig by 4, otig that there was a atural equivalece relatio o legs ( belogs to the same sheep ad each equivalece class has size 4. DO Derive part (a of Theorem 5.28 from Theorem Hit: Use the alphabet Σ = {0, 1} (so m = 2 ad ecode each -subset by its idicator fuctio viewed as a strig of 1s ad 0s. Theorem 5.32 (Biomial Theorem. (x + y = =0 ( x y Proof. (x 1 + y 1 (x 2 + y 2 (x + y = I [] After droppig subscripts, we get ( =0 x y Theorem 5.33 (Triomial Theorem. ( where = 1, 2, 3 (x + y + z =! 1! 2! 3! DO Multiomial theorem: ( where 1,..., r DO DO HW Express (6 poits (x x r = =! i!. ( 2 ( = 1, 2, 3 0, = i 0, i = ( i I x i j I y i ( x 1 y 1 z 1 1, 2, 3 1,..., r ( 1 ( + 1.! ( ( 2 asymptotically as DO Fid costats a ad b such that a b x 1 1 x r r a b c. Fid the costats a, b, c. 4

5 The umber ( of terms i the biomial theorem is +1. The umber of terms( i the triomial theorem is. The umber of terms i the multiomial theorem is r 1 2 r 1 DO Prove the last statemet above. Hit. We eed to cout the solutios of the equatio r = i oegative itegers i. Use the stars ad bars method. (Loo it up. 5