Lecture on Prime Numbers CS300A

Lecture on Prime Numbers

CS300A: Technical Communications

Lecturer: Rajat Mittal Scribe by: Ujjawal Goyal

1. Why primes

(a) No one can divide it

(b) Every thing is made up of Prime numbers

2. No. of Prime numbers is Infinite Why? Lets see some proofs:

(a) Euclid’s (Proof from ”the book”)

Suppose they are finite and LetP1, P2, ..., Pn be all prime numbers.

What about the no. K=P1·P2·...·Pn+ 1.

IfPi divides K then its must divide 1 =>K is only divisible by 1 other than itself.

Hence, There exist another Prime no. not inP1, P2, ..., Pn. (b) Lagrange’s proof

Conclusion from Lagrange’s theorem: consider the sequence 2⁰mod q, 2¹mod q, ... 2^kmod q, if k is prime then all the elements present in the sequence must be multiple of k.

Suppose p is the largest prime and consider the number 2^p−1. Let q be a prime that divides 2^p−1.

2^p ≡1 mod q.

Also by Fermat little’s theorem, 2^q - 2≡0 mod q.

or, 2^q−1 ≡1 mod q, here we get another value q-1 in the sequence 2^k ≡1 mod q.

Therefore, by Lagrange’s Theorem p must dividesq−1. Sop < qwhich contradicts our assump- tion.

Preposition: a·b mod c = (a mod c)·(b mod c) mod c.

Proof: Let a = c·q1 +r1, where 0≤r1< candq1is some integer. a mod c =r1. Let b = c·q2+r2, where 0≤r2< candq2is some integer. b mod c =r2. LHS = a·b mod c, after putting a and b, we get LHS =r₁·r₂ mod c.

RHS = (a mod c)·(b mod c) mod c =r₁·r₂mod c.

Therefore RHS = LHS. proved.

(c) Erodes proof

Let the number of prime is finite. then the sequence

1

2+¹₃+· · ·+_p¹

n converges.

So there exist a k such that

P

n≥k+1 1 p_n ≤¹₂. or,∀ NP

n≥k+1 N pn ≤ ^N₂

1

Lets divide N in two set Nbig andNsmall. such thatNbig +Nsmall = N.

N is big if∃p_mm ≥k+1 andp_mdivides N.

N_big ≤P

n≥k+1 N pn ≤ ^N₂.

N is small if all prime factors are smaller than p_k+1. N_small can be written as a_n·b²_n. N_small<2^k·√

N. and since K is fixed, O(N_small) is√ N. From here we getN_big +N_small< N which contradicts our assumption.

Hence Number of primes are Infinite.

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