CS300 - Technical Communication Assignment - 2

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CS300 - Technical Communication Assignment - 2

Het Hitendrakumar Patel 200440

1 Introduction

This lecture was all about prime numbers and there is a bit of magic in them, innit? Prime numbers are numbers which are not divided by any number and every number is made up of them. In today’s lecture our aim will be to prove that are infinite prime numbers. We will understand 3 proofs for the same.

2 Euclid’s Proof

As the name suggests, it was given by the great mathematician Euclid and the proof is stated in ”The Book”, the famous text having all the elegant proofs in maths. In this proof, method of contradiction is exploited.

Let us assume that number of primes are finite (say n primes).

Let the set of primes beS =p1, p2, ..., pn

This implies all the numbers that exist are made up of only the above primes.

Consider the numberN=p1·p2·p3...pn+ 1.

None of the primes from our set divides this number. Also it is bigger than every primes number we defined.

Hence according to definition of primes it has to be prime but this contradicts to our initial assumption.

Therefore, number of primes are infinite.

3 Proof using Group Theory

This proof uses two important theorems - Fermat’s Theorem and Lagrange’s Theorem. This proof too ex- ploits the method of contradiction.

Let p be the largest prime that exists, which in turn means that number of primes are finite, but we would prove it in a slick way so be attentive !!

There exist a prime q such that 2^p when divided byqthe remainder is 1 using Lagrange’s Theorem.

Also from Fermat’s Theorem we can say that q−1 is divisible by p. Hence we can say that q > p but according to our initial assumptionpis the largest prime, thus we arrive at a contradiction.

Therefore, number of primes are infinite.

4 Erdos Proof

This proof is based on the fact that if Σ_i=1ⁿ _p¹

i diverges, where pi is the i^th prime number, then n i.e. the number of prime numbers will be infinite.

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Again we use the powerful tool, the method of contradiction !!

Let us assume that the seriesΣ_i=1ⁿ _p¹

i converges.

⇒∃k such thatP

i 1 pi ≤ ¹₂.

⇒∀N, we haveP

i N p_i ≤ ^N₂.

We first define some special kind of numbers. If a number is divisible by prime number greater than pk

then it is called a big number. And numbers which are not big are small numbers.

Letrbe a small number less thanN.

Also, letr=an·b²_n, wherean is the product of distinct small primes less than equal to pk.

We can easily calculate the number of possible of values of a_n, and for every a_n the maximum number ofb_n is√

N. Thus, N_small<2^k·√

N Nbig <^N₂

Also we can always choose suchN that -Nsmall< ^N₂.

Thus the total sum of both will be less thanN. Here we arrive at a contradiction.

Therefore, the number of primes are infinite.

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