Technical Communication (CS300)

Technical Communication (CS300)

Nishi Mehta September 2, 2022

Abstract

The lecture is a discussion on Prime Numbers. Further we try to prove the claim that the set of all the prime numbers is an infinite set. This claim is proved using 3 different methods.

1 Introduction

Prime numbers are the numbers that are perfectly divisible by either 1 or the number itself.

The prime numbers are interesting because

• no other number can divide them.

• all the natural numbers are made up of prime numbers.

Let, the set P represent the set of all the prime numbers:

P ={2,3,5, ...}

the first question that arises is - is the set finite or infinite? So, the aim of this lecture is to prove that the set P contains infinite elements.

2 Proofs

2.1 Euclid’s Proof

This proof was given by Euclid and the proof is from ”The Book”.

Let’s assume the set P is finite such that P = {p₁, p₂, p₃, ..., p_n}. This means that all the other natural numbers are divisible by atleast one prime number. Let, k = p₁p₂p₃..p_n+ 1.

Then k is not divisible by any prime number. This contradicts the assumption that there are only n prime numbers.

Hence proved, that the set P is an infinite set.

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2.2 Proof 2

Again, let’s assume that there exist finite number of prime numbers andpis the largest prime number amongst them, also q is a prime number such that 2 < q < p and by Lagrange’s Theorem,

2^p−1 = 0 mod q Let k be the first number such that 2^k−1 = 1 modq From there the series will repeat

⇒ 2⁰ modq, 2¹ mod q, 2² modq ...

⇒ 2^k mod q, 2^k+ 1 mod q, 2^k+ 2 mod q ...

. .

⇒ 2^m mod q

This means that m is divisible by q−1 (Fermat’s Theorem). Thus p is also divisible by q−1, since 2^p−1 = 0 modq

⇒ q > p

This contradicts our assumption that p is the largest prime number.

Hence proved, there are infinite number of prime numbers.

2.3 Erd˝ os’ Proof

This proof was given by Erd˝os and takes into acount the idea that when Pn i=1

1

pi diverges, n→ ∞, where P ={p₁, p₂, ...p_n}i the set of prime numbers.

Let’s us assume that the series Pn i=1

1

pi converges.

⇒ ∃k for which P

i+1 1 pi ≤ ¹₂.

⇒ ∀N,P

i N pi ≤ ^N₂. Let’s assume ∀n≤ N :

⇒ if, ∃p_m;m≥k+ 1 and p_m mod n= 0, then n∈N_big orn is a big number.

⇒ else, n∈N_small orn is a small number.

Clearly, N_big+N_small =N

There exists a number that is a small number ≤N, which is represented by ab² where a is product of small prime numbers less than equal to p_k.

Since there are K prime numbers less than or equal to p_k, the number of distinct values of a≤2^k. The number of values of b≤√

N. Therefor, N_small≤2^k√

N.

⇒ N_big ≤ ^N₂

⇒ so we need to chhos anN for which N_small ≥ ^N₂

⇒ ^N₂ ≤2^k√ N

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⇒ N ≤2^2(k+1)

This contradicts because N cannot be less than 2^2(k+1). This means thatP

i 1

pi diverges.

Hence proved, that n → ∞or there are infinite number of prime numbers.

3 Conclusions

Based on these 3 proofs we can conclude that there exist infinite number of prime numbers or the set containing prime numbers is an infinite set.

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