Introduction to Mathematical Thinking 2#:Proving the Infinity of Primes Using Mathematical Thinking

This proof belongs to ancient Greek mathematician Euclid who lived around 350 BCE. We will show that if we list the primes p1, p2, p3, etc.Will the list continues forever ?

But first we need to be familiar with terminology of numbers…for example what is integers?

What is the difference between numbers and integers that we are familiar with throughout our school life?

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A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1,2,3,4 and so forth.

The numbers ….,-3,-2,-1,0,1,2,… are called the rational integers or simply the “integers”.

The numbers 0,1,2,… the non-negative integers,and the numbers 1,2,3 the positive integers.The positive integers are often essential to regard them as a subclass of the integers or some larger class of numbers.

What is the divisibility of the number?

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An integer a is said to be divisible by another integer b,not 0,if there is a third integer c such that a=bc.We express the fact that a is divisible by b,or b is a divisor of a,by denote as b|a thus 1|a ; a|a and b|0 for every b but 0.

When we denote b∤a it means contrary of b|a.

It is plain that b | a and c | b ➡️ which implies c | a. (from transition property)

And if c | b and c | a which means that c is also divisor of linear combination of b and a so from algebra properties c | ma+nb if c not equal to 0 and for all integer m and n.

What is the mystery of prime numbers?

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DURİNG Middle School, prime numbers always seemed mysterious to me🧙🏻‍♀️.

When I was holding numbers, I kept the prime numbers because they were mysterious and weirdly made me feel better.

Prime Numbers is a sub-class of peculiar importance,the class primes.

Prime Numbers satisfies if a number p is prime if p satisfies (i) and (ii)

(i) p>1

(ii) p has no positive divisors except 1 and p.

A number greater than 1 and not prime is called composite.

Why is the number 1 not a prime number?

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I found a convincing answer to a question I asked my teachers in my high school years.Why 1 is not a prime ? 👀

(i) If it was regarded as a prime number, then, every other natural number would be a multiple of it.

We called (ii) as a Fundamental Theorem of Arithmetic states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors for example instead of 6=2x3 being the unique factorisation of the composite number 6, an alternative would be 6=1x2x3 or 6=1x1x2x3. so if 1 would be prime, factorisation won’t longer be unique.

(iii) It would become rather difficult to define what a prime number is. After all 2=1x2 would show that 2 has a prime factor other than itself, namely 1. Yet we would still want 2 to be regarded as a prime number, so we would have to exclude “itself” as a legitimate factor. And still 1 would only have “itself” as a factor.

Now that we understand the concepts, we can look at the proof.

Why… such an important question? Maybe it’s a question that causes Mathematics.

Why are there infinitely many prime numbers?

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We start with a list of all of the primes and we assume we’ve reached some stage n, n could be a ten, a million, a billion, a trillion, whatever.

We reached some stage and we show that we can always find another prime bigger than the last one.For any finite set n= (p1 x p2 x p3 all the way up to pr) of primes we consider the number n=p1 x p2 x p3 x..x pr+1 is certainly bigger than the last one in that sequence.

So if n is prime, then it’s a prime bigger than pr. We’re not saying that n would be the next prime after pr, in fact it almost certainly wouldn’t be because n is a lot bigger than these numbers.

So n has a prime divisor p,but p is not one of the pi’s otherwise p would be a divisor of n and of the product p1x p2 x p3 x..x pr and thus also of the difference n- p1 x p2 x p3 x..x pr = 1 which is impossible.If it’s a different than one, it’s bigger than pr .

We’ve found a prime that’s bigger than pr.Is this prime p ,the next prime after pr? ..yes.💡

So a finite set {p1,p2,….,pr} can not be the collection of all prime numbers.

The point is we found a prime bigger than n, so once again, the list can be continued. Defining n that way to make sure that if there’s a prime dividing n, it won’t be equal to any of those.

END of the Proof. ⬛️

There are infinitely many primes.

To sum up ,it is very useful to learn by producing, I recommend it to you😁

Love

Şevval👩🏻‍🌾

Reference:

• An Introduction to the Theory of Numbers — Godfrey Harold Hardy