Proving There Are Infinitely Many Primes

Check out the main channel @polymathematic !

All positive integers greater than 1 are either prime or can be written as the product of prime numbers (that is, composite). But one thing humans have wondered about from almost the beginning of the natural number system is whether there are infinitely many primes, or if there are only finitely many.

Fortunately, it’s relatively easy to show any finite list of primes must be incomplete. Today’s video follows Euclid’s proof that there are infinitely many primes.

[alessiodimaria]

Few days subbed to this channel, all things i knew, I studied, but the simplicity of his explanations is amazing thxx

[jewulo]

Such a brilliant and simple explanation. Very clearly specified.

[amadddd]

Discrete Math rushing back to my brain

[DerKiesch]

What should be mentioned though is that even though this is a proof that there are infinitely many primes, hit might be easily misunderstood as a prime number generator, which it is not.
Even though it works for the first examples up to 11, for 13 the multiplication of all primes yields 30030, add one you get 30031 which is divisible by 59.

Basically the problem with this as "prime generator" is that the numbers square root outgrows the used primes quite quickly and then it's not guaranteed that you ruled out all possible prime factors.

[drakerobinson]

Thank you. I’ve been struggling on this for a while

[DhairyaPandya]

If only teachers taught us something like this

[tyronejames]

I don't doubt that there are an infinite amount of primes.

My thing is, what if you are trying to find out which number is prime individually by going up by 1 on the number line? Yes, 6 factors into 2 and 3, both prime numbers. And with 10, you have a different prime factor of 5. But with 16, there are four factors of 2. With 18, there are 2 factors of 3 and 1 factor of 2. In other words, could there theoretically be a point where all the composite numbers start to only have the same prime numbers as factors?

[Haveuseenmyjetpack]

My teacher tried to teach us this, I was just too stupid

[Raminagrobisfr]

Now i'll demonstre than there is an infinite number of composite numbers.

Suppose there is a finite number of composite number. Multiply all of them together and don't add one.
You got an extra composite number.

[adolphgracius]

The only Prime I would acknowledge is Optimus Prime

[Phillold]

How did showing that one list didn't account for all primes show that no list could?

[orterves]

Fantastic explanation any kid could understand

[chillyman]

Doesn’t this prove the twin prime conjecture never stops? If the primes don’t stop, then surely there’s got to be infinite configurations of prime gaps, therefore infinitely many gaps of 2

[apusapus]

It is simpler to say that because the lowest divisor greater than 1 of n!+1 must be a prime number and must be greater than n, the supply of prime numbers has no limit.

[jmiogo]

What does "infinitely many"? Is that not just infinite?

[agnidas]

polymathy doesn't mean mathematics ... why you don't do anything else in these shorts ?

[johndododoe]

That's a very informal statement that gets to concrete too early and then argues only the example . As an educated person I do know the generalized proof : Imagine you have the complete long list of primes . Multiply all of them to get an enormous composite number . Add 1 to get a number that doesn't divide by any of the primes just multiplied . But if those were all the primes, we just created a new number that doesn't divide by any of the primes, hence a prime that's not on the list, which means any such complete list is wrong .

[abolfazlabasnatj]

Cause
If you plus1
And then if you
2÷7
It will have 1 remain

[mike]

Is this kinda related to Godels incomplete thereom? Sounds just like it.

[adamantine]

But in order to generate that infinite list you'd need infinite energy and such is not the case so an infinite list, in reality, could not exist.