Proof: There are infinitely many primes numbers

What a clear way of explaining!

Legendary!

What I love about this proof is it is simple, ancient and completely ignores the practicality of calculating p1...pn + 1 and of checking its divisability.

Sir please do videos about zeta functions and properties

thank you

This video is too long. All you need to say is that because the lowest factor greater than 1 of p!+1 must be a prime number and must be greater than p, the list of primes has no limit.

must be a prime number and must be greater than n, the list of primes is endless.

There is double negative in sentence: "c is composite if it is an integer > 1 that is not not prime"

i get it now 🎉

Amazing🎉🎉🎉🎉❤ sir❤

Hands down the best explanation. I like how you defend every step. Everyone seems to just gloss over the factorization. Showing that there's a fraction, if you only use the numbers on the list, means you are missing a prime factor(s). Love it. This proof has always felt unsettled in my mind.

How don we get step 4 that p is a composite? I know p1p2…pn is a composite, so are you saying a composite plus 1 is a composite? I don’t get it.

I think from step 3 we can conclude that p is a prime because it is not divisible by any and all existing primes, p1, p2, …, pn. We end up with an immediate contradiction because we assumed that the largest prime is pn but p > pn.

don't you have to divide both sides of the equation?

I don't understand why p not divisible by any of the primes implies that p must be prime. All we get is that it's divisible by some prime q not among p_1, ..., p_n. But why does that prime factor q have to be p? I have never understood this claim.

hehe you said 'my pee is prime'

Still wonder why he added the 1 in the end, anybody can explain? Please

Wow! One of the best explanations I've ever seen

What a beautiful explanation! Thank you!

Why is 1 added the end of the list? What is the rationale for this?

Thanks,

Mark

Simply genius...

I have this question:
Prove that there are infinitely many primes of the form 4n+3.
(I hasn't found any satisfactory ans)
Thank you.

can someone explain why we have to plus one to p?

fucking incredible

Where does the +1 come from

You didn't prove that P2...Pn+1 is not divisible by P1

For example, if the largest prime was 5 then you would have 2x3x5+1 on top of your equation
Divide by (in this case 2) so you get 3x5+1 on top of our fraction and your P1=2
3x6+1=16 which IS divisible by 2 (your P1)

Superb Explanation!

Impressive 😍😍😍

Thank you. Best explanation.

I believe that there is a case missing here which is that it is composite, but is composed of some primes which are not in the original set that you assumed was all the primes.

orrr just use bertrands postulate

THIS IS A WONDERFUL EXPLANATION, THANK YOU SO MUCH
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Wow, you made the proof easy. Thanks. Could someone highlight the rationale being us adding 1? Because of course that makes the number p indivisible by any primes. And I am also still wondering how it becomes composite given that it is not perfectly a product of primes.
Thanks

This guy is good

Love from indiaaa❤

Superb best explanation!!
thanks for it

excellent explanation. thanks much!

I remember in my first year of uni i took a discrete maths class and we had to prove this in our final exam. My proof definitely made no sense lol

dr. cooked here

A small correction to the explanation.
The assumption that P is a prime is wrong!
It is either Prime or that it is a composite that is divisible by other primes not in our finite group.
For example : 2*3*5*7*11*13 +1=30031
This number is not prime as it is equal to 59*509=30031 !

Another example (simpler)
2*7+1 =15 which is of course not prime and divisible by both 3 and 5 , primes not in our group.

Those remarks don’t change our proof as we added new prime/primes to our finite group , which contradicts our assumptions and proof that the group is infinite .

Thanks but your voice is annoying

I just couldn't understand why we assume p=p1.p2...pn+ (1)?
What's the point of adding 1?

WHy doesn't this mean the product of primes + 1 is prime?

Awesome 👌

Bro is a saviour, hands down

Best explanation

Excellent video I now understand it.

there could be a prime factor of p > than pn

Excellent explanation 🎉🎉🎉