Proof: There are Infinitely Many Primes (There is no Largest Prime)

In this math lesson we go over a proof by contradiction that proves there are an infinite amount of primes. It is a very important and interesting result, but it is quite easy to prove! The proof supposes there are finitely many primes then considers the number that is the product of all the primes plus one. From here, using the fundamental theorem of arithmetic, a contradiction follows. Thus, we know that our original assumption is false and there are instead infinitely many primes! Enjoy!

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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[lucaaaa]

This video is great. I recently started relearning math in a logic and beautiful way and your video did a great job of explaining this.

[raeyap]

I have a question:
Why are we assuming that Pm | P when P equals to all sums of prime number +1? When we assume prime numbers have a finite number n, the assumption Pm | P is already incorrect at the first place

[jamesdesantis]

I had to listen to this a couple of times, but I think I got it. Thanks!

[davidwalker]

the proof looks good but it assumes that p1, p2, p3 .... pn contains all the primes to that point. EX. P= 2, 7, 11 ... well 2.7.11+1 = 155, which is divisible by 5 and 31 which can then be added to your list. It still gets the job done tho because if pm +1 is still divisible by another prime, then your list of complete primes is incomplete. Just saying y'all.

[DingleBerrieLol]

The first video that has actually helped! Thank you.

[sakurastv]

Fundamental Theorem of Arithmetic, yeah...

[Calverkristina]

Hello, I really like the simplicity of this video on screen paired with your explanation.


Could I please ask for some clarification about how you would conclude the proof as I have to be able to explain this for class. As P is the product of all primes+1, does this mean that the prime factor you called Pm (a multiple of P) cannot actually be a multiple of P because of the +1 and therefore P itself is a prime number (bigger than Pn)? This would contradict the idea of Pn being the biggest prime. Would this not mean that P has at least three factors (1, P and kPm+1_. This would make it not prime. Or is the conclusion that Pm being a factor of P is contradicted by a factor of P having to be Pm+1 and therefore the the assumption is disproved?


I would be grateful for your help, thank you.

[nadred]

what laws exactly says that multiplying primes and adding 1 creates a prime?

or how are you defining p to be a prime, in which there are infinitely many primes?

[Charles-xcsr]

Merry Christmas, is there any way i could contact you. I've been self studying math and i find it challenging to seek and connect with like minds.

[apusapus]

A shorter way to put it: Because the lowest divisor greater than 1 of p!+1 must be a prime number and must be greater than p...

[bazcalkin]

I don’t understand. Surely you can simply do other things such as squaring the primes to get to P1P2P3...Pn+1.

[ericpham]

Very clever. A job well done. Thank you

[adityasinghania]

Can we not use this to find all prime numbers? Just keep on multiplying up till the next biggest prime and add 1

[shubhambhartiya]

Sir I like your way of teaching...
but sir why we are adding 1 to it please reply sir

[atheistnon-stampcollector]

You can't prove a negative. You are committing a logical fallacy. If there really are infinitely many you cant look at all of them to tell if there is no greatest.