Proof: There are infinitely many primes numbers

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We use proof by contradiction to prove the wonderful fact that there are infinitely many primes.

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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.

ScrupulousAtheist

Dr. Bazett: I spent an entire day looking at this problem and always got stuck on the "remainder" issue, where it is just thrown out that dividing p by some prime results in remainder. Out of many dozens of articles and videos I've looked at you are the first one to actually explain this. I actually came close to your explanation at some point, but was incredibly perturbed that no other article backed up my intuition (and yours) about the resultant fractional component when dividing p by some prime. In other words, this was a valid observation, but I didn't know if this actually factored into Euclid's proof, or if it was something else that completed Euclid's proof. I was an engineering student at school and have started on a long road to re-learn stuff so I can learn more stuff. I immediately subscribed to your channel. Kudos.

maxwellchiu

Best video on this topic hands down, brilliant explicit showcase of this. Thank you kindly

samuelhawksworth

The remainder concept that you explained has really sunk in my mind. Thank You So Much

_jayachaubey

Amazing explanation! It’s cool how you write in reverse on the mirror.

mohammedshoaib

good job!! your videos are extremely helpful! please carry on with your work!

minhaj

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.

yongmrchen

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 .

matirachamim

Best explanation on youtube, thank you!

eleanorwj

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.

andrewharrison

Who would've thought that garble nonsense could be so elegant! I would love to see you explain through contradiction that the square root of 2 is an irrational number :)

jenniferwilcox

THANK YOU ONE HOUR BEFORE MY FINALS IT FINALLY MAKES

tesnyme

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

mugayamaddox