Euclid's proof that there are infinitely many primes! Classic math proof!

This is Euclid's proof that uses contradiction to show that infinitely many primes exist. This is a must-know proof for all math majors or anyone interested in math!

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#blackpenredpen #math #calculus #apcalculus

[willFALL]

The proof is trivial and is left as an exercise to the reader.

[t.s]

It's insane how much content you make. I love your videos I really learn a lot from them.

[backyard]

I think it's just simpler to say that q gives a remainder of 1 when divided by any prime p(i), i=1, 2, ..., n meaning that it's not divisible by any p(i) and is therefore prime itself

[Cannongabang]

alright I'd say..
the contradiction comes also from the fact that
in the definition of Q, division by pk gets remainder 1, and since pk divides Q (as it is one of his prime factors) it has also remainder zero, so this is in contradiction with the uniqueness of quotient and remainder in the algorithm of division :)

[joshuazelinsky]

There are a lot of fun variants of Euclid's proof as outlined here. Here's a cute variant (which requires knowing already that 2, 3 and at least one other number are prime): Suppose there were only finitely many primes; then we can set q as the product of all odd primes, so then q+1 and q-1 must both be powers of 2 (since they have no odd prime factors). But q+1-(q-1)=2 and the only powers of 2 which differ by 2 are 2 and 4, so q=3, but obviously q>3. Contradiction.

Here's a very different proof which is highly silly for being unnecessarily overpowered: consider the Riemann zeta function. zeta(2)=pi^2/6 and zeta(2) has the Euler product representation. If there were only finitely many primes then zeta(2) would be rational since there would be finitely many terms in the Euler product each of which is rational. But pi^2 is irrational and hence so is pi^2/6 which is a contradiction. This one is fun for requiring that pi^2 is irrational, that zeta(2)=pi^2/6, and that zeta has the Euler product representation. There may be more convoluted proofs of the infinitude of primes (the one using DFAs is also almost as bad) but I don't know them.

[ayyythatguy]

I love your channel, keep up the good work!

[pxlseyy]

Explained very well for my UK A level maths :) Thank you

[lpfan]

Can someone please explain to me why we add a 1 to the product of all primes. Is it just there so that we get a fraction at the end or what purpose does it serve?

[jackhandma]

This video is so old he still used chalk and blackboard. Definitely filmed during 300 B.C.E.

[HeraldoS]

Cool... If you are open to ideas, it looks to me that those are some of the tradicional Internet proofs but also some of the most clean proofs (and somewhat not like most proofs iykwim) and there are a lot more and more profound proofs outhere... Just within the basics of number theory there are: the division algorithm, Bézout's identity, (which use the good ordered theorem). You could also try the Fundamental Theorem of Arithmetic, and from that the number of divisors of a number... Or proof that if p prime and p|ab then p|a or p|b (to show how to proof an or statement)... Just some ideas...

Very nice channel, btw.

[filipsperl]

wait so if you multiply all the prumes and add one, doesn't that make the new number a prime? seems like the proofs can stop there

[FroggytheCamper]

Please do #15 on the 2017 MIT integration bee qualifying exam,

lim n->infinity of I n where I1 = ∫ 1/1+sqrt (x, I2=∫ 1/1+(1/1+sqrt (x)), I3=∫ 1/1+(1+1/(1/1+sqrt (x))) ...

[absol]

i have a question what if you can evenly divide 1 by one of the p's

[sciencestararvinsinghk]

I love your videos. They are so intriguing and even make sense to someone who is only currently in Math II

[DjVortex-w]

_Finally_ somebody explains why the product of all the primes plus one has a prime factor that's not part of that list of primes. When this proof is presented by people (and sometimes even books), it almost always lacks that explanation.

[gracehu]

thanK YOU SM! my textbook was really bad at explaining this important proof

[dappermink]

If the set of prime numbers is finite, then:
0 < product for all primes p of sin(pi/p) = product for all primes p of sin(pi*(1+2*(product for all primes p' of p'))/p) = 0
QED

[andreguimaraes]

Make a proof for fun video explaining the infinitely many Pythagorean primitives! :D

[rivenoak]

lectures by the real Euclid in ~300bc must have been cool as eff. infinite primes, irrational numbers and fun with triangles etc. :)

[apusapus]

A shorter way of putting it: The list of primes is endless because the lowest factor greater than 1 of p!+1 must be a prime number and must be greater than p.
This is not actually a 'proof by contradiction'.