Euclid's Proof There are Infinitely Many Primes

[theboombody]

I was thinking about doing a video on this topic, since almost every proof of primes involves a proof by contradiction, but I think yours is as good as anything I could do. Well done.

[apusapus]

A simpler way of putting it: Because the lowest divisor greater than 1 of p!+1 must be a prime number and must be greater than p, the supply of primes is inexhaustible.
That is Euclid's proof in a nutshell and, no, it is not a 'proof by contradiction'.

[fireflylectures]

Because since A is composite, it must be made up of a product of prime numbers (that's the def of composite). We know that none of our original set of primes are factors of A, so there *must* be another factor of A. If this mystery factor is prime, we're done. If it's not necessarily prime as you suggest, then that composite factor can be broken down to *his* prime factors, which would also be a factor of A. Either way, A has a new prime factor that we'll call 'q'. Hope that helps.

[franknakasako]

Thank you. The two times I encountered this proof in books, I couldn’t make sense of it. This did it.

[fireflylectures]

Not really... We cannot assume this or that must happen along the way, just because infinity is a long ways off.

For example, a^2+b^2=c^2 has tons of solutions. However, you would intuitively guess a^3+b^3=c^3 also has tons of solutions, especially with infinitely many options for a, b, and c. Turns out... not a single solution - ever.

Intuition is great for leading to a rigorous proof, but without a rigorous proof our intuition isn't worth much mathematically. Thanks for the kind words!

[NewcharTG]

As for the stop motion, try to speed it up a little or ad more cuts to make it a lil bit more fluid. It will take you more time but It will come out much better. Keep up the good work!

[albertrenshaw]

I don't understand this proof. 2*3*5*7*11*13+1 is not prime, it's composite. But how do you know that it's prime factors are not in the initial list, as opposed to it's factors just being multiple primes from the initial list (i.e. 2*2*2*2*3*3*etc). Obviously you can just check in the example I gave and see that the prime product is indeed not in the 2-13 list but how do we prove that that's always the case? I'm sure I'm missing something, can you help me understand that part of the proof, every video seems to skip over it—thanks!

[malekbr]

Great attempt :) Keep up the good work :)

[DiffEQ]

Way too much lead-up. GET TO THE POINT!