Theory of numbers: Euclid's theorem

This lecture is part of an online undergraduate course on the theory of numbers.

We discuss Euclid's proof that there are infinitely many primes, and give a few variations of it showing that there are infinitely many primes in certain arithmetic progressions.

A couple of typos pointed out in the comments:
Sheet 1 at the bottom: 13x39 should be 13x139
Sheet 4 (14:00): 2^2^6+1 should be 2^2^5 +1

[brettaspivey]

I agree so much about his lexicography discussion

[erichahn]

Thanks Prof. Borcherds. Please keep on doing lectures like that. They calm me down during these strange times.

[neilruston]

I notice "Euler" is mentioned instead of "Euclid", several times. It's a mistake I make and nice to see one of the top mathematicians in the world has the same issue :)

[wangjason]

Another mistake at 14:00 : 2^(2^6) + 1 is not divisible by 641. It should be 2^(2^5) + 1 (which equals 641×6700417). Instead, 2^(2^6) + 1 can be prime factorized into 274177×67280421310721.

[apusapus]

So much jargon. Is it not easier to say that 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?

[dneary]

I hate to correct Prof. Borcherds, but 1807 = 13*139 (39 is not prime, or coprime to 1806).

[peiruhan]

for 3(p1 to pn)-1, if it's a prime, then it's also 3n+2, that's good. if it's not a prime, it must contain a prime factor greater than (p1 to pn), if the factor is in 3n+2 that's also good. if it's in 3n+1, then we have (3m+1)k = 3(p1 to pn) - 1 3mk+k=3(p1 to pn) - 1 so k is -1 in Z3. Since 3m+1 is greater (p1 to pn) so k is no more than 3 and k is positive integer. k can only be 2. then 3 *m*2 = 3(p1 to pn) - 3. m*2 = (p1 to pn) - 1 . left is even right is odd since p1 =2 the 1st prime. so contradiction. is my idea right? i hope anyone can help to check

[kusy]

8:21 "A lot of examples in politics that I'm not gonna mentioned... is Pluto a planet?" I think I almost hear "Is a fetus a human being?" lol

[Michael-chhq]

Why are you using numerals and glyphs, thats not euclid you silly goose