Your number theory series is great! In fact, this is really my favourite UA-cam channel at the moment. So many lectures still to look forward to. I find it particularly remarkable that you chose such a good mixture between working out examples and giving rigorous definitions. The examples are very good and you waste no time writing down too many definitions but use your voice instead for those details. That makes the length of the individual lectures very well consumable.
Wow. Euler's proof of prime infinitude is very sweet (and short)! Never seen that before. Your lecture series is very enjoyable, Richard. Please keep it up.
Euler's manipulations of series always leave me awestruck. It's fortunate that many of the manipulations are vindicated by modern rigor (like, here I'm thinking of the sin(x)/x power series solution of the Basel problem, and I guess this Riemann-Zeta identity too---I had never seen it laid out so simply, so thanks!). I'm reminded of the remark that "all math is either trivial or impossible"---once you see the insights (expand 1/(1-x), and then realize that products of these series correspond exactly to getting all prime decompositions) it's more or less obvious, but to have come up with it in the first place seems downright mad.
14:31 "... omitted some integers, and if we add some extra integers like the ones of the form 4n+1", he meant 4n+3 (so 49 is no longer prime being divisible by 4n+3 for n=1 and the example of non-unique prime factorization into numbers of form 4n+1 e.g. 21x21=9x49 (9,21,49 all "4n+1-primes") becomes the unique factorization 3x7x3x7 = 3x3x7x7 (up to order) )
Great tutorial - thank you. As someone not trained in mathematics to university level, I am having some fun learning these fundamental ideas (and blogging/youtubing about it myself). My question is - what is the reference book you use with the translation of Euler's work? Thanks
To Mr. Richard E. Borcherds. Half the number of whole number factors of an integer greater than or equal to two are prime. All prime numbers have one. Eighteen has six whole number factors and three prime factors. Fifteen has four whole factors and two prime factors. I can prove.
Borcherds: "if you use a computer to calculate the sum of the reciprocals of the primes it will appear to converge to some number like around 3, but it is actually infinite", me: ROTFLOL !!! (as a computer scientist / software engineer its good to know there are still some things even computers can't do).
I would be strongly against. It's not about "looking nice", but about a bit of human contact. I never used handouts or prepared blackboard. I did my programming during my lessons "in flight", because there are all this, seemingly negligible (infinitesimal?) elements of personal contact, which, taken together, form a bond between teacher and pupil, even over the internet. I would say, this small bits are even more important when other forms of human contact are, by necessity, missing.
@@richarde.borcherds7998 As far as I know there are handwriting recognition and "enhancement" algorithms, but these form an "eerie valley" separating the audience from source. (Phenomenon much discussed after premiere of The Polar Express) Last, but not least, I have heard accounts of pupils feeling comparatively less emotional contact with these teachers, who use electronic white/glass boards, even though both sides of interaction are in the same classroom. Some teachers do avoid contact with pupils and try to keep it strictly formal ("Himmler of the lower fifth"... words spoken by Michael York among other greats...), but even then... As for handwriting, mine is definitely worse, be it Latin, Greek, or Cyrilic script.
Your number theory series is great! In fact, this is really my favourite UA-cam channel at the moment. So many lectures still to look forward to. I find it particularly remarkable that you chose such a good mixture between working out examples and giving rigorous definitions. The examples are very good and you waste no time writing down too many definitions but use your voice instead for those details. That makes the length of the individual lectures very well consumable.
Wow. Euler's proof of prime infinitude is very sweet (and short)! Never seen that before. Your lecture series is very enjoyable, Richard. Please keep it up.
Euler's manipulations of series always leave me awestruck. It's fortunate that many of the manipulations are vindicated by modern rigor (like, here I'm thinking of the sin(x)/x power series solution of the Basel problem, and I guess this Riemann-Zeta identity too---I had never seen it laid out so simply, so thanks!).
I'm reminded of the remark that "all math is either trivial or impossible"---once you see the insights (expand 1/(1-x), and then realize that products of these series correspond exactly to getting all prime decompositions) it's more or less obvious, but to have come up with it in the first place seems downright mad.
14:31 "... omitted some integers, and if we add some extra integers like the ones of the form 4n+1", he meant 4n+3 (so 49 is no longer prime being divisible by 4n+3 for n=1 and the example of non-unique prime factorization into numbers of form 4n+1 e.g. 21x21=9x49 (9,21,49 all "4n+1-primes") becomes the unique factorization 3x7x3x7 = 3x3x7x7 (up to order) )
Great tutorial - thank you. As someone not trained in mathematics to university level, I am having some fun learning these fundamental ideas (and blogging/youtubing about it myself).
My question is - what is the reference book you use with the translation of Euler's work?
Thanks
The book "Number theory: An approach through history" by A. Weil has many comments on the history of Euler's work.
I would chatacterise these videos as surveys, not tutorials.
To Mr. Richard E. Borcherds. Half the number of whole number factors of an integer greater than or equal to two are prime. All prime numbers have one. Eighteen has six whole number factors and three prime factors. Fifteen has four whole factors and two prime factors. I can prove.
Borcherds: "if you use a computer to calculate the sum of the reciprocals of the primes it will appear to converge to some number like around 3, but it is actually infinite", me: ROTFLOL !!! (as a computer scientist / software engineer its good to know there are still some things even computers can't do).
You should use a wacom tablet. It will look nicer.
I would be strongly against. It's not about "looking nice", but about a bit of human contact. I never used handouts or prepared blackboard. I did my programming during my lessons "in flight", because there are all this, seemingly negligible (infinitesimal?) elements of personal contact, which, taken together, form a bond between teacher and pupil, even over the internet. I would say, this small bits are even more important when other forms of human contact are, by necessity, missing.
"all these" of course, sorry
@@Suav58 agree with Slaw
I tried writing tablets briefly. Apart from the problems mentioned by Slawomir Wojcik, they made my handwriting even worse than usual.
@@richarde.borcherds7998 As far as I know there are handwriting recognition and "enhancement" algorithms, but these form an "eerie valley" separating the audience from source. (Phenomenon much discussed after premiere of The Polar Express) Last, but not least, I have heard accounts of pupils feeling comparatively less emotional contact with these teachers, who use electronic white/glass boards, even though both sides of interaction are in the same classroom. Some teachers do avoid contact with pupils and try to keep it strictly formal ("Himmler of the lower fifth"... words spoken by Michael York among other greats...), but even then...
As for handwriting, mine is definitely worse, be it Latin, Greek, or Cyrilic script.