I wish many people took such initiative so that people who do not have the privilege to study in premier universities still could access the same lecture. Knowledge for all and above all. Thank you Professor.
Dear Professor Richard, these lectures are so exciting to me. I really appreciate that you upload these kind of stuff. Thanks a lot and greetings from Medellin!
Finally, Number theory! Someone pinchs me please. These lectures have been the best thing ever of this pandemic. Hopefully Mr. Borcherds can cover class field theory, and takes us all the way through Takagi's theorem and Artin's Reciprocity Law
Professor Borcherd. I want you to know that I really love your videos. I’m a neurosurgeon and not a mathematician and over 40 years old and these videos keep me challenged. Many of these ideas are challenging for me but the process of understanding them is a lot of fun and I think it will keep me busy for the rest of my life long after I can’t operate anymore.
Hi, professor. I am really inerested in number theory and your course is wonderful. Since I have no experience in studying modern number theory, I want to know what prerequisites are needed for the course. Thanks for answer my question.
@@zihanwang9262 At the end of the video he recommends some books and mentions that An Introduction to the Theory of Numbers (by Niven and Zuckerman) has exercises
29:00 mandatory project euler shoutout for anyone wanting to challenge themselves with pure, recreational math. most of it is number theory but theres plenty graph theory/pathfinding, game theory (iirc), geometry, algebra, and some other pretty niche topics.
Thank you very much professor, this is a very informative and easy to understand explanation of this branch of mathematics, im a mathematics enthusiast however lacking the interest in the theory of numbers and this 40 minutes almost changed my mind, but there is still too many other areas that i would rather study first, including some of your other playlists subjects.
Hi professor, what is the difference between this course and the other one you have on your channel (Math 115) ? That one seems more rigorous, so would I be missing out on anything if I follow this lecture series instead?
Prof. Borcherds, thank you so much for posting this course! I'm hoping someone from the community can answer my question: Regarding the problem mentioned at x² = 5 + 101m, I wrote a script to find the solution, but it doesn't terminate. I'm wondering whether I made some mistake or the solution is indeed too big for my machine to handle.
Number Theory and discrete math topics should be taught alongside algebra and geometry . It's probably changed alot since I went to high school so this could be the case today but a number theory class in High school would have changed my life earlier than finding it in my 3rd semester Computer Science degree
Was curious to search for an explanation and definition of number theory and found this arithmetical pandora box ... LOL. I now understand the naivety of my curiosity and realize my curiosity cannot be answered as asked !
I think there are improved results to that mentioned at 26:20. See ua-cam.com/video/pp06oGD4m00/v-deo.html beginning at minute 3 : 40 of that video. Looking forward these number theory lessons.
Does anybody have any references to combinatorial number theory? Last summer I was calculating path lengths for random points on a grid and ended up dealing with a similar series. (I'm not a mathematician, I just got carried away)
You could try Combinatorial Mathematics, by Douglas West. It deals with a lot of lattice path counting problems, and generating functions (series). Also consider looking into the Catalan and Delannoy numbers, if you are unfamiliar with them.
No he didn't forget. The first problem at 42:47 is "can you express a prime as a sum of two squares ?" The one you mention "can you write a square as a sum of two primes ?" is the second problem.
Chen Jingrun got famous in China for a long report titled "Goldbach's Conjecture" which told the story of Chen's achievement under incredibly hostile environment during the so-called Cultural Revolution that lasted from 1966 to 1976 (the Ten-year Disaster). During that period of time, he was living in a 6-square meter room that can barely hold a bed and a desk. He was not allowed to carry out his research as number theory was deemed "useless" to the public by those in power and he had to do it secretly. He had little if any resources accessible to him, not even a calculator. The living condition took a toll on his health and in 1984 an accident when biking made it even worse. Another accident months later rendered him handicapped and he had nevered fully recovered. 12 years later he died at the age of 62. It is said that he was working on the Goldbach conjecture, hoping to eliminate the 2-prime summand in his earlier work but we will never know if he would succeed. On the other hand, there is a mistake in the Wikipedia page of Chen: he did not have a Ph.D. as there was no such programs in China before 1983. Hua Look-Geng was not his Ph.D. advisor, then.
Certainly, the report made Chen famous among non-mathematicians. But I believe that Chen's result also surprised experts. It was a common belief that the sieve method had reached its limit and could not provide further improvements. Chen's result overturned such an impression completely.
@@feiqi1975 I am no expert in number theory. Not sure what it exactly means by "the Sieve method had "reached its limit", especially in what sense.is that "limit". What I heard of is that the Sieve method cannot be used to prove the original conjecture, the so-called 1+1 case. But that was after Chen's work was published.
@@huaizhongr Before Chen's work, everyone believed that 2+2 is the best result sieve method can achieve. That's what I meant by "limit" (which had no mathematical meaning)
@@feiqi1975 When you say "everyone", could you please name a few of them? Obviously Chen was not among the "everyone" or he would not even try, would he? Also, I am not sure if Chen's mentor, Prof. Hua Loo-Keng would agree with that "common belief". He was known for not afraid of challenging authorities.
This man has a fields medal and is making videos about number theory that people can watch for free. Blessed day.
Amazing!!!!!! Can't believe it
this video just made my day
I wish many people took such initiative so that people who do not have the privilege to study in premier universities still could access the same lecture. Knowledge for all and above all. Thank you Professor.
I am glad that you are making a course on "theory of numbers"
Dear Professor Richard, these lectures are so exciting to me. I really appreciate that you upload these kind of stuff. Thanks a lot and greetings from Medellin!
Finally, Number theory! Someone pinchs me please. These lectures have been the best thing ever of this pandemic. Hopefully Mr. Borcherds can cover class field theory, and takes us all the way through Takagi's theorem and Artin's Reciprocity Law
Professor Borcherd. I want you to know that I really love your videos. I’m a neurosurgeon and not a mathematician and over 40 years old and these videos keep me challenged. Many of these ideas are challenging for me but the process of understanding them is a lot of fun and I think it will keep me busy for the rest of my life long after I can’t operate anymore.
excited for this series
I'm looking forward to watch this course
Hi, professor. I am really inerested in number theory and your course is wonderful. Since I have no experience in studying modern number theory, I want to know what prerequisites are needed for the course. Thanks for answer my question.
High school math and an interest in number theory should be enough.
@@richarde.borcherds7998 Thanks and where can I find some exercises for practicing?
@@zihanwang9262 At the end of the video he recommends some books and mentions that An Introduction to the Theory of Numbers (by Niven and Zuckerman) has exercises
@@richarde.borcherds7998 sir your playlist is enough for IMO ? For number THEORY
29:00 mandatory project euler shoutout for anyone wanting to challenge themselves with pure, recreational math. most of it is number theory but theres plenty graph theory/pathfinding, game theory (iirc), geometry, algebra, and some other pretty niche topics.
My new favourite channel. Thank you Prof. Richard E. Borcherds!
Thank you very much professor, this is a very informative and easy to understand explanation of this branch of mathematics, im a mathematics enthusiast however lacking the interest in the theory of numbers and this 40 minutes almost changed my mind, but there is still too many other areas that i would rather study first, including some of your other playlists subjects.
Hi professor, what is the difference between this course and the other one you have on your channel (Math 115) ? That one seems more rigorous, so would I be missing out on anything if I follow this lecture series instead?
We are grateful Professor.
Glad to learn such a lecture.
These videos are gold. I am quite lucky to find your channel Sir.
Prof. Borcherds, thank you so much for posting this course! I'm hoping someone from the community can answer my question: Regarding the problem mentioned at x² = 5 + 101m, I wrote a script to find the solution, but it doesn't terminate. I'm wondering whether I made some mistake or the solution is indeed too big for my machine to handle.
Hard to say without looking at your code, if it was on GitHub I'd be willing to take a peek
I think there is an error in your code because x=45 and m = 20 works.
Number Theory and discrete math topics should be taught alongside algebra and geometry . It's probably changed alot since I went to high school so this could be the case today but a number theory class in High school would have changed my life earlier than finding it in my 3rd semester Computer Science degree
Was curious to search for an explanation and definition of number theory and found this arithmetical pandora box ... LOL.
I now understand the naivety of my curiosity and realize my curiosity cannot be answered as asked !
I think there are improved results to that mentioned at 26:20. See ua-cam.com/video/pp06oGD4m00/v-deo.html beginning at minute 3 : 40 of that video.
Looking forward these number theory lessons.
Does anybody have any references to combinatorial number theory? Last summer I was calculating path lengths for random points on a grid and ended up dealing with a similar series. (I'm not a mathematician, I just got carried away)
You could try Combinatorial Mathematics, by Douglas West. It deals with a lot of lattice path counting problems, and generating functions (series). Also consider looking into the Catalan and Delannoy numbers, if you are unfamiliar with them.
I think he forgot to talk about the second problem of algebraic number theory he wrote in 42:47: when a square is a sum of two primes
No he didn't forget. The first problem at 42:47 is "can you express a prime as a sum of two squares ?" The one you mention "can you write a square as a sum of two primes ?" is the second problem.
@@netrapture ??
Chen Jingrun got famous in China for a long report titled "Goldbach's Conjecture" which told the story of Chen's achievement under incredibly hostile environment during the so-called Cultural Revolution that lasted from 1966 to 1976 (the Ten-year Disaster). During that period of time, he was living in a 6-square meter room that can barely hold a bed and a desk. He was not allowed to carry out his research as number theory was deemed "useless" to the public by those in power and he had to do it secretly. He had little if any resources accessible to him, not even a calculator. The living condition took a toll on his health and in 1984 an accident when biking made it even worse. Another accident months later rendered him handicapped and he had nevered fully recovered. 12 years later he died at the age of 62. It is said that he was working on the Goldbach conjecture, hoping to eliminate the 2-prime summand in his earlier work but we will never know if he would succeed.
On the other hand, there is a mistake in the Wikipedia page of Chen: he did not have a Ph.D. as there was no such programs in China before 1983. Hua Look-Geng was not his Ph.D. advisor, then.
Certainly, the report made Chen famous among non-mathematicians. But I believe that Chen's result also surprised experts. It was a common belief that the sieve method had reached its limit and could not provide further improvements. Chen's result overturned such an impression completely.
@@feiqi1975 I am no expert in number theory. Not sure what it exactly means by "the Sieve method had "reached its limit", especially in what sense.is that "limit". What I heard of is that the Sieve method cannot be used to prove the original conjecture, the so-called 1+1 case. But that was after Chen's work was published.
@@huaizhongr Before Chen's work, everyone believed that 2+2 is the best result sieve method can achieve. That's what I meant by "limit" (which had no mathematical meaning)
@@feiqi1975 When you say "everyone", could you please name a few of them? Obviously Chen was not among the "everyone" or he would not even try, would he? Also, I am not sure if Chen's mentor, Prof. Hua Loo-Keng would agree with that "common belief". He was known for not afraid of challenging authorities.
Sir can you make a video about Mock theta functions and hypergeometric series.
Thank you sir.
Insects are perfect!
Thank you a lot !
Count me in.
i know this is titled introduction but everything about math is titled that way. is it really an introduction for an undergrad?
Question: Why do you not consider 2 a Fermat prime? It is 2^0+1. I have heard others omit 2 too but nobody has said why.
Fermat primes are of the form 2^(2^n)+1, note the extra 2 in the exponent.
@@MartinPuskin That is not entirely satisfactory since 0 = 2^-inf.
There is no strong reason why 2 in not called a Fermat prime. It is just a historical accident.
@@richarde.borcherds7998 Thanks
Is number theory by ivan niven good book to begin with. And thank you sir .
This is a very good book, I recommend you read it! It was written by three great mathematicians in number theory
I found the online number theory course by Michael Penn very good. Worth checking out.
@@thaygiaocsp110 is it beginner friendliest?
Thankyou
Cool~
legend
ye
*Theory of numbœs, innit?*