Introduction to Higher Mathematics - Lecture 10: Number Theory

In this lecture we delve into number theory, one of the oldest branches of mathematics that still has unsolved problems to this day.

[BillShillito]

That's the beauty of YouTube - you're always more than welcome to pause and go back to any parts of the video you need to hear again. I recommend doing that to let things sink in.

That being said, if there are any topics you need me to clarify, I'd be happy to help out. :)

[AbsolutGB]

Thank you for these lectures! They're really interesting

[amydebuitleir]

Thank you so much for this series. I'm finding it particularly useful for filling in the gaps in my knowledge. I am familiar with most of the material you cover, but with the help of your lectures I am strengthening my foundations, learning bits here and there that I either never learned, or didn't understand at the time.

[OmS]

This was magical, illuminating and enlightening!!!

[mattlm]

So you only need to do the prime sieve up to the square root of some x because all of the multiples of the primes beyond the square root up to x will be multiples of numbers below or equal to the square root which are either the primes we have already done, or multiples of the primes we have already done. It took me seemingly far too long to figure that one out, but I'm happy I did in the end. Many other algorithms only require you loop up to the square root, which is helpful!

[totasalam]

by far the best teacher ever >> i've been trying to understand this since ever .
thank you much

[KannanNambiar]

Thanks for the beautiful video.
In the lecture I particularly appreciate
1. your practice of reviewing previous lecture
2. terse recap of unavoidable terminology
3. making clear Internet can bring out a billion da Vincis
4. Beautyspot :) at 20.35 (the exponent of p_2 is typo e^2? )
Aside: Is "ashura" of Japanese the same as
"asura" of Sanskrit?

[jcropcho]

If I recall correctly there is a polynomial-time (or constant-time?) bounded-error probabilistic algorithm for testing primality.

[joeproozen]

What a very clear way of explaining this subject! Thank you!

[Maryseight]

Prof. Bill, I confused on how did you get 3=3 x 309 - 44 x 21... can you please help me out!

[gabrielsperez]

Amazing videos, you make lessons easy to understand and you also add some humor, thanks, it helps a lot

[jerrymahone]

you have 3 groups here. group 1is (309 -14 (21), group 2 is -2 (21), and group 3 is -1 (309 - 14 (21). 2 is distributed in groups 2 and 3. -1 just change the signs in group 3. distribute -2 through group 3. you get -2 (-309 + 14 (21)) = -2 (-309) - 28 (21). add group 3 to group 1 and you get 3 (309) - 42 (21). now add 2 (21) to get 3 (309) - 44 (21).

[KannanNambiar]

Apropos beauty spot: I wanted to make it clear that I did not consider the typo a blemish. Thanks again for the video. It is my opinion that there is no such thing as difficult mathematics, unless a teacher makes it so. The plain fact is that sometimes the teacher has not understood and appreciated the subject enough to teach it.

[kirnhans]

When finding the value of the totient function, how does the system avoid overcounting non-coprime numbers? Example: 6 is a multiple of both 2 and 3 so is already removed from the list of [12] when we start looking at multiples of 3.

[multimicky]

According to the formula you have given how is the reimman zeta function zero at -2, -4, -6 (time 30:50) . eg. for -2 shouldn't that be (1^2 + 2^2 + 3^2 + 4^2 ....) ?

[pellythirteen]

If Complex numbers are a generalization of the Real numbers what happened to comparing numbers ? In all "lower-order" number systems we can ask : is a>b, a<b and a=b. In the Complex numbers, we can only ask a=b ! What happened to the inequalities ? Does it make sense to ask if the complex-number a is GREATER than another complex-number b ?

[BillShillito]

I'll be posting more starting next week. I had to take care of some stuff, and actually restructure the course a bit ... I'm going to be doing real analysis before abstract algebra instead of the other way around. :)

[jebre]

what if you are trying to find the highest or lowest power of a prime factorization  and you find a tie?  do you take the one that will have the the highest or lowest value (respectively) after the exponent is applied to the prime?

[pendragon]

How does letting 1 be prime destroy uniqueness? How are 1×3×5 and 1×1×3×5 different factorizations? 1^n=1. Isn't that kind of like saying 3×5 is different from 3×5+0?

[fouadhejazi]

at 22:15, what if i decide not to have anything to the power "0" in my factors. that completely changes the gcd(18, 24, 20). It would then look like:
gcd(18, 24, 20) = 2^1 . 3^1 . 5^1 = 30. 
we know this is not true. So what is the logic behind adding numbers to the power of "0" and when do you use them? 
Also, looking at the lcm(18, 20, 24) why do you 5^1 not 2^3?