Complex Quadratic Integers and Primes

[@1.4142]

Bro was trying not to wake anyone up at the end

[@Patrickoliveirajf]

A master movement

[@KrasBadan]

I'm a guy but that was kinda hot tbh, whispering math is sexy

[@NicolasMiari]

This is a really, really great complement to the material by prof. John Stillwell on the subject! Looking forward to the next video!

[@JasonHise64]

I’d be curious to see primes over the icosian ring. It’s 4D which would make it tough to visualize, but perhaps it would be feasible to show a rotating cross section? I’m imagining you might even pass through the Gaussian and Eisenstein primes as your cross section evolves!

[@polfosol]

I enjoyed watching this series a lot. It's really illuminating and tackles the topic from a different, albeit interesting, angle. Also thanks for the cool web app! Good luck and godspeed.

[@tcaDNAp]

I'm so excited to learn that the Euclidean algorithm works on complex numbers! How interesting that most of the complex Euclidean domains are prime and 3 (mod 4)..!

[@tcaDNAp]

It's almost intuitive seeing all the gaps in O(-4), and it makes me wonder if D could be a complex number too 🤔

[@rujon288]

Love these prime videos

[@MichaelDarrow-tr1mn]

What if, on the ones without unique factorization, we could switch between seeing primes and irreducibles?

[@TheGrayCuber]

That would be interesting, or even a four color system - unit, prime, irreducible, composite. Unfortunately it's not always straightforward to determine what is prime vs just irreducible, but I may look further into that at some point

[@tomkerruish2982]

​@TheGrayCuber And then there's zero, my hero. Such a funny little hero...

[@MichaelDarrow-tr1mn]

@@tomkerruish2982 zero is the only number that is prime but not irreducible

[@MDNQ-ud1ty]

In the pattern at 0:32 the lines seem to form from a lattice and there is either ellipses or parabolas inside. What about inverting the coloring so black is colored(maybe based on surrounding pixels) while the colored is blackened. I've noticed in many number theoretic functions there seems to be some type of complex of linear patterns and sometimes curved(usually something logarithmic/exponential). E.g., gcd(n,m) is a good example. here the heights, while changing with n and m exhibit many linear patterns in it. E.g., gcd(n,a*n). IIRC this is somehow related to tan(theta) and rational theta give lines or something. In any case, one typically see's these types of patterns a lot and it seems like there is a deeper mathematical structure that describes them(or rather which one can order the patterns). Basically something based on the protective plane.

[@neon_Nomad]

Yee I've been working with quaternions: D specifically gimbals and Stewart platforms.

[@frba9053]

Fascinating

[@40watt53]

Do you use Hyperlegible for your font?

[@TheGrayCuber]

Yes I do!

[@chixenlegjo]

I think the idea of primes can be generalized to any set A with some operation & (formally, a semigroup) by defining u as a unit iff there is some number of repetitions such that x&u&u&u&…=x is true for all x in A, then you can define some element p as prime iff there are no pairs of elements (x,y) in A without units that satisfy x&y=p.

[@weirdredstone42]

is O(cube root(D)) a thing??

[@TheGrayCuber]

Yes, cubic integers! They have been studied less than the quadratic integers

[@JM-us3fr]

Looks like there's some sort of Ulam spiral-like pattern when you zoomed out

[@angeldude101]

5 ≡ 1 (mod 4), so O(√5) would have ω = (1 + √5)/2 = φ. Can't wait to see the video on Real quadratic integers to see if this comes up.

[@JonBrase]

Not necessarily in the context of primes, but it would be really interesting to see a treatment of "integers" for which the units are +/- e^(in) , where n is an integer.

[@drdca8263]

These aren’t algebraic. If you use a fixed non-zero n, this won’t be closed under multiplication, but if you make it closed under multiplication by allowing n to take each value, then the result will be dense in the complex plane, and therefore not something that would be visually nice?

[@JonBrase]

@@drdca8263 TBH, I'm more interested in the general properties than necessarily visualization. But yes, the idea is that the value of n is arbitrary, not fixed. Even just the set of units is interesting, because you can use it to establish a weird ordering for the integers (7 radians is congruent to 0.71... radians, so if you rank the integers by how far around the unit circle e^in is, 7 comes in between 0 and 1).

[@charleskolozsvary8714]

This is incredible. Well done. My Algebra II professor would be somewhat disappointed by how much of this stuff I no longer grasp, but it is fascinating nonetheless. Are you currently studying as an undergraduate or graduate student? I’m just curious. Also, are there any particular resources or materials you would recommend or that you have found helpful for revisiting undergraduate level algebra (including UFDs, cyclotomic polynomials, Galois theory, etc.) and or also stepping beyond into graduate level stuff (Representation theory, Lie algebras, I don’t even really know what else)?

[@TheGrayCuber]

I'm currently self-studying various areas of math. I started with Contemporary Abstract Alg by Gallian, which was great, although the inspiration for this series came from a book on Number Theory that covered Gaussian integers. I recommend the Gallian book, and/or finding quality books in areas that interest you. Galois Theory by Stewart was decent

[@vincehomoki1612]

Hold on. 19, 43, 67, 163? Weren't those the numbers from some new-ish Numberphile video? The "cboose numbers" video? Ok what.

[@TheGrayCuber]

en.m.wikipedia.org/wiki/Heegner_number

[@ingiford175]

Yep the video Tree house numbers and Caboose numbers from about a month ago

[@pyros6139]

At 14:34 I don't like how you switched from diameter to radius as the meaning of "size". Maybe that's a nitpick, as I guess the radius would've covered just as much in the case of integers.

[@DeclanPeterson-s8n]

This a “prime” example for applications of quadratic primes. Like, what is Optimus prime’s favorite number, 17. Let’s see if anyone can figure out how it is his favorite number.

[@DeclanPeterson-s8n]

This is used in algebraic geometry.

[@tomholroyd7519]

7:00 or Both!

[@Han-b5o3p]

Deja deja vu vu

[@theunknown4834]

I got lost at the very end. Why is there no infinite unique factorisation especially /-15 or so on? Doesnt the circle cover everything?

[@stanleydodds9]

No; even for most of the cases where he mentioned that there was unique factorisation, the "circles" do not cover everything (they are not euclidean domains). Note how he says that being a Euclidean domain is stronger than being a unique factorisation domain. You don't need it to be Euclidean, but it is certainly sufficient. In fact, like he says, for D

[@theunknown4834]

@@stanleydodds9 I see, thanks for your explanation

[@Filup]

Is O(√-3) equivalent to Z[√3, 1/2]?

[@lucasdumetz5306]

i is an element of Z[i,√3,1/2] but not of O(√-3), so it's not equivalent.

[@Filup]

@@lucasdumetz5306 Oop, thanks for that. That was a typo. I've updated my comment!

[@mathcookie8224]

Z[1/2] is the dyadic fractions; because you can multiply halves together, it also includes fourths, eighths, sixteenths, and so on. Z[√3, 1/2], then, would also contain the dyadic fractions. The difference here is that O(√-3) contains 0.5+0.5√-3, but not 0.5 on its own; only numbers where BOTH parts are non-whole halves are in O(√-3).

[@Filup]

@@mathcookie8224 Ahh, you're right. Thanks for that clarification!

[@JMaChrisGCarters]

It is equivalent to z[w] where w is the third root of unity ((1+sqrt(-3))/2)

[@YouTube_username_not_found]

Not sure why the Euclidean division of 4 by 4i isn't 4 = 4i×(-i) + 0.

[@YouTube_username_not_found]

At 12:50

[@TheGrayCuber]

-i is not an element of O(sqrt-4) so we can't use it in the Euclidean alg

[@YouTube_username_not_found]

@@TheGrayCuber I see. Thank you!

[@YouTube_username_not_found]

@@TheGrayCuber Something about the Euclidean division of 4 by 4i still boggles me: Why is the remainder 4? isn't the remainder supposed to be constrained? For the Euclidean division in naturals we all know the remainder must be > 0 and < the divisor in order to get a unique pair (q,r) . The question is: what is the constraint on r in general. I am tempted to say that the constraint should be the following: The Norm of r is strictly less than the norm of the divisor, and thus, 4 cannot be the remainder. which leads to the conclusion that 4 and 4i cannot be divided for lack of appropriate remainder.

[@YouTube_username_not_found]

@@TheGrayCuber Hello?

[@jursamaj]

15:50 I don't think this is a valid application of the algorithm. By allowing 0 as the multiple, *anything* could be the divisor, thus the remainders won't get smaller.

[@roneyandrade6287]

Yo yo yo soy Jessie y Joi, I mean cool math

[@lyrimetacurl0]

-1 * 2 * i 2 divides -2i 😂

[@Filup]

If you are working in a domain of elements of the form a+bi, where b is odd, then you cannot factor 2. That is, there is no such b' in the domain such that 2*(a' + b'i) = a+bi. So, in some domains, 2 does not divide 2i or -2i.