Gaussian Primes Visually

[stanleydodds]

It's worth noting that your definition of "primes" is actually the definition of irreducibles. They happen to be equivalent for sufficiently structured rings, like unique factorisation domains, but in that case it's a bit circular to say that this general definition of a prime gives rise to unique factorisation. Really, it's the true definition of a prime that gives rise to unique factorisation, that being: p is prime, by definition, if p|ab => (p|a or p|b). In words, a prime is a number such that whenever it divides any product, it must divide at least one of the factors.

Also a formal statement of unique factorisation would allow for an additional arbitrary unit as a factor - this makes it clear that both 1 and -1 (and any other units) are an empty product of primes. Furthermore, only non-zero elements have unique factorisation, since zero is often the special case when it comes to multiplicative properties of a ring.

[StupidusMaximusTheFirst]

"That's beautiful man..." - The Dude.

[tomkerruish]

Subscribed!
Cool tidbit: If you mod the complex plane by the Gaussian integers, you get a torus. This isn't too surprising if you're old enough to have played the arcade game Asteroids. However, the same thing happens if you use the Eisenstein integers, meaning that gluing together opposite sides of a hexagon also produces a torus. (An octagon produces a two-holed torus, but then you need to introduce a hyperbolic metric so the corners will fit together, and it just gets more complicated from there.)

[matthijshebly]

I wonder how the Game of Life would evolve on that grid

[pierrerioux]

This video made me so happy. I've been thinking about complex prime numbers on and off for the past two years. I wanted to explore this without looking it up first. I figured out about 80% of what you say in this video, but I still had a few snags in building my set of primes. Anyway, a few minutes ago I decided to see if anyone had made a video about this, and there it is. And I learned some stuff too! Thank you so much!

[artis.magnae]

Nice video, I like the reasonable pace compared to most other math videos nowadays.

[benjfr]

Great work. I love how it goes back the basics of arithmetics to give definitions.

[graf_paper]

Seems like primes that are 3 (mod 4) are gaussian amd primes that are 1 (mod 4 are not) for the reason that primes 3 room 4 cannot be expressed as the sum of 2 squares.

[emilyrln]

Excellently-structured video for facilitating comprehension! You've earned my sub and I'm off to check out those Eisenstein primes now 😊

[stevemcwin]

This is a really interesting topic and the video is well made. I hope to see more of your content!

[dcterr]

Excellent video! I have a PhD in algebraic number theory from UC Berkeley, so I'm well-versed in everything you discuss in this video, Gaussian primes in particular, but I must say that you give a very good, intuitive description of them! I look forward to your next video on Eisenstein primes and I hope you make even more! (You might want to go into general number fields, UFDs, and perhaps even prime ideals and class numbers, but I don't know how advanced you want to get!) Meanwhile I'll have to check out your app!

[trummler]

„Here is cotton candy spirals“
caught me highly off-guard

[jennycotan]

The Gaussian prime plot looks... aesthetically amusing.
Gonna see if someone make it a tile pattern...

[santerisatama]

Word "unit" is a mistranslation of Euclid's term 'monas' (from which Leibnitz 'monadology' originates from). The word actually means 'unique', meaning the distinctive mutual property between coprimes and their lesser cousins the "solitary primes". The fundamental elements of Euclid's number theory are coprime fractions, from which also the lemma of "solitary" primes and their unique gcd multiplications are derived from.

Thanks, this presentation was really helpful for me, giving also some basic intuition of what Gaussian rationals could mean, when we replace the mistranslation "unit" with each of the unique coprimes as the uniques, instead of limiting our perspective only to "natural numbers/integers" and filtering out mathematics with that sieve.

I'd love to hear your perspective also on Gaussian rationals.

[Dravignor]

I can't believe I'm now learning about rings from the guy that used to solve twisty puzzles blindfolded a couple of years prior

[FunkyThousandMiles]

This what i need after a day at the office, thank you!!!

[themathhatter]

It's also worth noting; if a+bi is a Gaussian Prime, so is a-bi. You can combine this with the rules for units multiplication to see you only need to check Gaussian integers of the form 0<=b<a. This creates a "wedge" between the lines y=0 and y=x, which can be thought of as mirrors, reflecting to all the other primes.

There is a similar symmetry with the Eisenstein Primes, but I don't know how to describe it mathematically; visually, it's equivalent to flipping the primes across the x-axis. If we take w to be the cube root of unity in the positive y-direction, then we only need to check the Eisenstein integers of the form a-bw, where 0<=b<a. This creates a 30 degree "wedge" with one edge as the positive x-axis and the wedge extending into the negative y direction, and the same mirror trick works. I just don't know the formula while staying in the Eisenstein integers.

[graf_paper]

Great color scheme for this video. Chefs kiss!

[SirLightfire]

6:17, that graph looks highly symmetric. I wonder if this relates to the Reimann Hypothesis (via Fourier transforms)

[johnchessant]

The norm map N : Z[i] -> Z given by N(a+bi) = (a+bi)(a-bi) = a^2+b^2 is multiplicative, so N(x) being prime in Z guarantees x is irreducible in Z[i]. The converse is not true since N(x) could be composite but none of its proper divisors are the norm of any element of Z[i]. Since the sum of two squares can never be 3 (mod 4), any 3 (mod 4) prime p is irreducible in Z[i].