Some 'Prime Numbers' Are Not Always Prime

In one fundamental corner of mathematics, some prime numbers stop being prime.

0:00 -The Typical View of Primes
1:37 - Two-Dimensional Numbers
3:57 - Multiplying Special Conjugates
4:46 - Hidden Factorizations
6:42 - Modular Square Patterns
8:54 - Which Primes Are Which Type
9:55 - Conclusion

Clarifications:
- Around 6-7 minutes in, I say we are allowing all complex numbers as possibilities. I meant just the Gaussian integers of those (similar to how typically only integers are considered as factors).
- Some people commented that they thought the Eisenstein integers (an alternate triangular-grid-like way of making "integers" on the complex plane, as opposed to the Gaussian integers which are square-grid-like)... Although I do think the Gaussian integers can be seen as more fundamental in some ways (not all ways) I do understand your points. Perhaps in the future I'll return to this topic in an episode to do a deeper comparison involving both types.

Catch up on the earlier episodes of Grade -3 with this chronological playlist:
(the "playlists" section of this channel also has playlists for Grade -1 and Grade -2).

Bonus channel:

This episode of Combo Class was created, directed and edited by me (Domotro) and I also made all of the music in it. It was mostly filmed by Evan Clark, with some clips filmed by Carlo Trappenberg, and some filmed by me.

Mathematical topics in this video include: prime numbers, factor pairs, integers, real numbers, imaginary numbers, the complex plane, complex conjugates, multiplying complex numbers and getting real numbers, Gaussian integers, Gaussian primes, modular arithmetic, quadratic residues, square numbers in mod 4, patterns with which numbers can be expressed as the sum of two squares, surprising hidden factorizations of prime numbers, and other interesting mathematical subtopics and fun facts.

Disclaimer: Do not copy any potentially dangerous actions you may see in these videos, such as any actions related to fire, glass, or tools.

[ComboClass]

In one fundamental corner of mathematics, some prime numbers stop being prime.

0:00 -The Typical View of Primes
1:37 - Two-Dimensional Numbers
3:57 - Multiplying Special Conjugates
4:46 - Hidden Factorizations
6:42 - Modular Square Patterns
8:54 - Which Primes Are Which Type
9:55 - Conclusion


(See video description for more links and info)

[onymou]

it's always so comforting that people still care about math in the post-apocalypse and this guy is somehow able to keep his time machine functioning

[joje]

I think this video triggered a fugue state and I now feel very, very confused. The fast clips, the slow but shouty speech, the ominous half-music in the background, the 90s kids-show aesthetic... Why is there a fire? Why is the man smiling at me and explaining math during the fire? Where is this? During what season is it even filmed? Why is it suddenly 5.44 am? Who am I? Where am I?

[MelodiCat]

I was expecting clickbait, but instead the extension talked about in this video (primes but with complex numbers) was so natural that I feel satisfied. Wonderful!

[christopherwellman]

A girl at the bar gave me an imaginary number.

[Papa_Ceasar]

What I like most about imaginary numbers is that their translation into my language is basically "delusional numbers"

[imjay]

I think it's crazy that the logic and knowledge he speaks is so fire that it physically combusts and has to be extinguished in these videos

[vaibhavmishra]

"Every square number is a multiple of 4 or 1 more than a multiple of 4" This blew my mind 🤯

[timothyallen]

The video could have mentioned that (1) multiplying a complex number by its.conjugate gives the square of its length, (2) primes that are not gaussian primes correspond to the radii of certain circles in the complex plane whose circumferences pass through complex integer points, and (3) those points also correspond to pythagorean triples: a 3-4-5 triangle, 5-12-13 triangle, etc, etc. Of course, this video -- nicely done -- has to draw the line somewhere, but this would have let it link together several other generally familiar concepts as well.

[JimWitschey]

Your cat was very helpful with those slides!

[sebmata]

Ah I love the smell of number theory in the morning

[smwiasph]

Somehow unrelated but reminds me of a line from the ‘Animal farm’, “All men are equal, but some are more equal than others”!!

[rosiefay]

Seeing as you're changing the rules, it bears stating that your definition at 0:34 is of "irreducible", not "prime".

[hkayakh]

man your cats really like gausian primes

[XMgamePlays]

5:00 Brilliant!

For me, it also proves why a number like 13 and 17 pops up so often in several (RTS) games. Not for it being the "unlucky" number or the sum of 30. But for the 4 and 9 or 1 and 16 to be present (in the shadows) all the time.

[personperson]

Yay algebraic number theory. One of my favorite things in math is solving integer/diophantine equations by generalizing them to a different number system such as the gaussians, eisensteins, etc. to make them solvable. Fermat's last theorem up to exponent 22 can be solved this way, and some higher exponents too if you use ideal factorization.

[typical]

This man made me slightly afraid and also amazed I'm sure he's a mad man i will subscribe

[Henrik-jlgq]

But in this extension, there isn't any unambigous prime factorization any more, right? You can factorize 5 as (2+i)((2-i) or as (1+2i)(1-2i), all of which are prime.

[michaelallen]

Not just any complex factors but complex factors with integer coefficients.

[AKFunGamesEntertainment]

7:07 how did bro just spawn on a tree?