Ah yes, Euler's totient theorem. We usually use this to calculate large modulos, say 7^1989 (mod 100). Let phi of n be Euler's totient function. If n is a positive integer, phi of n is the number of integers in the range 1, 2, 3 ... n which are relatively prime to n. If a is an integer and m is a positive integer relatively prime to a, then a^phi of m is equivalent to 1 (mod m). How is this useful to calculate modulo by hand? Well, another way we can use the phi function is, if you have the prime factorization of any m taken a (mod m), written as x^k * y^l ... using Carmiachel's theorem we can state that phi of m is equal to LCM( x^k - x^(k-1), y^l - y^(l-1)...). This is very helpful, as we already know a^phi of m is equivalent to 1 (mod m). Again we can go back to the previous example, where we want to solve 7^1989 (mod 100). We take the prime factorization of 100, which is 2^2 * 5^2, and place them into our function, so we get lcm(4-2, 25-5), which is lcm(2, 20), obviously 20. Thus, 7^20 is equivalent to 1 mod 100. This is very helpful, as we can go back into our previous equation, simplify it, and solve easily. Man I never expected a 3blue1brown vid on this channel but it was very good!
Very cool indeed. I'd like to point out that at 2:32 it may not be so obvious that each rotation of a certain coloring will produce a different coloring. Lets assume that there is a coloring that, when rotated by r/p*360 r<p, produces the same coloring as the one we started with. This means that there is a pattern in the ring which repeats itself every r beads. Thus, the p beads can be subdivides into some number of groups of r. In other words, r divides p and since p is prime, r has to be 1. So the only example of such a coloring is one where each bead is of the same colour, ie one with all blue beads. All other colorings will belong to a family of p different colorings generated through rotation. QED
@everyone pls read the whole thing:With more proof here is a vid I made that 3x29$’.iwkdiremvergniueartveRut ntVUI?nuivggvrinhevrgnuertvnvn5;lmlkcdcdcidejedjdjdeijeeedmiedd. -chromahunter frfrfrfrfrfrrffrrfrfrfrfrfrfrfrffrfrrfr ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-XJey9uEGtHg.htmlsi=INb2uAJeCf8qnQHc a vid that i totally made about the sowwyfututut proven in to ways color and the misadventureas method of color theory explains at 2:32 the color wheel and how the color format of the wheel regerlates to the sanx of the color emotions of how 12 buckle my shoe is a sowwyfututut crèche polestar of the math theory of how the world can trade the wheel of shapes with godmas of the Cesar theory which is 47755x-a=f298 a=3. f=cm2 which is a small equation and if you don’t know it, your school is very very dum (joke) the rest of the sowwyfututut theory will be proven in the vid above or here ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-XJey9uEGtHg.htmlsi=INb2uAJeCf8qnQHc of the color sowwyfututut fut hope shake theory that’s all of the theory sowwyfututut goodbye!
This happened to me on the Login Screen the other day for a moment. 🙂 I noticed that the background is a collection of multiple map background sections repeated. One of those sections went black for me, it looks like for you all of them went black for a bit lol #FlorrGreen
