The great thing about number theory is that it is SO HARD to prove anything, and most die trying. But at its most basic, it's accessible to someone with very little knowledge past arithmetic. What fun exploration this could be made for kids. Something to think about, thank you!!
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9:44 The only possible remainders are actually 1, 3, 7 and 9, because since we’re dealing with prime numbers, suppose p = any prime number, p/10 will always give an uneven remainder inferior to ten, and the reason we don’t get 5 (the only missing uneven number) is because all numbers ending with 5 are multiple of 5. Therefore, we can only get the remainders 2 and 5 at the beginning (2/10 = remainder 2, 5/10 = remainder 5)
While exploring my interest in number theory, I was trying to think about what Fibonacci numbers would like like under mods. I saw the odd repeating patterns and decided to do some research, finding pisano sequences and then later stumbling upon this video. This was very insightful and I have learned a lot from this, one of the best math videos I've ever seen. Nerding out so hard to this one
The pedagogical outlook expressed in the introduction actually hooked me. Multiple/alternative modalities, recognition of many possible representations, lovely! Math content that treats students as curious humans rather than the "show your work" automata i recall from my school days.
Hi Jacob, first of all great video hats off !! A personal opinion at 08:05 the way the patterns alternate and repeat each second time looks like fractals, same pattern from small to big picture
One idea for going 3-d: Pick two moduli, lets call them u and v. Divide the sphere into u lines of longitude and v lines of latitude. Pick two sequences, x and y to form a sequence of ordered pairs. for each ordered pair, take x mod u and y mod v. for adjacent ordered pairs, draw a line connect the point on the sphere for the first pair with the point for the second pair. E.g. if our moduli are 4 and 5, we might have longitudes of 0, 90, 180, and 270 degreeswith latitudes of 0, 45, 90, -45, and -90 degrees(or maybe +/-40 and +/-80 to avoid values that are 0 or 4 mod 5 collapsing the mod on the sequeence generating longitudes collapsing. if the latitude sequeence has a modular period of 1(e.g. all the remaindrs the same), then the pattern degenerates to the 2-d pattern for the sequence used to determine longitude. Another idea using 2 sequences is to have a rectangle U-1*v-1 in size where you map out the remainder ordered pairs and connect the dots, which naturally extends to a rectangular box and three squences. Another 3-d idea is to use a torus and two sequences where on modulus determinesrotation around the hole and the other the rotation around the band. And of course the primes mod 10 only produce 1, 3, 7, and 9 as reminders, an even remainder would mean the number is even and 2 is the only even prime and a remainder of 5 would mean the number is divisible by 5 and 5 is the only prime multiple of 5... I suspect primes mod something else might be less restricted, though you would only ever get a single remainder 0 and only if the modulus is prime... and I'm pretty sure a prime mod an even can never be even.
Wait, did I it right that you study math? :( Try drawing squares around 0...n^2-1 .. And then just forget about the spiral/image and work with bloody quadratic polynomials.
I'm glad I came across this. Are you able to do me a small favour? Can you look at OEIS A068869 "smallest k such that n! + k is a square" and explain why for n = 4,5,6,7,8,9,10,11,13,14,15 & 16, k itself is also a square? And while you're there, can you explain why for when n = 12, k is not a square? I think it's pretty curious and would love to know what's going on there. I stumbled upon this stuff back in university when I was studying up on the fact that sqrt ((7!) +1) = 71 and some other religious ideas. I think about it all the time, and would really appreciate an explanation as to why it works out the way it does. Thanks in advance!
Interesting video. I have been discovering the beauty of math and how it truly weaves its way thought all of creation. I can already see application in the arts and will be applying this to some musical ideas that I have been exploring. Contrary to what I believed my whole youth, I am finding math to be quite beautiful, useful and not as scary as I thought.
The pattern of Recaman sequence seems match one Crop Circle, also some other patterns that with symmetry quality seems match some of Crop circles too. I actually think that our math might have taken numbers on a difficult side way. Every nature constant is irrational number, and seems to me that only irrational number could lead to constant circulation. They are the base of nature. And I think maybe geometry is an more fit way to work on nature and produce machines and develop math. Maybe the Crop Circle is the ET trying to point out the true way of math to us.
take the sequences and put into a matrix and you will create an infinity generator Of any wisual imange ever observered or to be onserved it a self generating record of its existance
TAKING A SYSTEM THAT CREATES A RECTANGLE , SPIN IT WITHIN CIRCLE.. QUESTION.. DOES IT REPEAT/OVERLAP OR NOT REPEAT AND IF SO, WHAT IS THE REMAINDER WITH EACH SPIN (360') OR CUMULATIVE REMAINERS OR AGAIN, DOES IT REPEAT ?
Very interesting video. I am doing research for an MS in Math regarding 2 dimensional numbers. A 2D number can be defined for every (x,y) coordinate, and then turned into a 4 digit Fibonacci sequence (ie. ((1,1), 2,3) for the first 2D number). It would be interesting to look into this further as connected to the wave pattern characteristics of 2D numbers. Each of these numbers define a Pythagorean triangle and other integer based figures.
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I may be looking too much in to this, but the modulus chart with all the dots, before adding the straight lines, reminds me of the double slit experiment. Anyone else get that feeling?
There is a reason why primes end up like that. Once I wrote prime number generator, that was no sieve of erathles, but using modulo, and it generated every prime.
@@jacobyatsko I'll make video about it when I find my time. I don't remember exactly how it was, but it was using differences between primes, stack and modulo operator and when there was mod = 0 stack was reseted.
