41 and more Ulam's Spiral - Numberphile
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- Опубликовано: 20 ноя 2024
- More on prime numbers and Ulam's Spiral - this time focusing on 41 and Arthur C. Clarke.
More links & stuff in full description below ↓↓↓
This video features Dr James Clewett. More Clewett videos at: bit.ly/JamesCle...
See our other Ulam Spiral video at: • Prime Spirals - Number...
And more to come soon...
The book discussed is The Garden of Rama.
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Наука
7:55 "it was Euler who first did this .."
should've seen that one coming. It's always Euler
Or Gauss
or the Greeks
Or Newton!
But why would u not hate him ofc...
Euler was such a boss. We're still printing his works. Instead of slowing down after going blind, he sped up. He had a team of scribes translate his math from his brain pan. You should you totally do an episode on Euler
But every other video would then be a Corollary of this one!
I found if you put 23 at the center of an Ulam-like spiral on a hexagonal lattice it generates 22 primes, just as 41 on a square lattice generates 40.
"Keep going, as long as you feel... inclined"
PUN! Because the lines are diagonal... hahaha.
Sadigziggi get out.
2^7 likes.
Nice.
I think you have a beautiful punnable soul..
That pun was di-agony to read through...I'm not sure if that pun is as bad or worse than yours...Cheers either way!
@@alansmithee419 3^3 * 5 when I type this.
the patterns are just a direct result of the initial pattern used. What is more interesting is the gaps, and the apparent 'ghost axis' in the center, which is common to both patterns
I think 41 is the biggest number where f=N^2+N+X gives you series of primes is because for X=42 you get the answer to life the universe and everything.
seriously, 90% of the time these things are just coincidences...
it's just everyone looks for them now!!!
hi numberphile from nine years ago
I'm slightly uncomfortable because @7:40 that marker is dangerously close to marking the book.
kurtilk it's ok, if it's Gentry Lee's book, it's worth using it for kindling or trashing.
LOL, that's the most random comment I've read in a while.
@@blackjack2150 Is it actually random though?
I was also worried! :D
@@00bean00 why?
ha ha - I didn't say WHEN I would make his dream come true!
Well think of it this way:
1/3 written in decimal is 0.3 repeating
2/3 written in decimal is 0.6 repeating
3/3 written in decimal is 0.9 repeating, or just 1
You crossed the 7! Finally someone who writes the number 7 properly! You are the best mathematician on Numberphile for me! Well, the one with the best 7 at least.
yes we're on the same page. The number two can be 2 or 10 in binary but its the same number. In base 6, the number seven would be written as 11. Base systems allow us to look at numbers from different perspectives and use them in different ways. I can count up to 1023 using binary on my fingers.
Rama series REPRESENT!
I have a whole new level of respect for James, the Rama books are my favourite thing I've ever read.
I don't think I got anything from this video other than the fact that you are a fine doctor
the prime numbers in base 6 all end in either 5 or 1. You could color them differently and see what happens. Nothing might happen or you might see something really cool.
That. Is. AMAZING.
I've seen a documentary about project Orion once. It was fascinating how ambitious the project was. A video about it would certainly fit into one of your channels.
Complete Respect... The Rama Books I can and have re-read over and over... I'd love a movie but I know It'd disappoint.
James's explanations are actually very informative and understandable, always liked videos with him, waiting to see more)
Be careful of the shape worship out there, but it is interesting to note that if you generate a hexagonal spiral starting with "1", you get a diagonal that goes, 1, 7, 19, 37, 61. If you look carefully, you can pull out a "+1" and some pattern with the number "6" or "2 x 3". This is the pattern of 1, 3, 6, 10... etc.. so it has the form of f(n) = n(n + 1)/2.
And, so g(n) = 6*f(n) + 1 = 3((n + 1)n) + 1 (n = 0 when g(n) = 1).
I don't think this diagonal is all primes though, just lots of them!
Whats really interesting is that 41-41 is a change of 0, 41-43 is a change of 2, 43-47 is a change of 4, 47-53 is a change of 6, 53-61 is a change of 8, etc... so many patterns :D
Numberphile vids should have a slot on TV.
Around 7pm, on week nights - y'know, when most people watch
The show could be called...
I met Stan Ulam when he was a visiting professor at UC Davis in the late 1970s. Fascinating guy. Got him to sign my copy of his autobiography: Adventures of a Mathematician.
I also see horizontal and vertical lines in there too, only it's every other number since there's no even primes (besides 2, of course).
The Rama series of books are bloody brilliant. Love em.
the number 41 in the Rama series first appears in Rama II. The prime series is General Michael O'Toole's password and it is explained in different (less formulaic) terms when General O'Toole's thoughts are narrated in that book.
He's right though. The series only works until 40, so he does indeed write down N=40, and the solution is indeed 1601. N=41 gives 1681 though, indeed, but that's equal to 41^2 (since the -N+41 would cancel each other out) and that's not a prime.
So, he's not making a mistake, he's just writing down N=40 as the last of the series of primes
Weird, this series is "very" prime. First, the series seems to NEVER have a prime factor smaller than 41. Checked up to f(100000). Could that be proven?
Another weird thing, even after the nonprimes start at f(41) there are still lots of primes and even the nonprimes are "almost" prime. By "almost" prime I mean they have few prime factors. (And again, they're all 41 or bigger.)
The first nonprime is f(41) = 1681 = 41*41. Another way to describe that point is the first member with 2 prime factors. All of the nonprimes after that also have only two prime factors ... all the way up to f(420).
The first member of the series with 3 prime factors is f(421) = 176861 = 47 * 53 * 71
The first with 4 factors f(1722) = 2963603 = 41 * 41 * 41 * 43
With 5 factors f(14145) = 200066921 = 41 * 47 * 47 * 47 * 47
+Bob Stein Only one of the outputs of that series has three *unique* prime factors, the 47*53*71? That number has just as many divisors as f(1722).
For a prime p, simply evaluate a univariate polynomial over p
consecutive outputs and it will cycle through all possible remainders
(mod p). If any of its values are divisible by p, 1 or more multiples of p will show up in those p consecutive numbers.
F(N) is never divisible by any prime less than 41.
For example, let's check it for 5. One can see that the difference F(N+5)-F(N) is a multiple of 5, therefore if there is a F(N) multilpe of 5, ve can add or subtract 5 into N until we got at 0,1,2,3 or 4. But F(0), F(1),... are all primes and bigger than 5 so they are not multiples of 5. So no F is multiple of 5.
(For example if F(235) is multiple of 5, F(230) is also, then F(225),... then F(0)=41 would also be, contradiction)
Same reasoning works for 2,3,... 37.
+ Bob Stein
f(n) = n² - n + 41 = n(n - 1) + 41
As Andrey is hinting at, you can do a modular arithmetic analysis of it, using successive primes as moduli - 2, 3, 5, 7, etc. You will find that 41 is the smallest prime that can divide f(n).
Interestingly, as you go through n-values, it begins to look as though the only primes that will ever divide f(n), are primes that appear among f(n) values, until you get to
f(82) = 6683 = 41·163
f(83) = 6847 = 41·167
and a number of later values. [163 and 167 are not among f(n) values. They fall between f(11) and f(12).]
It's depressing that most primes which may divide f(n) are never generated by it, but can only be
discovered through factorization or modular cycles. My favorite simple equation is 2n^2 - 199.
OH YES A NEW NUMBERPHILE VID!!
Oh wait watched this one after the last vid and it was just hidden until now...
The function was n2 - n + 41, not n2 - n + 1. Therefore, f(40) = 1601.
You are right about every second number not being prime. Nontheless prime numbers appear to follow some kind of pattern in those Ulam's Spirals. Otherwise the dots marking prime numbers would be distributed randomly (maybe even Gaussian) in the graph. The thing is: They are not, and that is remarkable, considering the importance of calculating prime numbers in everyday's life.
What do you mean "0 and 41 again"?
The two coloumns are N and the formula...
"1 and 41" in the video is that N=1 and the result is 41. (again).
Well if you think about it , it's not that outstanding. It is easy to see that :
f(N+1)=f(N)+2N. Now use induction. Imagine that you walk along the spiral. Given that you are in the diagonal in the N-th step ,you walk 2N numbers as the sequence relation says. But walking 2N steps takes you to the diagonal again, because tou walk N horizontally and Ν vertically. Hence by induction you always stay in the diagonal.
I thought of this, too. Try just writing the odd numbers. You will still see more than random numbers. But of course. It could all just be random. But Math is too beautiful for that^
there is - that is my sixtysymbols channel!
I think people perceive 7 as random because when they are in primary school they see usually the following sequence:
07
14
21
28
35
42
49
56
63
70
The numbers in the end of each member of this sequence are 7, 4, 1, 8, 5, 2 9, 6, 3, 0 (all numbers from 0 to 9).
I think this is why people THINK 7 is very random: because there seems (wrongly) "there is no pattern" in their multiples, so when multiplying by 7 we usually HAVE to memorize the multiplication table for that number because we use base 10 and it seems that "no easy pattern exists" - we think that because we probably always thought about it when kids.
If we were taught modular arithmetic from a very young age, this probably would be less prevalent.
7 would not be picked in base 7. I'm willing to bet on it.
I had memorized in the past multiplication tables until 20², because when doing math for exercises with pencil and paper before going to the University (in which numbers per se are not THAT much popular), numbers from such table were particularly popular.
I had also memorized all square numbers till 30² as well (all squares until 1000 are somewhat popular as well).
Needless to say, after those tests this proved to be a big waste of time because in fact I do not do such math in daily basis, as I'm not an accountant or a person that crunches raw numbers daily.
Ha, in a prime base such as 7 (but larger than binary), any non-zero digit would have the maximum period length.
Sadly, 2 is the largest base whose only totative is 1, though the only totatives in base 6 are 1 and 6 - 1.
It's because of the n^2 term, you're adding the same numbers no matter what. I.e. 0, 1, 3, 5, 7, 9, etc. These numbers make up the sum of the next two side lengths of that spiral at any given point.
As for the primes, in a subject with infinite possible functions, some of them are bound to have extraordinary properties. Not that that makes them any less fun.
I cannot get over how much this story sounds like something Vi Hart would tell.
41 is my favorite and lucky number thank's a lot for those great videos, it helps me to learn english too. Great Job !
How could you have forgotten this video????
I'm 41 and am more ill than I've ever been. So far, my life since turning 41 has been worse than ever in my life between health issues and losing loved ones.
6:53 upper right hand corner is a heart
A is also the 3rd letter from the last in both illuminati and Brady.
"Happy nerdiversary, honey! You're a prime number of years old. One could even say, you are in your prime! "
"And ... it's not just *any* prime - it's the *Lucky Thirteenth Prime*!"
I'm going to have to remember that joke 4 years from now for when my age turns prime again.
Next year will be great because I will be 25 = 5^2 years old! 5 is the 2nd odd prime number and the first prime number is 2 The difference between the 2nd and 3rd prime numbers (3 and 5) = 2
Intresting indeed! Well I wondered it a while and conclusion is this:
Few things to remember: (-1)²=1 and -(-1)=1. We got 41 on the both sides so we can drop it out and get: n² + n = (n+1)² - (n+1). Luckily I found easy graphical way to prove that (I try to say it understandably) : Think 9 dots (3x3) at the square and add one side +3 more dots, you get rectangle with 4x3=12 dots. Now this should ecual to 4x4 square minus one side 4. And this is 4x3=12!
Or just algebra: ->n²+n=n²+2n+1 -n-1 -> 0=0
Sounds like how I talk when I'm trying to explain things to people; and yet, people tell me that I elucidate things very easily.
41 is 1 more than 40 and 1 less than 42 (which is the meaning of life) so 41 is 1 less than the most important number so is quite important.
Could these prime spirals be linked to why we see the Fibonacci spiral in nature?
By the way this is an invitation to see an artist theory on the physics of ‘time’ as a physical process!
right so the formula he's using is f=n^2-n+41. If you use negative numbers then the n^2 will remain positive, but n will be negative, so it will be like -(-n) = +n, so instead of subtracting n from the answer you add it, so thats why the answers are bigger. At least that's why I think it does that
but you could play with base dependent primes like palindromic primes who only work in specific base systems - palindromic primes would be that pretty since there are not that much in the common sequences listed at OEIS
because you came up with different values for lim x^2 while approaching from 1 and -1
my mind is blown. interesting stuff here
Vi Hart has a lovely video on Ulam's Spiral.
I think that it would be kool to design a city based upon Ulam's prime number spiral where every marked prime denotes where a building is to be placed.
7:20; I did the circle with squares on the right and lined up the twin primes on the left. They went 3, 6's, 12's, 24's. ... Orbited with the most primes in the circle.
0^2 is just 0*0 which equals 0. I think you must have misheard something in the video because he doesn't claim that it's undefined.
7:18 The plans to the Death Star have been in front of us this whole time
I think 41 also represents the finiteness of our knowledge: however close we get to the Ultimate Answer, we'll never get all the way to 42. Not that we're really very close.
I noticed something I find interesting. For this equation, given N and f(N), f(N+1) = f(N)+2N. In other words, to find the next f, take the last N, double it, and add it to the last f.
If you're willing to toss out 0 from the list of N, you could simplify the initial equation by calling it "N * (N-1) + 41."
IT ALL ADDS UP! ...LITERALY!
The Archimedean spiral reminds me of the Death Star.
I don't know if this has been mentioned before but I found this spiral pretty much by myself yesterday using excel, the "gaps" you get kinda bummed me out but I came to an explanation! I'd call them Half-Primes, as they are only divisible by primes! 9 for example is only divisible through 1,3 and itself. the same goes for the rest of the gaps try it out. Wish I could publish this somehow :D
That's cool. What you are talking about might be semi primes btw.
It's also a bummer when you rediscover something that has already been discovered many many times before but unfortunately 90% of your discoveries would have already been discovered, it's just you haven't found another guy who discovered it yet, be it because they did not publish their results or you are not skilled enough at googling concepts which you name yourself.
@@Zenovarse ill have to check that out, thanks!
@uuurgaah zero is sometimes called nought ( rather than not, which has a different meaning,) sometimes expressed as naught, or pronounced as nowt. These may be a morphological shortening of nothing.
Can you film the mathematical proof that 41 is indeed the biggest number that works in this equation?
Keep up your fantastic work Brady!
When he read the excert the only sound I heard was WHOOSH as it was going over my head
I did a spiral today and found that the all of the squares of the integers fall on a diagonal. All of the even integer squares go up on a diagonal to the top left, and all of the odd integers squares on a diagonal to the bottom right. Only the squares are on the diagonals, NO Gaps! And Every square!!
it's not so amazing that they are diagonal. What's amazing is that there are lines showing up at all.
It was never a problem numberphile even has a video about it
I feel so special seeing this before it showed up in the sub box :D
It should be expected that the primes come in diagonals, as every other number is going to be even, and therefore, 1/2 of the numbers are out of the question.
They should go through each number all the way to 100, a bit like how they talk about each element in periodicvideos. That would be cool, Information about my favorite numbers and the history of them, and how they relate to each other and their importance, etc.
I would legitimately watch that...
I think you just won the internet... Congratulations!
it's an en encoding, so it depends on the Unicode standard and the number of chars in the private area you want it to have
In the maths department I studied in, all the pure maths people were a bit weird, so yeah I can easily see them speaking like that in casual conversation!
what do you mean? numbers are numbers... different base systems express it differently but they are all the same sequence. which is what matters (sequence) in the spiral. if you look at this spiral and see 41 in it, it is the 41st number from the spiral. if you wrote 101001 BIN instead, it would still be the 41st number...
"No one talks like that" uhh I don't about that… sometimes my calculus professor talked like that. He's *super* into discrete math.
I know they turned a bit "soap-opera in space", but they just got more and more interesting as they went on. The Octospiders were just awesome.
Thing is, numbers an extended knight's move (1 vertically or horizontally, 3 perpendicular to that - a camel's move if you're into fairy chess) are also all odd, but I'm not seeing much in the way of 22.5 degree diagonal lines on that 200x200 plot. Just 45s.
41
4-1=3
Half Life 3 Confirmed!
IndieGO Your comment was posted three years ago.
Half Life 3 confirmed?
yellowcrash10 Your comment was posted three weeks ago.
Half Life 3 confirmed?
Mads Marquart your comment was posted 21 hours ago. 2+1=3.
Half Life 3 confirmed?
VCaIÜS your comment was posted 3 days ago. 987 * 83=3. Half life 4 confirmed?
The Pybro Your name is "The Pybro"
Pybro sounds like Piebro
Pie is delicious, and bro rhymes with Three
Half Life 3 Confirmed
I always thought that Douglas Adams got it wrong when he chose 42. He should have used a prime.
on the way
8:45 "...you don't need a patent lawyer, you need a weapons system."
How about a patent lawyer WITH a weapons system?
Naked Snake if he went through law school
The Arthur C Clarke (non-fiction) book Ascent to Orbit has lots of really great maths stuff in it.
found it just yesterday after watching your spiral... Now I feel cool :)
Sorry, obscure references. 42 being the ultimate answer in Hitchikers guide to the galaxy, Douglas adams. Einstein was quoted saying 'god does not play dice with the universe' in response to Shrodinger and Bohrs uncertainty principle. There are 42 dots on a pair of dice.
Yeah, just like language only exists if people exist. The stuff language describes is pretty real, people or not, but the words we know, those things are just a way of describing reality made and used by us.
Exactly, it's like saying "o" and "nill" as well.
James is easily my favourite of the Nottingham scientists :P
We've proven that there is an infinite number of primes separated by a maximum of 246.
Then a saw prime numbers interestingly organized on the sides of an hexagonal pencil in another video.
My instinct then told me to divide 246 by 6 which equals to 41... Then i see this video and i'm wondering if this is somewhat related...
I'm tempted to say that this is a coincidence, since I believe that number has gone down from 246 to something like 12 now.
@@MMrandomdude12 Can't primes be separated by larger numbers than 246? If you took the LCM of say 248, then a sequence of numbers above that ranging from LCM of 248 + 2 to LCM of 248 + 248 will all be composite.
@@quevda Primes gaps can be arbitrarily large if you search long enough like you've said, but the proof is that the are infinitely many prime gaps smaller than 246
While reading your comment, the audio of the video synchronized perfectly with my "inner voice" :D
I'm not sure about this but because you can write the following equation
0^2 = (0^1)/(0^1) which is undefined
so yeah i don't think 0 should be able to take any power because you do the same thing i did above with any power of 0 even 0 to the power 1
i actually want Brady to make a video explaining this issue.
Well, it is not quite true that _nobody_ talks like that. One mathematician told me, "I don't pick up hitchhikers because I refuse to integrate their weight over distance in fuel."
The story continues as N increases over 41. The prime series continues, but there are also composites now made up of series of new primes, and then ever more series, all interwoven together.
Next do one on Primes in Pascal's Triangle.
Could this make progress in solving the Riemann hypothesis, and could you do a video on the Riemann hypothesis since you seem to love prime number theory.
From 5:55 and on I think he made a mistake; sure the pattern emerges no matter where you start the spiral, but he seemed to imply that this particular diagonal also showed up. When you start the spiral from 1, this sequence does not line up at all.
then how did you comment?
The one minute there was right out of my calculus script... That's what I think all the time :P Nobody talks like that!
Douglas Adams chose it because he felt like there was nothing special about the number. It may have some properties, but isn't that true of most numbers?