Paterson Primes (with 3Blue1Brown) - Numberphile
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- Опубликовано: 18 сен 2024
- Grant Sanderson (from 3Blue1Brown) discusses the briefly brilliant discovery of Paterson Primes. More links & stuff in full description below ↓↓↓
See all three videos in this series - Grant's Prime Pattern Trilogy: bit.ly/PrimePa...
Grant's own false pattern video: • Researchers thought th...
Grant's channel is 3Blue1Brown: / 3blue1brown
More Grant on Numberphile: bit.ly/Grant_N...
Grant on the Numberphile Podcast: • The Hope Diamond (with...
With thanks to Michael Colognori for computing the big prime list.
The Paterson Primes on OEIS: oeis.org/A065722
Correction at 1:20 - 5 (not 17) is 11 in base 4, - still prime of course.
And with thanks to Patrick Paterson - of course.
Numberphile is supported by the Simons Laufer Mathematical Sciences Institute (formerly MSRI): bit.ly/MSRINumb...
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Special thanks to our friend Jeff for the accommodation and filming space.
Part 2 is at: ruclips.net/video/NsjsLwYRW8o/видео.html --- And Grant's own false pattern video at: ruclips.net/video/851U557j6HE/видео.html
Sir could you please tell me how and from where I can learn to code a program to check any conjecture or check any pattern in my laptop just like you...∞
Fun fact: The factor 11 does not appear in your list, the first one is for prime: 9011 which is 2030303 = 11 * 379 * 487 in base 4.
The factor 101 does not appear until prime 16992067 which is 1000310131003 = 79 * 101 * 125367857 in base 4.
Part 3 when? 🥹
.. can you just release the unedited video of part 3..?
Patrick Paterson and his patented primes were a Parker precursor. He gave it a go, and got pretty close.
The scam bot got one thing right, that @elflfa does deserve congratulations for this comment. 😆 👍
"Parker" and "Paterson" both start with "Pa." I conjecture that there is a connection between the two.
I read this in Parker's voice
So Matt even Parker Squared, making the concept of a Parker Square. Poor guy it never ends.
??.
I will never not be amazed by Grant's seemingly natural understanding of complex patterns in mathematics. And it helps that he is able to calmly and precisely explain it.
What seems natural on video likely took a lot of understanding off camera.
His intuition of derivative products and vice versa was a game changer for me.
That's same for every mathematician
@@pomelo9518 they both are same specie
Emphasis on calmly
Whenever I see the word "prime" or the name "3blue1brown" in a Numberphile video, I feel the urge to watch immediately, so I dropped everything for this one. The traffic behind me can wait until I'm done.
ha ha
criminally underrated comment
I'm a simple guy, I see 'prime', '3blue1brown' and 'Numberphile', I click :)
Bro I am waiting behind you 🙁
@@ahmedyawar31 lol
I love that as an aside Grant explained the rule for finding if a number is divisible by 3 or 9. I've been using that fact for almost two decades and had never thought to ask why it was true.
1) If you don't generate the hypothesis then you have no chance of getting a theorem.
2) When you test a hypothesis you will get a deeper understanding. Even while disproving it.
3) and it's fun.
Thumbs up to all concerned.
Indeed. Just the proof that the process generates numbers which are not multiples of 2, 3 or 5 is interesting enough in its own right!
Exactly, follow the null hypothesis to the end, you will learn, no matter what. That's what science is truly about.
??.
Makes me think of my 10 years old self, so proud of discovering that the hypothenuse of a 3 and 4 units sided right triangle is 5, and that it works for 6,8 and 10 too.
I remember being so proud of myself for finding out that it works for 30, 40, 50, as well as 300, 400, and 500, and 60, 80, 100 and 600, 800, 1000
Some questions that come to mind:
- Are there infinitely many "Paterson primes"? (I do think so but can't think of a straightforward way of proving it rn)
- How exactly does the ratio between "Paterson primes" and "non-Paterson primes" behave for larger and larger numbers?
- Is there a longest consecutive run of "Paterson primes"? So, could it theoretically be all "Paterson primes" after a certain number? If so, from what number on is that?
If not (which is probably more likely), what's the longest consecutive run of "Paterson primes" we know of?
This is some numerical data on the distribution: ("primes below n as input", "number of input primes", "number of output primes", "ratio")
2 0 0 0
4 2 2 1.00000000000000
8 4 4 1.00000000000000
16 6 6 1.00000000000000
32 11 10 0.909090909090909
64 18 15 0.833333333333333
128 31 24 0.774193548387097
256 54 38 0.703703703703704
512 97 58 0.597938144329897
1024 172 97 0.563953488372093
2048 309 157 0.508090614886731
4096 564 244 0.432624113475177
8192 1028 392 0.381322957198444
16384 1900 633 0.333157894736842
32768 3512 1049 0.298690205011390
65536 6542 1788 0.273310914093549
131072 12251 3048 0.248796016651702
262144 23000 5375 0.233695652173913
524288 43390 9506 0.219082737958055
1048576 82025 16920 0.206278573605608
2097152 155611 30351 0.195044052155696
4194304 295947 54939 0.185637968960658
8388608 564163 99811 0.176918727389070
16777216 1077871 182365 0.169190005111929
33554432 2063689 330601 0.160199041619159
67108864 3957809 601667 0.152020221289102
134217728 7603553 1102217 0.144960783465309
268435456 14630843 2035882 0.139150013433949
536870912 28192750 3763838 0.133503755398108
I tested all primes between 2 and 100000, and the ratio just seems to keep decreasing. It ended at about 0.3481377, but it doesn't seem like it has a reason to stop there.
I also started thinking similar questions! Commenting to follow this thread
I am testing up to a million now, and it has already dropped to about 0.299
@@want-diversecontent3887 Did you try plotting the ratio?
None of the Paterson composite numbers shown in the video are divisible by 11. For those wondering, the first one is 9,011 -> 2,030,303 = 11 × 379 × 487.
Also, no coincidence that it lasts that long: divisibility by 11 in base 10 can be checked by looking at the alternating sum of the digits. The same happens for divisibility by 5 in base 4. So if the alternating sum in base 10 is zero, then the starting number was divisible by 5. As an example, 231 in base 10 is an 11-fold since 2-3+1=0, in base 4 the number is 32+12+1=45, a 5-fold. So this Paterson method can only give an 11-fold if the alternating sum is an 11-fold, but non-zero. Which takes a while, if you can only use 0, 1, 2 and 3.
I was absolutely wondering :) I was also wondering if there's a largest Patterson prime, but I suppose no one knows that
@@jkid1134 I imagine it's very likely that there is no largest Patterson prime. My reasoning is that there's no largest prime, and of those infinitely many primes, some, in base 4, would probably result in a larger prime. Then again, i was surprised by how low a quantity of the first 1000 primes churned out a Patterson prime, so maybe it does continue dwindling.
@@jkid1134 I assume that there are infinitely many Paterson primes.
Thank you kind stranger :)
That ending (the first 1,000 primes checked) was therapeutic (although it almost felt like Patrick’s obituary)
This is great! I love seeing my favorite RUclipsrs entering each other's worlds. I did notice a typo (others probably did too): at 1:17, the graphic indicates that we are writing 17 in base 4, but the prime in question, as Grant just stated, was 5 (11 in base 4).
I noticed that error also and was about to comment.
I'm jealous of Grant for having had friends like that in high school, who could just talk about nerdy math stuff. That's the coolest kind of kid.
Oh hey, look at us breaking into the numberphile "prerelease vault"
Are you a time traveler?
@@saberxebeck patreon
@@eldhomarkose8330 I am in fact not a Patron, I got here via the link at the end of Grant's latest video.
@@fuuryuuSKK okay
After some thought, I've come up with an extension to Paterson Primes.
Consider a set of primes {p1,p2...pn} and a small base A. To find a larger base B such that, when you take a prime in base A and interpret it in base B, will not divide any of the primes in the set, B must be subject to the following conditions: if a prime from the set p is larger than A, B=k*p for some natural number k, or in general, A=B (mod p).
Let's do a small example. For the set {2,3,5,7}, and starting base 4, B must be a multiple of 2, one more than a multiple of three (which combine to require B is congruent to four mod six), either have a residue of four mod five or be a multiple of five, and either have a residue of four mod seven or be a multiple of seven. The smallest B which satisfies these conditions is 70. Thus, if you write the primes in base four and interpret them as base 70, you can be ensured that the resulting numbers will not be divisible by 2,3,5, or 7, which is neat, but far less elegant than P. Paterson's original result.
I really enjoy the work of 3Blue1Brown. He has a way of explaining things that just intuitively makes sense.
So much editing for part 3. I bet it's going to be amazing!
So what's the longest known "Paterson chain" (i.e. repeatedly plugging in the result to get another prime)? Will all chains eventually end?
This is a question that MUST be answered!
2 and 3 are the longest ones 😅
I feel like it's almost certain all chains will end, because there's no polynomial that can only produce primes, and I don't think any recurrence formula could either. I have a feeling the longest chain could be six, if the remainders cycle mod 7.
@@l.3ok
2 and 3 are loops, not chains.
5 is actually pretty long:
5 -> 11 -> 23 -> 113 -> 1301 -> 110111, 6 steps
I'm curious if there is a 7 step number or if all other numbers are tied or below
A numberphile and a 3b1b video on the same day?! This week can’t get better
Grant's eloquence and conveyance of mathematical principles is near unmatched.
The add-the-digits test for divisibility by 3 was my first experience of discovering a proof of a result. It was a bonus exercise my older sister had been set in school. Addictive experience.
Waited for this collab for ages.
Bonus Numberphile video with 3b1b?!?😍😍
So a schoolkid came up with that idea? Doesn't matter that it doesn't hold eventually, that was smart thinking! I thought I was doing well as an adult for having come across the divisible by 3 rule myself. (I also figured out whether a number can be divided by 11 too, so I call that a win! 😜)
We weren't taught anything like this at school (40+ years ago). Obviously we were taught about primes and how to do "long division" (which I promptly forgot after realising that writing it out like a fraction and dividing that way was far simpler and quicker!), and I have the vaguest memories of binary - this was when computers were being coded using punch cards. Only the really smart kids got to do an O level in computing, and they had to go once a week to the only school in the region to have a computer. Binary was of "no use" to anyone who wasn't going to go into STEM subjects. Actually, they didn't even have an acronym back then lol.
We didn't even have calculators. I still have my "log book" with the charts of logarithms, cos, sin, tan etc, squares & roots and yet more (can use most of them still if I need to. Just...) My little sister, doing her exams 2 years after me, was in the first year to be allowed to use calculators. Us "oldies" were horrified by the "cheating" 😂.
I can still do quite quick mental arithmetic (that was walloped into us in primary, especially our times tables!), including area, volume (unless it involves π, then I need paper, pen and - if I'm not using my calculator, which I usually do now - the log book), percentages and the like. Basically, if it's arithmetic based, I'm hot. One step beyond anything I'm ever likely to use in "real life", I'm clueless! 🤷 To be fair, I got a certificate in mathematics from my uni as an adult, and that was hard work, but I've forgotten everything except the quadratic equation formula (if I didn't already know it). A bit of revision and I'd be great with statistics again - I love playing with numbers. It's just remembering equations and which ones to use when that gets me. I only understand Pythagoras' theorem because of an old joke about fat squaws and a hippopotamus hide. Don't ask, it was barely acceptable in the 70s (even as a kid I squirmed) but it did teach me how to do that!
All in all, I'm trying to say how darned impressed I am by that chap as a youngster. I hope he's gone on to success in whatever he does now.
Once you've mentioned the joke, convention states that no matter how acceptable it is or isn't, you have to tell it!
This begs these questions though: What is the longest string of Patterson Primes? (A string being a prime number goes in, and a prime number comes out as the seed for the next Patterson Prime) Does it happen in the low numbers? Does it exist in the 'big' numbers? Is there an infinitely long string of them? are there an arbitrarily infinite number of infinite Patterson Prime strings?
Thinking about patterns I always think about density and if there’s anything to learn for it, like if there’s a point where you run out of primes by using this method or if there are infinite Patterson primes and they just get more and more sparse, also if there’s a relation between the distance between primes and if different bases affect the spread, etc.
5:00 I've used the "add the digits" trick to check for divisibility by 3 for years...but never knew why it worked.
I like thinking about other bases. Great video, as always.
I feel like we have definitely observed an increase in Grady’s mathematical abilities/confidence over the years of him conducting all these wonderful interviews. Love to see it!
A reasonable conjecture would be: given any m>n positive, there exists a prime p such that the n-ary expression of p interpreted as m-ary, is not a prime.
1:15 Whoops! Editing mistake.
There very much might be a mod 4 or mod 8 aspect to primes, since there IS one for the bijective multipliers within mod (2^n) spaces .. such that if x*y = 1 then the 4th bit ("eights place") of the binary expansion of x is not equal the 4th bit of the binary expansion of y .. always .. a fact used to calculate modular inverses faster than newton
to be more specific, for a given x, its 2^n modular inverse y will always be the same in the first 3 bits (ones, twos, and fours places) and always be different in the 4th (eights place) .. while after that, it depends
I love 3Blue1Brown ❤
This is absolutely astounding! I have been working on a very similar version of this for several weeks now. Except it's much bigger in scope. I am fairly certain I know why 31 fails.
I have been studying what I call zones of Naomi. A RUclips comment is a touch too small to go into detail.
My current record prime found is over 12k digits in length and it looks very cool indeed.
I suppose it's about time for me to start making videos.
Sounds cool! Definitely make a video
Paterson primes are the stuff that Parker squares are made of.
I hear you. ;)
Numberphile teaming up with 3Blue1Brown forms a kind of nerd supergroup. Good for everyone!
On top of 2, 3, 5, there are no 11s in the factors as well - because any number divisible by 5 in base 10 is divisible by 11 when converted into base 4 (since 5a = 4a+a = aa in base 4).
Grant crushes the math problems with his biceps
Just like for 3 there's a divisibility rule for 7 that you can use on 1211. Since 10 is 3 mod 7 then 10^2 is 9 = 2 mod 7. So you have 12 * 2 + 11 mod 7 -> 5 * 2 + 4 = 14 = 0 mod 7.
Or just subtract twice the last digit:
1211 => 121-2 = 119 => 18-11 = 7
The 3Blue1Brown channel dropped a new video only a couple of hours ago and now we get THIS TOO today??? Christmas came early!
Would be interesting to see whats the longest recursive chain of paterson primes you can generate.
Aside from the trivial infinite chains (primes less than 4), I have the same question.
The longest I've found so far start with 5, 29, and 73. These end at 1301, 200133233 and 10301133301033, respectively. I've checked all the starting primes below 2500.
Update: Checking the other comments, chains with one more number (but maybe not two) exist. But the smallest starts with a 9-digit prime, so I'm done.
a + b (mod 3) = a (mod 3) + b (mod 3) isn't strictly true. You still have to mod it again at the end. For example, 2 + 2 (mod 3) is not equal to 2 (mod 3) + 2 (mod 3), which would equal 4.
My thought as well!
I think it should be:
(a (mod 3) + b (mod 3)) (mod 3)
@@hebl47 in theory yes, but that would defeat the point, as you needlessly do 3 operations now, instead of 1.
@@m.h.6470 What else is math if not theory? You have to be precise in phrasing your functions. And doing 3 easy operations instead of one hard is still a win.
@@hebl47 I would postulate, that - unless you work with an incredibly large number - you exchange 3 easy against 1 barely medium operation.
gosh, that would be pretty cool if i had a math friend like paterson back in school
🧑💻don't mind me just haxoring into the vault of unreleased vids.
FYI I got here from 3B1B's latest vid, endcard linked to this vid.
endgame: We had the best crossover ever!
Numperphile and 3Brown1Blue: Hold my brown sheet, please!
Not only is Grant one of the greatest math educators out there today, but he's also getting hella swole.
Yeah I love his videos to teach myself things but I didn’t think he’d be so conventionally attractive lol
Ikrrrr
There’s a mistake at 1:17 in the video. It says 17 is “11” in base 4, but you were converting 5 to base 4 at the time.
So there are two possibilities for the result: either it's a Paterson Prime, or it's a Parker Prime 😆
Yes, indeed. You have to be very careful if you think you have found a pattern, because there is so much room for coincidence. Always look for counter-examples!
My favourite way to visualise the connection between the cross-sums and divisibility is this:
1000 * a + 100 * b + 10 * c + d
= 999 * a + a + 99 * b + b + 9 * c + c + d
= [an obvious multiple of 9] + a + b + c + d
64 * a + 16 * b + 4 * c + d
= 63 * a + a + 15 * b + b + 3 * c + c + d
= [an obvious multiple of 3] + a + b + c + d
In general, the difference between a base-N number and its cross-sum is a multiple of N-1.
i actually discovered this while I was messing around during math class. all the primes that i input seemed to output a bigger prime, so I was disappointed to realise after checking on google that not all of them were primes
1:40 Seeing the scrolling stop just before 31 was pretty funny
I suspect (and would love to see) a proof that every "Paterson sequences" have to eventually become composite must be possible. Meaning p --> f(p) --> f(f(p)) --> etc. Eventually must produce a non-prime term.
You have to set the condition that p >= 5 because 2 and 3 are trivial counterexamples.
I do have one question about this.
would it be possible to find a base that improves this method so it excludes 2,3,5 and 7?
I doubt it. but I had to ask since there's a chance that it exists.
That's fascinating.
Is it generalizable? That is, can you choose what numbers you want to exclude and then pick a base or, if given a base, can you figure out what it will screen out?
2 Three blue one brown videos in one day!
@Numberphile What is the outro song, please? It has a really chill vibe. Neither Shazam nor Google Sound Search had any luck.
Also curious about this :)
Im imagining someone using the biggest discovered Mersenne prime and then stumbling upon a new prime by pure luck.
Now here's a novel followup puzzle:
What's the longest "chain" one can build by using the Patterson Prime Method. We saw 5->11->23->113->1301. 1301 converts to 110111, which is not prime in base 10, so that is a chain of length 4 (or 5 if counting from 1 makes more sense than counting from 0).
I suspect the longest chain starts from a small value, but it isn't inconceivable for there to be an arbitrarily long chain somewhere out there. Just incredibly unlikely.
5 gives length 5, then 101495533 gives length 6, but i can't find 7 or more.
We know very big Mersenne primes. But, I assume, not all the primes before it are known.
What is the highest prime number where all the primes smaller than it are known?
Your question cannot really be answered, because if I told you the answer is p, you could very quickly use known algorithms to find a slightly larger prime number, and then that would be the new highest prime number where all the primes smaller than it are known. And you could keep doing this forever. Just not very quickly.
We have found all primes up to about 10^18 but not yet 10^19, according to Chris K. Caldwell at UTM. Using best available techniques and all memory storage in the entire world for the sieve, with heavy optimizations, we could conceivably get all primes up to about 10^25. Beyond that, lacking the memory capacity to sieve, you'd have to switch to much slower algorithms that would spit them out one at a time. You could go on finding primes for millions of years that way and never stop.
@@chiaracoetzee Come to think of it, that's actually quite small, considering that, IIRC, the largest known prime is on the order of 10^2000000.
How do the lengths of unbroken Paterson prime chains behave as the value of the initial term increases? What is the limsup thereof?
Grant is simply amazing at explaining things and Brady (almost) always asks the right questions - Love this video, wish I could upvote it more than once!
Even if it were foolproof, it still wouldn't be a very useful test of primality for the number you started with because you'll have to know if the larger, more "difficult" number is prime or not. As a way to generate primes from a known prime though, it would be pretty great.
Finally, a worthy opponent for the venerable Parker Square!
As soon as I see a video with the love of my lif- I mean 3blue1brown I have to click immediately
Okay, we need Neil Sloane to get to work finding the longest string of Patterson Primes he can. The rule is: start with a seed prime, do the Patterson Conversion, and if it's prime, convert that, and so on until you run into a composite. What's the largest number you can find that can be reached this way?
I love 3 blue 1 brown, especially as it was a spinoff of 2 girls 1 cup
omg
By the way, you said it doesn't have the immunity from 11, but the divisibility test for 11 implies that having the larger number divisible by 11 requires either at least 9011 (with the larger number equal to 2030303) or for the larger number to be divisible by 5. (If you can't see why, remember that 5 is 11 base 4).
That was a really fun video. Very relatable.
Loved that outro music on there to the Paterson Primes scrolling by :D
It's good how the modern nerd hits the gym now.
Brady and Grant collaborating again: great! 👏🏻
It isn't mentioned in the video, but I suspect that the prime distribution of the output follows a log scale.
Haha, I remember doing that exact same base 4 conversion in middle school and thinking I had found a formula for larger prime numbers. I was very disappointed when I finally stumbled on a counterexample.
This was so much fun!
Does anyone know how long the longest string of paterson primes might be? in other words how many primes can you possibly generate using some prime number as a seed using this method?
me when my favourite numbers (607 and 67) for reasons unrelated to math happen to be both prime numbers
The Patrick Paterson Patented Procedure for Procuring Primes
Is a specific BaseNumberSystem forbidding us to think in different way? The reason why I ask is because I've seen the movie that says our language system forbids us to think out of the language.
So, used language is matching to our decimal basicnumbersystem, my opinion.
I wonder how one would transfer thus idea over to something like the surreal numbers?
not divisible by 2 by 3 by 5, so there is a big chance to maintain some occurence in both bases not a wow video. but the music at the end of the video is amazing. what is the track name
7:07 I think there is almost immunity from 11. Let's say s is the digit string which is the base-4 representation of p and the base-10 representation of q. Sum those digits that are in s's odd places; sum those digits that are in s's even places; let d be the difference between those two sums. If 11 divides q, then 11 also divides d.
Now if d happens to be 0, then (by a similar argument) 5 divides p. 11 is indeed a Paterson prime produced from 5. But you'll only get a Paterson pseudoprime divisible by 11 if d is divisible by 11. And it takes a few digits for that to happen. The first example is 2030303=11*379*487, which comes from 9011.
It makes me wonder if there are some number that used in the please of four in this algorithm you get more prime numbers?🤔
I don't know if anybody has pointed that out already, but there's a mistake @1:16, they are talking about "5", but the video is still showing "17" from the previous example.
as the non-math paterson of the family, i understand none of this but love that my brother and grant do
5:12 well a + b mod 3 = (a mod 3 + b mod 3) mod 3 actually, both a mod 3 and b moc 3 can give you for example 2 then you'd get 4
So some of the numberphile videos with Ben are shot at Brady's place. But these seem to be over at Grants.
Now I desperately want to know if the Paterson Prime chain can be infinite, and if not, what's the maximum length.
Great question.
prime 5 has length 5, then the smallest prime that has length 6 is 101495533. I haven't found 7 or more yet.
I suspect it's just a question of probability and search depth. If each prime may or may not generate another prime, and it's never a zero percent chance, so if you check long enough, I don't see why you couldn't find any finite length chain. Doubt it's infinite tho...
Got here early from the new 3B1B vid, it seems!
this comment proves i reached the end of the 3b1b video
Are there an infinite number of Paterson Primes? If not what’s the biggest? What about bases? There are more questions to answer!
This is more personal. I like it.
Woah, am I supposed to be here?
what if grant trolled brady
Fascinating and now I'm wondering about the relationships between other bases. Is there any base for which this holds? Or a base for which the list of eliminated factors goes much higher?
Might have to go write some code...
People: *Invents numbers*
Also people: Bah gawd, the numbers.
I have an estimation off the ‘prime guesser’ it’s (n^(n+1))-ceil(n/2) if contains 7 it’s probably prime (except 5 it doesn’t work)
The strong law of small numbers is strong here.
I come from the “ pattern fool ya”
Converting from base 24 to base 5634 works for the first 17 primes. (4 to 10 only for the first 10) and produces some pretty large primes, like:
277 to 61987 to 715685960827 to 6093125672235600646607486318497
9:50 Paterson is Cameron Frye from Ferris Bueller's Day Off
Why do both PPs and composite numbers appear in such significantly strong waves, even at very high input values?
At lower input values, waves of successive composite outputs seem significant.
And at higher input values, waves of successive PPs still keep coming surprisingly frequently.
Refreshing Video to watch. A wealth of numbers , fluently if not lyrically narrated and keyboard soundtrack. From what your saying, increase the base and you increase the number of primes ?
Seems more like "The Patrick Paterson Patented Process for Producing Primes" to me.
Presumably Patent pending