In 2003 We Discovered a New Way to Generate Primes

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There is a Fibonacci-like recurrence that seems to generate primes! It was discovered in 2003, but at the time no one understood why it worked. A few years later, I plotted the primes in a way that reveals some hidden structure. This is a tale of logarithmic scale.
Followup video on this sequence: • Why Does this Algorith...
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References:
Fernando Chamizo, Dulcinea Raboso, and Serafín Ruiz-Cabello, On Rowland's Sequence, The Electronic Journal of Combinatorics 18(2) (2011) P10 (10 pages).
doi.org/10.37236/2006
Benoit Cloitre, 10 conjectures in additive number theory (2011) (46 pages).
arxiv.org/abs/1101.4274
Eric Rowland, A natural prime-generating recurrence, Journal of Integer Sequences 11 (2008) 08.2.8 (13 pages).
cs.uwaterloo.ca/journals/JIS/...
Serafín Ruiz-Cabello, On the use of the least common multiple to build a prime-generating recurrence, International Journal of Number Theory 13 (2017) 819-833.
doi.org/10.1142/S179304211750...
Open access: arxiv.org/abs/1504.05041
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0:00 Recurrence
2:59 Doubling relations
4:03 Plotting locations of primes
6:24 Clusters of primes
9:49 Predicting primes in each cluster
15:22 Answers to burning questions
18:19 Changing the initial term
20:08 Cloitre's lcm recurrence
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Animated with Manim. www.manim.community
Thanks to Ken Emmer for supplying the microphone.
Web site: ericrowland.github.io
Twitter: / ericrowland

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26 май 2024

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Комментарии : 506

@sobertillnoon Год назад

We haven't used the Richter scale since 1970. The current measurement scale is called the moment magnitude scale.

@studytime2570 Год назад

btw whats the notable difference between richter scale and moment magnitude. And today will still say "that was a 6.3 magnitude earthquake". So is it not richter?

@sobertillnoon Год назад

@@studytime2570 the Richter scale was designed to be used in California. For reasons that are beyond my level of geological knowledge it didn't map onto other regions. So a global scale was created. To the last question: yes.

@Bob-ik1jj Год назад

@@studytime2570 Only the moment magnitude scale is capable of measuring magnitude 8 and greater events accurately. Additionally, the Richter scale was calculated for only one type of earthquake wave.

@multiarray2320 Год назад

i just read in wikipedia about this. this blew my mind.

@dannygjk Год назад

Richter is still used. You speak as if Richter has never been used since 1970. What is your first language?

@vnever9078 Год назад

Bro so based he makes expository math videos based off of his own research. Chad.

@MalcolmCooks Год назад

when you don't get invited to Numberphile "Fine, I'll do it myself."

@iamjohnrobot Год назад

Chad-adic and fantastic

@nickmaiorino4744 Год назад

@@iamjohnrobot You mean, 'p'-Chad-adic and fantastic! 😄😎

@spellignerror8998 Год назад

I'm pretty sure that's not the meaning of based 🤔 still a good video

@anntakamaki1960 Год назад

Vishwaguru math video developer

@AbyssPog Год назад

Damn, I wish every research paper could be explained in a digestible video format like this. Great video!

@GuzmanTierno Год назад

Next step is having chatGPT generate videos like this for every paper ...

@abj136 Год назад

@@GuzmanTierno That would be awful. Because (if you weren’t aware) ChatGPT is very bad at math.

@GuzmanTierno Год назад

@@abj136 yeah, you're right ... luckily ...

@dumbidiot1119 Год назад

@@abj136 that kinda makes sense tho, chat-gpt is a language model

@suomeaboo Год назад

@@abj136 For now. Give it a few years.

@curious_one1156 Год назад

20 years later, congrats Eric ! This is awesome. Your own theorem.

@EricRowland Год назад

Thank you!

@bothieGMX Год назад

@@EricRowland Lol, I don't know your channel, didn't even realize, it was you who wrote the paper ;) Chapeau!

@maximkosey5549 Год назад

@@EricRowland did somebody prove this theorem ?

@EricRowland Год назад

@@maximkosey5549 Yes, I proved it.

@maximkosey5549 Год назад

@@EricRowland so you can definitely generate all prime numbers, without gaps ?

@demonicdrn3037 Год назад

Yooooo! This is the best way to read papers; by not reading them at all and forcing the author to tell you, in what I assume to be an excruciating lack of detail, what they proved and how. Thank you so much!

@rosiefay7283 Год назад

I got the impression that it was more an excruciating overabundance of detail, some of which we could easily have worked out for ourselves.

@oncedidactic 8 месяцев назад

@@rosiefay7283perhaps, but it’s a 22m video and we have the whole picture and more, save for some rigorous steps. Tradeoff, sure, but I definitely used my 22 minutes better on the video. That’s probably true for most, even researchers? Thoughts?

@drjacovanniekerk Год назад

Why so few videos? You had me on the edge of my seat from start to finish. Video quality/explanation is spot on. This is "a million subscribers" content.

@EricRowland Год назад

Thanks! They take a long time to make, but more to come!

@drjacovanniekerk Год назад

@@EricRowland I know! I have a channel for university content and one for Numismatics. Hours and hours of editing. Will keep watching yours.

@michaeldamolsen 11 месяцев назад

@@drjacovanniekerk Checked your main channel, and subscribed immediately.

@kcthomas9531 9 месяцев назад

How about do a collab with 3b1b? I feel like that would be the quickest way to get a lot of subscribers! @@EricRowland

@snowfloofcathug 5 месяцев назад

… I didn’t even realise but apparently I’ve seen all the videos, it was just long enough between them for me to not notice

@_Wombat Год назад

I feel like there is a real question to be had about why humanity finds primes so incredibly interesting. I've watched so many videos about prime numbers and yet I am still hungry for more. Great video :)

@vnever9078 Год назад

It isn't for no reason that they are called _prime_ numbers haha

@mohammedsamir5142 Год назад

You should believe in number theory to realize how awesome are the prime numbers

@MasterHigure Год назад

This is a quote from a math book on a mostly unrelated subject, but I feel it fits here too: It's an intriguing mix of pattern and chaos.

@ttrss Год назад

crypto

@wauwau4896 Год назад

Primes have always fascinated me because they feel like the building blocks of numbers. It's remarkable to think that every other natural number greater than 1 can be decomposed into a unique product of primes. It's almost as if primes are the elemental components of the number system, much like atoms are the fundamental building blocks of matter. This fundamental property of primes is what makes them so intriguing and important for us humans. At least that is what I think.

@user-tn2dk2pg2p Год назад

I don't think anyone's posted the reason as to why 3 is the second number in every cluster, so for those curious, it's stems from the fact that every index with nontrivial gcd is either 0 or 2 mod 3. This comes from simple induction: the first index indeed satisfies the condition, and if the previous index n was 0 mod 3 then 2n-1 isn't divisible by 3, so the smallest prime p dividing it is -1 or 1 mod 6, leading to the new index being (p-1)/2=0 or 2 mod 3 more than the previous one; likewise, if the previous index n was 2 mod 3, then 2n-1 is divisible by 3, so the next index shifts by (3-1)/2=1, making it 0 mod 3. Because of that, when we get to an index t that makes 2t-1 prime, 2t-1 is also the index of that prime (since the index goes from t to t+(2t-1-1)/2=2t-1), and since the index is prime and more than 3, it isn't 0 mod 3, so it's 2 mod 3, leading to the next number 2(2t-1)-1 in the sequence being 2*2-1=0 mod 3 i.e. the number after the prime must be 3.

@oncedidactic 8 месяцев назад

Thank you for the nice walkthrough. The mod3 sequencing has the same flavor to me as Syracuse sequences. It seems like there is something about mod3 carrying information that pops up in recursion that’s not coincidental.

@johnchessant3012 Год назад

It must've been very cool to find out that the prime properties of this seemingly arbitrary sequence is related to a very active area of research, namely primes in arithmetic progressions. In particular, I find it really neat that these sorts of questions are playful enough that you could imagine Fermat or Euler studying them, but we can now describe them with our more modern techniques.

@AssemblyWizard Год назад

I first saw the last pair you highlighted, 121403 & 242807, then I went looking for the same relation and found the others

@EricRowland Год назад

Nice!

@oleble3317 Год назад

@@EricRowland first one I noticed was the 233 and 467 pair and I then confirmed on the bigger ones

@RSLT Год назад

Super interesting, high-quality, and creative video. Fantastic Job! I have been looking to see a beautiful method like this for many years.

@EricRowland Год назад

Thank you so much!

@andreasmaaan 7 месяцев назад

@Eric Rowland, in the three videos you've created so far, your ability to explain mathematical concepts with clarity and insight is remarkable. I really hope this there is (a lot) more to come!

@EricRowland 7 месяцев назад

Thank you so much! There are more videos to come. (They just take a long time to make!)

@andreasmaaan 7 месяцев назад

@@EricRowland very happy to hear (and completely understand!) :)

@sixhundredandfive7123 Год назад

Thanks so much for telling me to look at the patterns myself. Where "...5,3..." occur at such interesting intervals so does where "...7,3..." occurs as well. "5,3,11,3,23,3,47,3" is 8. "5,3,101,3,7,11,3,13,233,467,3" is 12. You can then write them as iteration numbers: "P3,P2,P5,P2,P9,P2,P15,P2" is 8.

@pizzarickk333 Год назад

very clever and excellent explanation. walking someone through the thoughts your brain went through when solving a problem is my favorite way of teaching.

@bobitsmagic4961 Год назад

That video was absolutly amazing, didnt expect such a high quality from a random youtube video. Well done

@francescodero8759 Год назад

it's always nice to see actual progress in abstract mathematics and number theory, keep it up, who knows, maybes someday humanity will discover some relation between these patterns and the riemann hypothesis

@romanbriggs2457 Год назад

Math educators like yourself have been invaluable to me. My eyes will glaze over reading the papers you cite, everything goes wavy and the nomenclature makes no sense without help. Watching videos like these, with explanation and animation, the information feels much more natural. I probably won't contribute to advancing the discussion on these topics, but to understand a little more about them without enrolling in a whole degree program makes me fortunate. Thank you

@vitoramim5346 Год назад

The even more amazing part is that you explained it in a way even I could understand. Great video and congrats for the theorem!

@The_True_J Год назад

I absolutely love this. At no point does it feel like rigorous mathematics. It feels like you're just playing around with a simple sequence and seeing what patterns appear. Awesome job. As of writing this comment, idk if you've made a follow up video, but I'm looking forward to it.

@EricRowland Год назад

Thanks, that’s the vibe I was going for! The follow-up video is still a work in progress. Hopefully soon!

@yudoball Год назад

@robertunderwood1011 Год назад

I am interested in the generalized REPUNT primes. In base two, these would be the Mersienrs

@tylerhaslam2083 5 месяцев назад

I love this video! Thanks for making it. I love how it shows the process of conjecturing by poking around in the structures and formulas of the patterns observed. Very nice window into the first steps of mathematical thought.

@elfumaonthetube Год назад

I love videos about patterns and primes, and this one is among my favorites. Great job, and congrats for giving a theorem your name.

@samuelwaller4924 Год назад

I can only imagine the satisfaction you felt when you discovered all of this. Great job, this is really cool!

@SomeoneCommenting 5 месяцев назад

If everybody who makes math videos was so concise, clear, and give visual examples that can demonstrate your point so simple and obvious as you do, we would be able to understand a lot of other things much better.

@vanhetgoor Год назад

While you were talking I had some wonderful ideas. You are an inspiration! Normally I listen to sequential music so that sounds don't interrupt with my flow of thoughts, but this works too! I do not want to give the impression that you are boring, but it comes close, in a polite and gentle manner. My attention drifted away after the first mentioning of Fibonacci, endless lists of numbers, all with a meaning and significance. It is a glorious day, summer is on it's way.

@EricRowland Год назад

This same thing happens to me at conferences. Listening to other people talk about their work (or rather, *not* listening) has given me some great ideas. Interesting social phenomenon!

@KrasBadan Год назад

Great job! At first I thought that this was too hard for me, but eventually I understood almost everything. So cool.

@mikecaetano Год назад

Observed pattern. In the first cluster, 5 is followed by 3. In the next 11 is followed by 3. In the fourth, starting with 47, 5 is again followed by 3. In the fifth, starting with 101, 7 is not followed by 3, but 11 is. 13 is not. Scanning down, it appears that whenever 5 or 11 appear in a cluster they are followed by 3. But this does not appear to hold for 7 and 13 -- which also appear to never occur as the first terms of any cluster. So perhaps for numbers that start clusters, if they reappear in other clusters, they do so followed by 3. And numbers that do not start clusters, if they reappear in other clusters, they do so not followed by 3.

@cryptogenik Год назад

Wow that's pretty mindblowing that you came up with that!

@Busterblade20 Год назад

What a wonderful video. I wish every math paper could be explained in such a wonderful video format like this.

@wilderuhl3450 Год назад

Only 3 videos, but they’re all fantastic. Thanks for sharing.

@tomascortespacheco5703 Год назад

Your exposition was superb. I really enjoyed the pace of the video, and how it was structured as a `story` that was easy to follow. Suffice to say that you have a solid understanding of manim. Have you considered posting the manim code? It would help a lot manim beginners to further learn how to use it!

@ethanlewis1453 Год назад

Hi, the most prevalent pattern in the prime sequence generated I noticed @ 3:00 seems to be 3 - 5 - 3 which occurs frequently but not quite predictably.

@MrDannyDetail Год назад

21:50 That sequence is interesting. If you take the first 2 to be in the 2nd position (so the sequence just has no first position) then all the primes, other than 3, seem to appear in their own numbered position (i.e. 2 in the 2nd pos, 5 in the 5th, 7 in the 7th). You then have other primes appearing, and at intervals corresponding to prime multiples of that prime (e.g. 5 in the 5th, (2x5)th, (3x5)th and (5x5)th positions) though it looks like possibly any given prime will only appear in the sequence a 'few' times (for some definition of few) then never again.

@Alex_Deam Год назад

Idk about the multiples, but your first point about the pth position being p is proven as Proposition 2.3 (Proposition 5 in the arxiv version) in the Ruiz-Cabello paper linked by Eric above!

@bolleholle Год назад

in the first 10000 terms, there are 5 instances of 5, the last one on n=25=5*5 one instance of 7. 3 instances of 11, last one on 33=11*3. 8 instances of 13 (7th on 91=7*13, 8th on 169=13*13). 17 appears three times, last on 51=17*3. 19 appears once. For the following the appearances along the sequence continue to be equally spaced: 23 appears five times. 29 five times. 31 once. 37 once. 41 three times. 43 five times. 47 five times. 53 three times. 59 five times. 61 seven times. 67 five times. 71 three times. 73 thirteen times, last one on n=949=73*13. 79 once. 83 three times. 89 appears 15 times, last on 1335=89*15. There is a nice pattern but it is a little disturbing how 13 appears at n=169.

@Asdayasman Год назад

It's pretty damn sweet that new maths is both happening, AND becoming popular and easily digestable on youtube, no doubt in no small part thanks for 3b1b's manim.

@siegfriedbarfuss9379 Год назад

Brilliant and perfectly paced 🙏🏻

@dimkadimon Год назад

Wow it's THE Eric Rowland! I have been amazed by this sequence ever since I saw it. Thank you for explaining it so clearly.

@joseph7858 Год назад

such great narration of your discovery process: thank you Eric! 😊

@EricRowland Год назад

Thank you!

@williamrutherford553 Год назад

I feel like this is related to Dirichlet progressions. I'm actually doing applied research into finding the upper bound of the first p of the form sn+1, which is MUCH easier to prove the primality of using a deterministic Miller-Rabin test. So far, it looks like p(s) < c*s^L, where L is approximately 2. However, it seems like if you pick an L value > 1, you can find an N such that the bound holds for s>N. I thought it was related, especially due to the clustering in a log-log plot, you get that same kind of behaviour when graphing the strictly increasing subset of s, p(s) (just like ignoring the 1s).

@user-ey2vv1dl3n Год назад

cool format, plz dont stop)

@monkeymathematician5896 2 месяца назад

What is said from 8:04 prevents from looping over all values of a cluster and sets its boundaries. It also means that the last value's index of the cluster is enough to describe it and averaging the values or the indexes could be unnecessary. It also says that there might be something hidden in the gap between two clusters. This saved me weeks, maybe months of work and much CPU time. Deserves the Fields to me. Thank you Professor 😁

@draido-dev Месяц назад

noticed this pattern while solving project euler #443, lovely video!

@EliederSousa Год назад

Congrats for you making this theorem. It's amazing.

@user-no9wi4vu3x 9 месяцев назад

Man, this is so amazing! Love it!

@takeguess Год назад

Nice work! I love this!!! Thanks for putting it together

@EricRowland Год назад

Thank you!

@RichardHolmesSyr Год назад

Thanks for this video, I just learned about this recurrence a few weeks ago from Wikipedia and found it very interesting!

@EricRowland Год назад

Glad you enjoyed it! That's a fun coincidence!

@IagoMartinsJ 9 месяцев назад

You blew my mind in a 10 Richter's scale' magnitude, that was awesome

@HasekuraIsuna Год назад

Wow, this is so cool! Finding patterns in primes!

@trapkat8213 2 месяца назад

Wow. Brilliant work and brilliant presentation.

@spaceyote7174 Год назад

Wow! I have an obsession with primes and I read about this exact theorem a few months ago, how surreal to have a video by the author of it to pop up in my feed

@kdicus Год назад

Absolutely beautiful work. Beautiful math. Beautiful thinking. Beautiful video. Someone will figure out how to build on your work.

@arhamshah71 11 месяцев назад

you are doing great work in making these videoes. It really helps a lot in visualising while studying maths concepts. I wish to see your videos more often and hope that your videos reach to those who need it and recieve much greater attention. you are going to be the next 3Blue1Brown.

@EricRowland 11 месяцев назад

Thank you!

@adamnagy4544 Год назад

I could die for videos like that for every publication!!!

@timsim83 Год назад

Thanks for the ending summary. I was hoping for the explanation about finding common divisors of 10 digit numbers being a computational hurtle.

@thewhitefalcon8539 Год назад

GCD is no problem. GCD 10^9 times is maybe 1 minute - 1 hour of computation (hard to estimate accurately). But I'm guessing we already know all the primes up to 10^9.

@artophile7777 Год назад

THIS CHANNEL IS UNBELIEVABLE

@crowdozer3592 Год назад

Very nice animation and narration

@rileycampbell5691 5 месяцев назад

This video was great. Really really clever.

@remziogultum6697 5 месяцев назад

this was really fun and educative to watch

@LorenzoAGJ Год назад

Really nice. A few months I was playing around with some code trying to find some relation between Fibonacci's sequence and prime numbers. I did it just for fun because I liked these two topics and I wanted to relate them some way haha

@austinclees9252 Год назад

What I saw first at 3:00 is that the 3’s are on opposite sides of other primes, like the twin prime conjecture

@algorithminc.8850 Год назад

Thanks. Enjoyed this (and your other videos). Great stuff! Cheers.

@EricRowland Год назад

Thanks so much!

@pirobot668beta Год назад

I've got a simple means using Prime Factoring and math to directly 'predict' the interval between primes. 'Paired Primes' like 17 and 19 seems to break the game (Pa+2=Prime)...skipping them for now. Take Pa+1 and Pa+2 as Prime Factored Composites, and add the first terms together to make 5. 1. Starting with 43 as Pa 2. Factoring Pa+1 = 44 = (2!*11) 3. Factoring Pa+2 = 45 = (3!*5) 4. 46 place holder 5. 47b = calculated Pb For extra fun, take the difference of the second terms, (11-5), which leaves 6. Counting backwards from 42 (since we are done with 43) 6 places leaves us at 37 = PrimeC. 6. 37 PrimeC, counting backwards. 5. 38 4. 39 3. 40 2. 41 1. 42 43 [skipped!] Working my way through Primes to 500; found a few spots where it doesn't work in both directions. Enjoy!

@washemoamadah4706 Год назад

Ey dude, at around 10:15 in the video, if you take the sum + the prime you wanted - 1 you get the next sum in the sequence. If you do that again with the new sum you get the next sum. But you surely already have seen that showed why, and I missed it. Great video man.

@schweinmachtbree1013 Год назад

Absolutely spectacular video! Bravo!!

@EricRowland Год назад

Thank you so much!

@christophergilbert5988 Год назад

Absolutely amazing video!

@EricRowland Год назад

Thank you so much!

@yukelalexandre8885 Год назад

Dude! Been working on this very problem for like a decade, mad respect for the explanatory work! ✊

@alien3.0c Год назад

This was really interesting and well explained

@aromeran 11 месяцев назад

Waiting for the next part. ABSOLUTELY GREAT video Eric!

@EricRowland 11 месяцев назад

Thanks! Hopefully the next part will be done in the next few weeks!

@richardfredlund8846 Год назад

@Eric Rowland, awesome video, and maths. After watching I was interested in the Cloitre's lcm recurrence, so wrote some code to generate it. What I found really surprising is when I looked at the set of numbers generated from the first 500 values. It's exactly the set of primes less than 500. Except there is no 3, but there's a 1. It's also true with first 50,000. (and my computer fell over when I tried on 100k cos my codes not super efficient). I'm sure I'm not the first to notice this, ... but seems rather remarkable.

@richardfredlund8846 Год назад

it's quite easy to prove that when n is prime P(n) =n for all odd primes >3 because C[n] is the product of numbers strictly smaller than n. It gets more interesting in the case of P(n*n) where n is prime. this requires that there exists some prime q which divides a*n -1 for some a in {2,3,...,n-1}. For example P(5*5) is saved because 19 divides 5*4 -1 which means that: If we make the hypothesis that for all odd primes p>3 there exists another prime q such that q = a*p - 1 for some a in {2,3,...,n-1} then this hypothesis is implied to be true if Cloitre's variant makes only primes. equally if this hypothesis if false, that implies Cloitre's variant doesn't only primes. (which is not an if and only if because if the hypothesis is true it doesn't imply Cloitre's variant makes only primes.)

@joeeeee8738 Год назад

Awesome walkthrough❤

@General12th Год назад

Great work, Eric!

@EricRowland Год назад

Thank you!

@danielwilms6919 4 месяца назад

Great visualization

@Scratchfan321 Год назад

I saw 3 repeating on every 2 numbers [EDIT: In some parts], 7 and 11 were appearing too often but I didn't see exactly where. Also great video!

@royvanrijn Год назад

The forward-moving algorithm works for any input number: Given a number n Calculate the target p=(2*n - 1) Find the smallest prime factor of p=>pf Update n += (pf-1)/2 For example, start with n=44: 44 p=87 pf=3 p=89 pf=89 89 p=177 pf=3 p=179 pf=179 179 p=357 pf=3 p=359 pf=359 359 p=717 pf=3 p=719 pf=719 719 p=1437 pf=3 p=1439 pf=1439 1439 p=2877 pf=3 p=2879 pf=2879 2879 p=5757 pf=3 p=5759 pf=13 p=5771 pf=29 p=5799 pf=3 p=5801 pf=5801 5801 (etc) All of these generate factors and/or prime numbers (obviously... when you think about it).

@cheeseburgermonkey7104 Год назад

This guy is underrated

@riadhalrabeh3783 Год назад

Your clusters graph for the primes(min 6:33) resembles the cluster of stable elements of the periodic table. This is a support of an idea I had published before that the growth of condensed matter follows the growth of primes. This makes primes the elementary particles of mathematics and of physics as well.

@usernameisamyth Год назад

thanks for giving so many things to think about

@blacklistnr1 Год назад

@2:43 [Pause the video], Ah yes, observing a great sequence in the wild, after hours of sitting camouflaged as a rock making Potoo mating calls, this unexpected beauty shows up. As I zoom out my telephoto lens and add a few beauty filters I can finally see.. nothing of interest. I'm here for cool math animations and graphs in my food break. After that great intro getting me hooked I'm most definitely not going to stare at some numbers :)) Edit: Great work! This is quite an interesting little set of interactions

@gdgiantdwarf9639 Год назад

an other somewhat interesting pattern i've noticed is that each new cluster of primes actually begins with a point where the index equals the value (noticed it at 9:18, might not hold up later on in the series)

@K9Megahertz Год назад

I noticed this as well. Though maybe I missed something earlier that would have made that seem obvious, but after reading this comment, I guess that's just how it ends up and yes, it is quite interesting.

@adacohen 5 месяцев назад

At 2:51, I noticed a more general version of the doubling pattern which seems to hold true everywhere (but I haven't proven it). If you let x and y be two "largest so far" primes in the sequence, then y = 2 * x + p_s - p_n - 1, where p_s is the sum of the primes in the sequence between x and y, and p_t is the number of primes between x and y. (Trivially, you can put the ones back in the sequence and use the same formula, since the ones are just canceled out between p_s and p_n). For example: ... 467, 3, 5, 3, 941 ... x = 467 y = 941 p_s = 3 + 5 + 3 = 11 p_n = 3 2*x + p_s - p_n -1 = 934 + 11 - 3 - 1 = 941 The 2*x + 1 case is just a special case of this: ... 5, 3, 11... p_s = 3 p_n = 1 2*x + p_s - p_n - 1 = 2*x + 3 - 1 - 1 = 2*x + 1 And you don't even have to do this with two consecutive "largest so far" primes. For example: ... 47, 3, 5, 3, 101, 3, 7, 11, 3, 13, 233, 3, 467 ... x = 47 y = 467 p_s = 3 + 5 + 3 + 101 + 3 + 7 + 11 + 3 + 13 + 233 + 3 = 385 p_t = 11 2*x + p_s - p_n - 1 = 94 + 385 - 11 - 1 = 467 I'm not sure how this relates to everything else, or if it's useful (it doesn't actually predict the jumps), but it's interesting.

@rasowa2958 5 месяцев назад

This is because every prime bumps R(n) up to 3*n. See it at 1:52. So: R(x) = 3*x R(y) = 3*y = R(x) + p_s + (y - x - p_n - 1) + y this is because R(y) is a result of adding: - R(x) - primes between x and y (p_s) - ones between x and y in amount: y - x - p_n - 1 - prime y Now solve it for y: 3*y = 3*x + p_s + y - x - p_n - 1 + y and the result is your formula: y = 2*x + p_s - p_n - 1

@snapo1750 Год назад

Might i ask what you use to create your videos? they look amazing...

@EricRowland Год назад

Thanks! All the animation is done with Manim: www.manim.community

@Matematikervildtsjov Год назад

Super cool video! Liked and subbed!

@EricRowland Год назад

Awesome, thank you!

@gadxxxx 10 месяцев назад

I look forward to more interesting videos.

@cthoyt Год назад

awesome video, super interesting.Thanks!

@AngelOfRepentance Год назад

Great vid. I think that most people interested in this sort of content don't need to be explained what a logarithmic scale is though :x

@AB-Prince 6 месяцев назад

there is a method of primality testing, called the witness numbers. where if a number fails the test, it's guaranteed to be composite. numberphile did a great video on this, and combining that with the formula that skips 1 should work.

@grandrapids57 Год назад

SUPER! This is really great work!

@EricRowland Год назад

Thanks a lot!

@jean-louisnouzille7545 Год назад

Thank you for your excellent video. Watching your video, I imagined an improvement of the AKS algorithm. Also, I'm thinking about Mersenne numbers with your video.

@jamesknapp64 Год назад

What stood out was the abundence of 3s. Also amazing video

@ben1996123 Год назад

I noticed the doubling pattern around 1:50 when I saw 23 and 47

@ernestoherreralegorreta137 Год назад

Very impressive. Got yourself a devoted new subscriber. Thx4 sharing!

@EricRowland Год назад

Thanks, glad you enjoyed it!

@bennyl9228 8 месяцев назад

The thing that jumped at me when you included the indexes was that the ones that were doubles plus one had the same index as their own number.

@denimator05 5 месяцев назад

I noticed the pattern looking at the very start. I didn't realize the +1 thing though and just thought "Eh, 47 * 2 is close enough to 101"

@Rkcuddles 5 месяцев назад

Ooo love the idea of mathematicians making content to explain their work.

@westonmarkham1294 Год назад

For the psychology survey: I initially started looking for patterns in the frequencies of low primes, but didn't see anything obvious. So I started looking at the higher prices and saw that each new record high was just slightly higher than twice the previous one. I continued the video at that point.

@dawiugamer2262 Год назад

Really nice video, super entertaining👍

@tulliusexmisc2191 5 месяцев назад

Since you ask, at 2:45 the most obvious pattern was the lack of 2s, followed by the sequence 3 5 3 being common. At 4:17 I was surprised you didn't mention the striking pattern that many primes first occur where n is equal to that prime. In fact, this is so prevalent that at 7:40 you highlighted the primes themselves when you were actually talking about their indices.

@robshaw2639 5 месяцев назад

I wonder if studying this sequence could shed some light on the Collatz conjecture.

@cyberjab Год назад

Thank you so much for videos like this.

@EricRowland Год назад

Glad you enjoyed it!

@benjames9153 Год назад

amazing work, love your videos

@EricRowland Год назад

Thank you so much!

@capitainerie 4 месяца назад

The problem is that we want a function that generate prime number without calculating a gcd(1, n) to see if it’s 1 , and in your démonstration you calculate gcd(n, R) so it’s basically the same but it’s recursively so the gcd have to be calculating N times, so it’s look like a O(n + n^(n-1)) or something similar due to the recursive terms, the idea might be good to think that with this function we got wave like prime number but in fact it’s because you use the prime number sequence and you make thing with it not the contrary