In 2003 We Discovered a New Way to Generate Primes
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- Опубліковано 19 кві 2024
- 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 Generate...
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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
We haven't used the Richter scale since 1970. The current measurement scale is called the moment magnitude scale.
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?
@@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.
@@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.
i just read in wikipedia about this. this blew my mind.
Richter is still used. You speak as if Richter has never been used since 1970. What is your first language?
Bro so based he makes expository math videos based off of his own research. Chad.
when you don't get invited to Numberphile "Fine, I'll do it myself."
Chad-adic and fantastic
@@iamjohnrobot You mean, 'p'-Chad-adic and fantastic! 😄😎
I'm pretty sure that's not the meaning of based 🤔 still a good video
Vishwaguru math video developer
Damn, I wish every research paper could be explained in a digestible video format like this. Great video!
Next step is having chatGPT generate videos like this for every paper ...
@@GuzmanTierno That would be awful. Because (if you weren’t aware) ChatGPT is very bad at math.
@@abj136 yeah, you're right ... luckily ...
@@abj136 that kinda makes sense tho, chat-gpt is a language model
@@abj136 For now. Give it a few years.
20 years later, congrats Eric ! This is awesome. Your own theorem.
Thank you!
@@EricRowland Lol, I don't know your channel, didn't even realize, it was you who wrote the paper ;)
Chapeau!
@@EricRowland did somebody prove this theorem ?
@@maximkosey5549 Yes, I proved it.
@@EricRowland so you can definitely generate all prime numbers, without gaps ?
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!
I got the impression that it was more an excruciating overabundance of detail, some of which we could easily have worked out for ourselves.
@@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?
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.
Thanks! They take a long time to make, but more to come!
@@EricRowland I know! I have a channel for university content and one for Numismatics. Hours and hours of editing. Will keep watching yours.
@@drjacovanniekerk Checked your main channel, and subscribed immediately.
How about do a collab with 3b1b? I feel like that would be the quickest way to get a lot of subscribers! @@EricRowland
… 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
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 :)
It isn't for no reason that they are called _prime_ numbers haha
You should believe in number theory to realize how awesome are the prime numbers
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.
crypto
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.
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.
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.
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.
I first saw the last pair you highlighted, 121403 & 242807, then I went looking for the same relation and found the others
Nice!
@@EricRowland first one I noticed was the 233 and 467 pair and I then confirmed on the bigger ones
Super interesting, high-quality, and creative video. Fantastic Job! I have been looking to see a beautiful method like this for many years.
Thank you so much!
@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!
Thank you so much! There are more videos to come. (They just take a long time to make!)
@@EricRowland very happy to hear (and completely understand!) :)
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.
very clever and excellent explanation. walking someone through the thoughts your brain went through when solving a problem is my favorite way of teaching.
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
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
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.
I love videos about patterns and primes, and this one is among my favorites. Great job, and congrats for giving a theorem your name.
The even more amazing part is that you explained it in a way even I could understand. Great video and congrats for the theorem!
I can only imagine the satisfaction you felt when you discovered all of this. Great job, this is really cool!
That video was absolutly amazing, didnt expect such a high quality from a random youtube video. Well done
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.
Thanks, that’s the vibe I was going for! The follow-up video is still a work in progress. Hopefully soon!
+1
I am interested in the generalized REPUNT primes. In base two, these would be the Mersienrs
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!
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.
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!
Great job! At first I thought that this was too hard for me, but eventually I understood almost everything. So cool.
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.
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).
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.
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.
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!
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.
What a wonderful video. I wish every math paper could be explained in such a wonderful video format like this.
Man, this is so amazing!
Love it!
I could die for videos like that for every publication!!!
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.
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.
Wow that's pretty mindblowing that you came up with that!
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.
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.
Brilliant and perfectly paced 🙏🏻
such great narration of your discovery process: thank you Eric! 😊
Thank you!
Only 3 videos, but they’re all fantastic. Thanks for sharing.
Nice work! I love this!!! Thanks for putting it together
Thank you!
Thanks for this video, I just learned about this recurrence a few weeks ago from Wikipedia and found it very interesting!
Glad you enjoyed it! That's a fun coincidence!
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
noticed this pattern while solving project euler #443, lovely video!
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
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.
Thank you!
You blew my mind in a 10 Richter's scale' magnitude, that was awesome
Thanks for the ending summary. I was hoping for the explanation about finding common divisors of 10 digit numbers being a computational hurtle.
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.
Absolutely beautiful work. Beautiful math. Beautiful thinking. Beautiful video. Someone will figure out how to build on your work.
Thanks. Enjoyed this (and your other videos). Great stuff! Cheers.
Thanks so much!
Absolutely spectacular video! Bravo!!
Thank you so much!
Congrats for you making this theorem. It's amazing.
Wow. Brilliant work and brilliant presentation.
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.
Waiting for the next part. ABSOLUTELY GREAT video Eric!
Thanks! Hopefully the next part will be done in the next few weeks!
Absolutely amazing video!
Thank you so much!
This video was great. Really really clever.
Dude! Been working on this very problem for like a decade, mad respect for the explanatory work! ✊
Great work, Eric!
Thank you!
Wow, this is so cool! Finding patterns in primes!
Very nice animation and narration
@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.
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.)
THIS CHANNEL IS UNBELIEVABLE
cool format, plz dont stop)
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!
I was looking for a large number the moment I saw 11-23-47. I was expecting 95 (which isn't prime) so was happy to see 101, but no idea on how that happened. Time mark 2:57 as you requested 🙂
Is it just a coincidence or does that last sequence from Cloitre (@21:55) generate all primes? I see all of them at their native index except for 3.
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 😁
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).
This was really interesting and well explained
@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
Awesome walkthrough❤
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)
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.
What I saw first at 3:00 is that the 3’s are on opposite sides of other primes, like the twin prime conjecture
this was really fun and educative to watch
How much of the journey to the theorem was playing around with relations, graphing to spot patterns and general exploration? The video gives a strong sense of chasing someone who is deliberately leaving a trail, haha.
Do you have any more insights about what the mod3 sequencing is doing? In your eyes does that relate to the primes being 6n +/- 1?
Very impressive. Got yourself a devoted new subscriber. Thx4 sharing!
Thanks, glad you enjoyed it!
Perhaps an interesting observation is that the first prime you still havent produced with your sequence if fairly close to the first non-prime at 19:43
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
Great visualization
thanks for giving so many things to think about
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.
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
What stood out was the abundence of 3s.
Also amazing video
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.
I'm following for the most part and it's pretty cool. Question: Suppose I wanted to calculate the 9,153rd prime number, without a prime number chart or brute forcing multiplication? Like in cryptography they use extremely large (hundreds of digits) prime numbers. How are such numbers derived?
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.
Hang on - the way we currently find new largest primes is by testing Mersenne numbers, which are not guaranteed to be prime and need to be tested. Couldn't the 2n-1s from this sequence be tested similarly? Or would composite numbers be too dense for this to be feasible?
Smallest primes are most common (3 by a wide margin, then 5 and 7), and there seem to be patterns related to 3. For example, if a 5 and 3 are beside each other, the 5 is always first *unless* it is surrounded by 3s on both sides. The same is true for 11 and 3, and 3 and 7 are similar but reversed (I.e. 7 follows 3 except when enclosed by 3 on both sides.
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.
I look forward to more interesting videos.
At the beginning, I was not looking for that kind of pattern at all. I was trying to look at the rhythm of how many numbers come between each iteration of the same number, starting with 3's.
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.
I love your research!
Thanks so much!
awesome video, super interesting.Thanks!
This guy is underrated
Really nice video, super entertaining👍
Super cool video! Liked and subbed!
Awesome, thank you!
Great video! I am interested in a follow up on the details, though I guess I could just read your paper... 😂
Thanks! I’m working on a follow-up video… Hopefully it will done in the next few weeks!
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!
SUPER! This is really great work!
Thanks a lot!
Fascinating, thank you!