How do you prove a prime is infinitely fragile?
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- Опубліковано 27 лип 2021
- Consecutive primes which are widely digitally delicate.
people.math.sc.edu/filaseta/C...
Data from the paper:
people.math.sc.edu/filaseta/C...
Here is the bonus footage. • Infinitely Fragile Pri...
And yes, the theme song (along with the remix) is on Bandcamp as 'pay what you want'. So you can download for free, or pay something to help support the channel. standupmaths.bandcamp.com/tra...
More than ever, thank you to my Patreon supporters for making this video possible. It is also their fault that the next SUM-007 will be called "The Spy Who Loved e". They got to vote on ten pun titles and "The Spy Who Loved e" beat second-place "No Time to π" 239 votes to 226. / 53485855
Help support and shape the videos I make! / standupmaths
CORRECTIONS
- At 00:33 there is a "33" which should be "53". Mistake first spotted by Andrew Foong but I'm going to claim it's some hidden message about the number 33.
- I say k is a "positive integer" a bunch when it should be "non-negative integer" to include zero. Like at 06:30, 08:48 and 14:35 (thanks Martijn Oostrom!). The formal statement on-screen at 17:15 is the official correct version showing that n is positive but k is non-negative.
- Robin Houston noticed that the on-screen text at 4:19 says Erdös instead of Erdős. Excuse me while I delete the whole channel.
- Let me know if you spot any more mistakes!
As always: thanks to Jane Street who support my channel. They're amazing.
www.janestreet.com/
Music by Howard Carter
Lyrics and vocals by Helen Arney
Animations by William Marler
Filming and editing by Matt Parker
Maths graphics by Sam Hartburn and Matt Parker
Colour grading by Alex Genn-Bash
Design by Simon Wright and Adam Robinson
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
US book: www.penguinrandomhouse.com/bo...
UK book: mathsgear.co.uk/collections/b... - Розваги
Apparently some Patreon People cannot find their Fragile Prime at the end of the video. They're all there! If you ask on this post I can confirm what yours is: www.patreon.com/posts/54207228
AND YES, 33 isn't prime. Should have been 53. But it happened at the 33 second mark, so maybe it's a big conspiracy.
33 is a prime. We're all being lied to.
in all seriousness, insightful video, amazing production quality too thanks for sharing :)
It happened at the 0:29 second mark. It was just there at the 0:33 second mark.
It may not be a prime, but it tried pretty hard and almost got there, even if it still failed in the end...
@@gibrana9214 Shhh...
The 007-inspired intro is a masterpiece. Also, the fact 007 is mostly leading zeros is pure magic.
Don't you mean double (infinite) O seven?
And its a prime
Too bad 007 isn’t a fragile prime
@@sleepycritical6950 LOL you're right... he's actually 000000(repeating)7
I know right? Nobody else (except maybe the writers of Bojack Horseman) would think of that pun and then decide to go through the process of hiring a singer and possibly an extra animator and put such an insane level of polish on it. That pun cost at least 1000 dollars and I say that's quite a good deal all things considered lol
That intro was a religious experience, I love it
Transcendental.
@@darrendarby1189 Too irrational, I think. Well, at least it wasn't too complex or even imaginary!
Best part of the video, totally. (Oh, besides 24:18- _that_ was definitely my favorite moment.)
You could hypnotize someone with that.
Was not expecting that. Definitely loved it.
I have a gorilla with very weak fingers.
He’s my digitally delicate primate.
Haha. Clever.
Still haven't discovered an infinitely digitally delicate primate.
Mine has the same issue but has a really broad hand span and is thus considerably more apposite to the video. He's a widely digitally delicate primate.
First off, you can’t have a prime 8…
@@hamblance5938 you can in an irrational valued non integer base
Matt: I will spare no effort on this intricate opening sequence, including James Bond remix of tune.
Also Matt: Let me explain this by writing on this pdf while the software turns all my lines into arrows.
Except when he tried to draw an arrow... and then it turned the arrowhead of his arrow into an arrow.
If he's James Bond, the animator is definitely Q.
@@gdclemo "Yo dawg I herd u liek arrows..."
I just think of the arrow as a Parker line.
Parker: "Here, it's really simple..."
his tablet: * _laughs in PDF_ *
Production quality of the titles tends to infinity as video number tends to infinity
... which doesn't tell anything about the quality of this particular video, but it was brilliant.
Hello time traveller
@@willwhite1987 *monotonically*
@@geekjokes8458 That still doesn't tell anything about the quality of this particular video. But it was still brilliant
Parker compliment
log(😅) = 💧log(😄)
ln(💕) = 🩷ln(🩷)
log(♂️)=♐log(⭕)
Hey, if anyone should have a 007-styled theme, it's Grimes. James Grimes.
Just so you know, it’s James Grime, not James “Grimes”
@@ragnkja Yeah, I was playing off "James Bonds".
:D
@@Qermaq You've committed crimes against movies and their characters. What say you in your defense?
@@nonachyourbusiness1164 #ParkerPost
@@Qermaq But the character’s name isn’t “James Bonds” either 🤔
I love the contrast between the high quality animated intro and the pdf annotation with letters awkwardly constructed from arrows.
Steve Mould for Q perhaps? Weaponized chain fountains and suchlike.
"We've proven they exist and that there are infinitely many of them but they are too huge to compute and we don't know a single one"
If that isn't the most maths thing I've ever heard I don't know what it! xD Great as always
If mathematicians could find the first one of them, it could be used to narrow down the search for large primes, since any number with the same ending as that would automatically be not prime.
Although I think the first one of them is already several orders of magnitudes above the largest known prime
@@caracaes I think that would be much less helpful than you think.
@@caracaes searching for large primes is actually pretty easy, checking for primality in general can be done with an algorithm that runs with a complexity of something like O(ln(n)^5) for an n digit number.
@@caracaes we know that the only even number that is a prime is 2. It narrows things down, but finding one widely digitally delicate prime (from now on wddp) would help only when we reach the next time it ends with the same digits. which are 9 cases for every x digit long numbers. (xxxx yxxxx y0xxxx y00xxxx). Checking every other numbers to that number would slow things down there. The question would be if it is worth the effort to check if the current number does not end to a wddp compared to checking those special cases to every previous prime. If the first wddp is 100 digits long it would mean that for the next set of numbers, half of a Googol -9 times, it would be an unnecessary check and only in those other 9 cases it would help things.
I like it when mathematicians work on things that generalize rather than being specific to base 10.
Well, you're in for a treat then.
I actually find number theory phenomena which are base-10-centric quite uninteresting.
@@yonatanbeer3475 General results are great, but imo it's also interesting when results only hold for specific bases, and seeing why that's the case
All bases matter. Don't be a baseist!
Its a good thing every base is base 10
Surely the same process would be simpler and have smaller primes for base-2... I wonder if a "binarily delicate prime" could be found within reasonable computation constraints?
Every time I come across some kind of interesting phenomena that relies on base 10, I always ask, “what’s it like in other bases?”
That's a really good idea. Maybe it's even doable by hand. There are only 2 values of d to cover
@@redpepper74 Same
@@Ricocossa1 I'm trying to do this, but it's actually harder than it looks, even for 2 values of d. I used mod powers of 2 to take care of one of the values, but the other one proves to be more difficult.
Okay, so the problem I'm facing is that 2^p-1 is only prime if p is prime (this is unique to base 2). We need a distinct covering system (with no repeated mod bases), and I'm pretty sure this is impossible when the bases are all primes....
Given a union of {n mod p, p prime}, any additional set will either be completely redundant, because already covered by some remainder mod some other prime, or it will be disjoint from the union.
There's a theorem that says that disjoint covering systems cannot be distinct....
I liked this video halfway through the "Digits are Forever" song. That was enough. I am sure everything else will be gravy, but you already have my enthusiastic applause. That was amazing.
The end song...*chef's kiss*
Damn. This new James Bond has me hyped!
James Bond, agent 00000...007
James Bond: The primes are not enough.
Or maybe
James Bond: License to factor.
Or is it
James Bond: Doctor Nº.
@@frechjo Of course, this one was "Digits are Forever". Other suggestions are:
- From infinity with love
- You only count twice
and of course:
- The Man with the Golden Calculator (working title until negotiations with Casio have concluded)
Tau (more 0) never Pi's ?
Live and Let Pi
Pi Another Day
Tomorrow Never Pis
No Time to Pi
After the famous grothendieck prime 87, we finally have the parker prime 33.
33 gave it a go to become a prime, and ended up just one factor short of being one. It fits too well.
Wasn't the Grothendieck prime 57?
Typographically 33 contains two brilliant primes!
Its their close proximity that breaks them.
@@petertaylor4980 Probably yes. I got confused between the two because both looks primish but isn't...as 87=3×29 and 57=3×19...both 19 and 29 being primes
@@petertaylor4980 87 is the Parker-Grothendieck prime.
at this point I'm certain Tao's mind has ascended beyond human evolution
I love how the standupmaths theme is just as recognisable and catchy as any movie theme
...?
What I love about these videos is I'll be totally following everything no problem and then I'll suddenly realized I have no idea what's going on or how far back I got lost.
...Ok, ok,... 1+1=2, got it. 🧐
...😵... _Infinity!_ When did we get here? 🤔 I'm so confused.
SAMEEEE
Same
I was immediately lost after the theme song, cause my brain was preoccupied with wanting Matt to release all these great variations (variants? call the TVA!) on Spotify (and a longer version of the orchestral version)
These numbers are just too g-dang big.
matt: calls 3 a great prime
earlier matt: '[2 and 3] are not real primes, i call them sub-primes'
What's next, he says Tau is better than Pi?
@@BenKonosky *gasp* HERESY!
Parker Primes?
@@dielaughing73But they weren't created by Matt, so he didn't give it a go creating them.
all primes are real tho
I forget how much I love watching Matt struggle just as hard with on screen annotation as I do. Also, the math are always way more engaging than any of my calc of diffeq classes I took through college.
8:48 - Technically, k has to be every _non-negative_ integer since you need to consider d×10^0
Okay. Gonna start a civil war. You mean that k has to be a natural number, right?
@@nickfaire Haha, not going there! But even the paper shows k in the union of the positive integers and the set {0}.
@@MCPhssthpok I saw it, the paper remains just neutral about the maths civil war. XD I just like the meme, and the number 0. •>•
@@nickfaire Well it depends where you're coming from, doesn't it?
Here in India we learn that the set of natural numbers = {1, 2, 3, ...}
and the set of whole numbers = {0, 1, 2, ...}
It really depends on whether you consider 0 "natural" or not.
You have to consider that most cultures didn't consider 0 as a number until an Indian mathematician Aryabhatta introduced the concept of nothingness being a number on it's own.
Sure some cultures used placeholders like we use 0 to fill in the gaps. Like "19_87_5" meant 1908705.
But I don't think any of them considered ascribing this nothingness to a single digit: 0.
The romans had an entire numeral system that did not involve 0 at all.
So surely 0 was not a "natural" number to them.
@@adarshmohapatra5058 In nowhere the set of whole numbers is {0,1,2,...}. You probably mean {...-2,-1,0,1,2,...}.
Also, Aryabhatta used a placeholders too, not a unic symbol for zero. The zero was implicit in his work, but was not mentioned directly. As far as I know, the arab al-Jwārizmī was the first to normalized the use of the number zero, and then it spread to Al-Andalus, from where the venezian Leonardo Fibonacci introduced it to the rest of Europe.
It was just a joke about the """war""" in mathematics about if zero is natural or not. I'm not talking about cultural-based definitions, which are not formal and therefore are not the subjecr of study of mathematics. But, as I said, it was just a joke.
Wow, I never noticed that 502,123,813 was a digitally delicate prime - and now it's mine! 😍
502,123,813
"...That can lead to 43 and the other 2 digit primes..."
*Shows 33*
Hol up
It’s a new type of prime! A Parker prime
Perhaps it was supposed to be 44 - another digitally delicate parker number
Oops, I missed your comment before I posted.
@@LukeSumIpsePatremTe 53?
This fucked me up tbh
I have a learning disability that makes maths extremely difficult for me to understand, yet I really enjoy watching these videos. They're really intruiging and sometimes I even feel like I've learned something.
Oh! Is it dyscalculia?
Intriguing indeed.
What is the disability
@@Osmium78 it's non verbal learning disorder
It stems from my IQ having an extreme gap in between subjects
Like I understand memorization work very well but math for example doesn't make sense to me whatsoever
I burst out laughing when the James Bond intro started. Bravo.
Finally, a Bond film I want to watch
Go see the title sequence to Casino Royale again :)
The production quality of this video is over the top, from start to finish. Well done!
You mean from start to 16:15, right?
Now we're waiting for 1M subs;)
I did not expect a James Bond style intro in a maths video, good job :)
Hello. For the intro alone you deserve a like and a million subs. So happy you continue to make videos for all of us.
With logic?
Well Mensa seems to think it, and I agree, you are a genius
Yeah right, like that would ever work. Low IQ individuals these days...
Pffft, who uses logic when you can just do proof by calculator?
papa
Big Brain
ive said it before, i’ll say it again,
helen arney is absolutely amazing, what an intro!
The intro is incredible! I would love to sing it, but the lyrics are hard for me to get right. Please, could you help me with that part "To the left is not (be-???) Zeros continue to (whenever?)"
This (the tube-like covering system [hence system], starting at around 12:00) would be a pretty cool way to generate musical motifs.
1) Choose a working system with the same amount of tubes as your target scale (say C major);
2) Assign each tube to a note within said scale;
3) Adjust the system chosen in step 1 to favor certain notes over others (by making some integers fulfill more than 1 possible requirement of each tube, and checking each tube in a certain order) if desired;
4) When an integer comes along to a tube for which it fulfills the condition, play the associated note;
5) Wait until it starts repeating again (which I think it would always do*), and take the set of notes generated;
6) That's the motif.
*If it doesn't, then just take part of it.
The prime numbers 2 and 5 are interesting in that if you change any of the infinitely many leading zeros, it will never be prime. But if you change the 2 or 5 itself, it could be prime. You have to make up a name for that too.
Hey i like your Norway Fluttershy pfp
1-10-left fragile
10^*1*
*100*
Left fragile
So we're acknowledging leading zeros now?
Everything is better with a lot of leading zeroes... Example: My 99 toyota has 00000000000000000000000000000000000000000085 horse power.
@@SuperPhexx you mean your 0000000000000000001999 Toyota?
@@SuperPhexx What horse power is it? This is why you should metric instead.
@@Liggliluff Wooden childrens toy horses.
This was such a high quality video. I personally love videos that just take a moment to discuss new papers in math!
Helen Arney has a great voice! Your video is good too, Matt. This kind of stretched my 45-years-ago maths education, but your videos are always fun even if I only understand them viscerally.
Where the hell did that intro came from!? That was amazingly beautiful and I can't wait to see more!
that intro made me ask my self, Am I watching a math video or a netflix series?
Would you believe they turned me down? True story.
@@standupmaths They don't deserve an edutainer of your quality Matt.
This is one of the best videos Matt's done - I feel like I genuinely understand a complicated piece of maths that I didn't before.
Really appreciating the production values, and a good topic to explain in depth as it's surprisingly understandable!
Did you just put a bond intro in an educational math video about grouping prime numbers?
You're my favorite
The intro is a masterpiece. I absolutely love it!
This is the best video on this channel, stunning work with the intro and a really good deep dive into concepts that help make this ridiculous sounding claim, digestible. Well done!
I think my favorite part of your videos, Matt, is I can watch a complicated math video for 30 minutes, and it's the perfect balance of humor to keep me interested but also actually in depth exploration of math.
ok so I wasn't expecting a music score in this
The James Bond intro was amazing
I love your videos! Even when I don't always understand what I'm learning I can still feel that knowledge being mashed in to my brain. And that James Bond-ian opening title was absolutely brilliant!
I mean.. Amazing intro. Cant express enough my both gratitude and excitement for just witnessing this undoubtably first-class piece of art! Hats off!
OMG. This video went completely above my head. Usually i can understand maths videos, and Matt is great at explaining stuff. But this… felt more flashy, hand wavy, documentary. 😰
I tried giving it another watch… and another. But… whoosh!~ 🌬️
I'd say when he comes back from the animation of covering systems factories, he starts going way too fast.
Wait a minute, 007 can be changed into another prime by changing it to 107. Maybe James Bond was truly Matt's "prime" example of inspiration for this video.
Not to forget 017!
Also 002, 003, and 005.
00,000,000.0000004 = 0,000,000.0000004 = 000,000.0000004 = 00,000.0000004 = 0,000.0000004 = 000.0000004 = 00.0000004 = 0.0000004 = .0000004
Wow this is one of the best videos you have made yet! The 007 title sequence was amazing as well!
This is my adviser and academic brothers' work! So cool to see you doing a video about it!!!
24:29 they're like graphics cards, we know they exist but we can't find any of them
The value for A is probably computable. "A" will have about the same number of digits as all of the primes summed together (so printing out A would take about the same amount of paper as printing out the list of its factors). The time complexity for multiplying numbers is almost linear so that also shouldn't be a factor.
According to Wikipedia, the time complexity of solving a system of congruences is quasilinear if you do it carefully so B might be computable? I'm not a number theorist though so I'm much less confident about B than I am about A being computable on current hardware.
The number A is about 750000 digits and can be computed in about 10 seconds with plain unoptimized Python. And given all the congruences B must be less than A. Unfortunately calculating B involves many more multiplications and divisions, which make this somewhat a nightmare to compute. In particular given the large value for A most multiplications apart from the very first ones operate on values that have several hundreds of digits.
A=(756576 digits, composite, odd)
B=(739341 digits, composite, even)
Took about 2 minutes to perform the calculation and verification steps …
That is, after the code worked correctly (which took longer).
Love the singing/effects. You found someone with quite a nice voice. This video goes into a seemingly more complicated topic but the general gist of what you were explaining made sense. This video is great for someone who is maybe more experienced at number theory problems. Modular math is one of my favorites.
Holy crap, the intro was incredible. The production value is going off the chain, great job.
I feel like Im watching a maths documentary, not a normal maths video.
P/S: I know that Helen Arney sung the opening title, but where can I get the background music?
We've not released the background SUM-007 mash-up music because it's not that exciting out of context. And without me talking over it: maybe strays a bit too close to the original.
@@standupmaths hmm, 'a bit too close' huh?
@@standupmaths But I think it's not too close. For me, it's far better than the original, especially with Helen Arney singing at the back. (Don't get me wrong, the original is still very good, this one is just better). It's still your choice to release it, though.
You should do a video on QR codes. You could talk about how to encode/decode them and how the error correction works. Maybe memorize the code of a link to your website do you can draw it out.
Absolutely top notch. I sincerely hope you can make a series out of this. On a slightly better release schedule than the source of the homage can manage lately.
This is the first video I've watched of yours that just completely went over my head. I'm not sure why I can't wrap my noggin around it. I need to sit down and work through it I suppose.
Two is the best prime, it became even against all odds.
I would like this comment, but it's currently at 3 and that just seems like the perfect response in itself. Why am I even leaving this message? I don't know.
Now it’s at five, another prime.
I felt like I was watching a 007 movie in the intro (or ...000000000000000000000000000..0000000007)
I wasn't expecting this level of production quality. LOL thanks for blowing my mind
Great visual style! And nice pun with Diamonds are forever.
"Though Matt explains, it hurts my brain!"
I felt that.
Dear Mat, that was the best title sequence in any math video on youtube!!! I am blown away!!! I am still locking for an answer to the following question: Up to which number do we know definitely every prime number? Thank you very much and best regards, Markus
Production quality has gone up by a factor for 10 in this one! That song and graphics were very well done.
Wow, the production value in this video is just amazing. The animations! The song! The maths!
Oh man, I love this guy, I wish he was my professor when I was at university.
As a huge fan of math, puns, and hype, this channel never fails to disappoint. Keep being awesome, Matt and Co.
congrats for your 007' style opening! amazing
This video is amazing. Thanks for the great content
Everyone: the intro!!😍🎶
me: the outro also!!🎶😩👌
Helen Arney knocked it out of the park and into orbit
Channelling some Ladytron there!
And now my eyes won't uncross, and I think my brain is morphing into a non-newtonian liquid. Thanks, Matt. That was an EXTREMELY dense video (or maybe it's just me that is dense). Loved the Bond themed music, would've loved to see you do the silhouette entry, but, hey, can't do it all.
There was so much effort here and that's pretty spectacular
Leaving both a like and a comment in recognition for that amazing intro song (and animation to match)
God: "Hmmm... I keep dogs endlessly amused by giving them tails that they can chase. What can I do that will have the same effect with mathematicians? I know, I'll make some of the numbers prime!"
..End if PI confirmed then by god, otherwise this would be needless. He knows they will find it.
I got lost halfway through the video, but still kept watching because it was interesting even tho i didn't get it
This has to be one of the best episodes ever. Love it!
OH my god! This video is ultra good!!!
Continue!
When they finally cast woman as 007, I would vote for Matt to play Moneypenny.
🤣😂
He'd have to be named Dollarydoo.
I wish they would cast a man to play Queen Elisabeth
@@oz_jonesthey have, didn't you hear?
The title sequence cracked me up. And then it just went on and on! Awesome.
Very nice walking through the paper! It shows how accessible such papers can be!
Fantastic episode. Thank you!
No, no, no... *headdesk* Euler clearly tells us that it's "the *pi* who loved e"... *sigh*
oh dang, the puns XD
Pi or spy, can't wait for the video. Keep up the always excellent content.
And let's not forget Golden i
@@frankbrockler Ohhhh deer gawwwd! Well played.
{painful eye-roll} Ugggghhhh!
The song at the beginning of the video is incredibly catchy
Hi, I'm just here to respect your deviantArt emoticon pfp 👌
The animation and editing of that title sequence was bloody brilliant.
GREAT VIDEO! Thanks for the awesome content!
ahh yes "overproving" something in maths, or as it's otherwise known, flexing. :P
No, it's not actually flexing at all, and I don't really want to explain why. It seems to be implicit in the video anyways.
who composes this music for you? I know you have loads of experience being ⅓ of the nerd show, but still…
the production value of your videos keep going up
the previous song that's stuck in my head is "total eclipse/ellipse of a chart" 😂
The vast majority of the music is done by Howard Carter. In this case in collaboration with Helen Arney. (Howard also wrote the Festival of the Spoken Nerd theme and a bunch of music for that so they are already frequent collaborators with Helen.)
@@standupmaths OMG it's you!~ 😱😍 thanks for all your efforts Matt!~ I'm a big fan of you 🎉
Tunes are nice… thanks shout out to Howard 🎶
This is my new favorite video of yours
Best Intro on any Matt videos till date.
(remember that actually every intro music on this channel is top notch)
0:37 since when is 33 prime?
Obligatory "Parker Prime"
53 is missing. So maybe a typo?
@@CodyDanielson Look in the description!
@@CodyDanielson He answered in the pinned comment, that that was the case,
though in that case, he is still missing the hexagram connections.
20:24 Wait, why isn't it a problem that several of the entries in the table have the same m and different r's? Doesn't that lead to incompatible constraints on B?
EDIT: got it, it's fine as long as the associated "p" is different.
Fantastic production !!!! Wow!!!
Snazzy new intro/graphic design - lovely stuff!
I would start with 2- it's prime, and adding any digit to the end results in an even number, and thus that number is not prime. Of course, I then realized this skips past the "changing the digits within the number" part, and you can indeed get to 3, 5, and 7 from 2 by changing only one digit.
Yeah, I thought of this as well ;)
I should add, the "adding any digit to the [left!!] end results in an even number" is some usage of covering systems, the first one, the "(0 mod 2) and (1 mod 2)" covering system. If the last digit is guaranteed to be 2, then the number is equal to 0 mod 2, and is even, and aside from 2 (and because of 2) there are no (other) even prime numbers.
So 2 and 5 are examples of two primes that can only be changed into another prime by altering only one digit (the ones place digit), 2 can be altered by single substitution to 3, 5 and 7 and 5 likewise to 2, 3 and 7. Any other number ending in 2 or 5 is composite so any change to any other digit (ie to any of 2 or 5's leading zeroes) results in a composite number and so they are fragile on an infinity of digits less 1.
Let us call a prime that can be turned into another prime by altering a single digit (including leading zeroes) robust. Therefore 2 and 5 are minimally robust, if they were robust on 1 less digit they would be infinitely fragile (as defined in this video, aka widely digitally delicate).
So are 2 and 5 the only minimally robust primes or are there other primes that can only be changed into another prime by altering one other digit (including leading zeroes)? If there were another minimally robust prime it seems likely it could only be transformed into one other prime, not three primes like 2 and 5?
Seems tricky to prove.
Regarding 5:05, what do you think of the term "prime terminus" or something along those lines. That would make it clearer that any number terminating with this sequence of digits is prime.
Love the "Diamonds are Forever credits" sequence! oh, leading zeroes! I see what you did there!
Love your videos. Been watching for years, but this is the first video that I have no idea what in the world you're talking about. Loved it, but was completely lost halfway though (when you started going through the PDF).