2.920050977316 - Numberphile
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- เผยแพร่เมื่อ 21 ก.ย. 2024
- Dr James Grime is discussing a new prime-generating constant.
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Extra footage from this interview: • Prime Generating Const...
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Hi everyone! I'm one of the authors of the paper. First of all, special thanks to James for helping a bunch of random friends from another country publish our first paper, AND making a Numberphile video about it!
If anyone's interested in a challenge here are some things we didn't manege to prove:
-Is the constant transcendental?
-What happens to the sequence if we pick our starting constant f1 to be a rational number? Does it always get "stuck" at a certain point?
Also feel free to ask us anything, we are very glad to see people commenting about their own research and experiments on the formula!
And if you feel you made any new or interesting discovery about the formula or constant, please do post about it!
"Also feel free to ask us anything"
How does it feel knowing you're famous now? :)
Did you check how many decimal digits are needed to generate a given number of primes? Let's call this number N. If the number of digits in your constant is something like log(N) or sqrt(N), then that would be awesome, because the constant could be used to efficiently compute a lot of primes on computers.
It doesn't matter how efficient this constant can calculate primes, because calculating the constant depends on knowing the primes
@@Simpson17866 Haha ;P Nah, I don't think anyone will remember my name after watching the video. But it really is exiting being featured in a numberphile video! Also I'm having a little bit of impostor syndrome, Juli was the MVP that came up with this brilliant idea! I just brute-forced some digits, looked them up in OEIS, and found a possible candidate for the number we were after, that turned out to be the average of the smallest primes that do not divide n. Then I wrote some Python scripts to find lots of digits using that formula.
Congrats guys.
James is always telling constants they're his favourite but he keeps dumping them for newer, hotter constants
He's cheating on his constants
"I'm gonna give you four words to live by: New is always better" - Barney Stinson ;-)
He loves constants but not commitments :'(
He is unconstant in his love of a constant.
His favorite constant is, in fact, a variable.
It's always a treat to see Dr James Grime know every constant to 10+ decimal places
Didn't expect to see you here
Jperm! Big fan of yours
Also jperm is my fav pll algorithm
Wow, you're up early! Hope you're well dude :)
DAMN! my fav youtubers on my 2 favorite activities together!!
@@gauravpallod4768 same!
“I’ve got a new favourite constant” (with a beaming face of joy). This is the purest form of numberphile and I love it 😍
numberphile greentext
His joy for numbers is so wholesome
I bet you’re American and spelled “favourite” with a “u” just because you’re that pathetic. And, before you ask, it’s because I enjoy it.
@@honorarymancunian7433 Everyone would have a "joy for numbers" if you skip the decimal part...
This guy is a hack...
What's with the weird (and aggressive) comments in this chain??
This title is so classic Numberphile
*Tongue taps the like button
Nice 👍
Good thing they have not done a video on the 10billion numeral expansion of pi.
Haha true
Yeahh
Thought it was a re-upload for a minute
Dr. James Grime still looks like the age when we used to solve puzzles on his channel
Mathematicians age a lot slower than others. That's why they live so long; as long as they don't get stabbed, shot, contract a fatal disease or commit suicide (like Archimedes, Abel, Galois, Eisenstein, Riemann, Clifford, Ramanujan, von Neumann, Taniyama).
maths is an easy job
As I was recommended to 4 year old video with him, I came here. Even now he look same..
I was thinking the same as you before coming to comment section!!
@sidarthur8706 make a constant that gives all truntactative primes
??
You've seen elf on the shelf, now get ready for James Grime on primes
Grimin’ with the primes.
*this is your brain on primes*
[cracks egg into a pan]
*Jame Grimes
@@Muhahahahaz
No, it’s just one James Grime.
??
His smile never gets old
He has a formula for that
@@dArKoMeGa89 JOYmes' baby formula
OK cheeseball
false.
The sobering real-life side of research: ... "Received 16 Sep 2017, Accepted 29 May 2018, Published online: 30 Jan 2019"
Published in American Mathematical Monthly (121:1): November 2, 2020.
Yup.Just yup
@@lonestarr1490 Covered by Numberphile in November, uploaded November 26th, 2020, replied to you December 6th, 2020.
replied to @@aadfg0: 18 February 2024
"Pretty important junk"
"We need this junk"
- Haran & Grimes, 2020
We'll save this junk for later, when it stops being junk lol
I got this "junk" to work on my calculator, when I wrote a FOR loop. Sadly, this "junk" broke down after the 12th prime number. :)
Just one Grime.
??
For a moment I though random guys solved one of the most difficult problem of all time. Even if this is not the case, they were very smart!
You would definitely hear about it from everwhere
Random high schoolers no less.
@@icisne7315 They're very clearly above-average high schoolers, but yes it does look somewhat more impressive than it actually is. By the way they have a comment thread here where they answer technical questions about it. They're very aware it has limited applications but you can tell they're smart.
They might have, at least in part.
Point is, up until now the primes generate the constant. But the constant actually _can_ generate the primes, as it was shown in the video. So, if someone manages to re-find this constant elsewhere where it might be representable in a closed form or at least computeable to some ludicrous precision, then we've won. (Apparently, the average of the sequence of smallest primes that do not devide _n_ doesn't do the trick.)
??
I can't imagine being smart enough to see a maths video on TH-cam and go "you know what, I can do better than that" and then find a new, seemingly very useful, constant.
It's not really that it's useful, since you can embed an infinite amount of information in a decimal number. More of a mathematical curiosity. It's conceptually similar to the number 0.2030507011013017019023029...
Not really useful, primes with a reasonable number of digits are easy to calculate already. But a lot of pure math is just stuff that's mildly interesting.
Yeah I get that it's not incredibly useful after watching the rest of the video, but it seemed like it is so my comment is still valid
Everything is useless until it's not
169th like
A disadvantage for Numberphile is that nobody will write that number in the search bar even by mistake and find this video
Here come the "let's be honest, you didn't search for this" comments.
The search also includes the description and the transcript of the video.
If you aren't searching 2.920050977316 at least once a week, then are you really living? Very glad to see Numberphile FINALLY post about this.
Sad :’(
😂
Engineers: Three take it or leave it
@@Dducksquad no, 5 is for military purposes
Safety factor, 4
tbh, we usually use 1.5 and for well understood stuff like the fatigue limit 1.2
Or you can
To be fair, 3 is not bad... We could also say : "It is in the order of magnitude of one" :-)
“I used the primes to calculate the primes”
-Thanos
A Numberphile viewer
OMG ....It's James Grime💚💚💚💚💚....It's soo good to see him back making videos with Brady....
Numberphile you are my favourite channel 💚💚
I found this constant to be regular-level of interesting for a Numberphile video, and then when he pointed out that it turns out to be the same as the average of that easy-to-describe sequence, my mind was blown. That's why I keep coming back to this channel!
Anybody else instantly click when they see James grime?
YEP
Yes. To be frank, I hadn't clicked in a while lol.
*James Prime
Yup
Lebron James Grime
Me to Mill's constant after watching this video: I don't want to play with you anymore.
Mood
@@takatotakasui8307 it's a Toy Story 2 reference.
I did recognize the meme
Seems like a relative of 2.3130367364335829063839516.., whose continued fraction is all the primes in order. i.e. take off the integer part and take the reciprocal repeatedly and this generates, 2, 3, 5, 7, etc. Again, made from the primes, so isn't predictive. Here's another number whose continued fraction gives the primes in a slightly different way: 2.7101020234300968374157495... (Hint: add the integer parts you get.)
That is so cool
Math teachers in primary school: prime numbers have no pattern.
Every mathematician ever:
You’re wrong but I have no proof
*yet*
On US Thanksgiving Day and I wake up to a new video from James. Now that's something to be verythankful for!
Framed demonstration of Graham's number, by Graham himself, on the wall. My jealousy knows no bounds.
Yeah, that IS pretty cool.
Graham and Grime,
They almost rhyme,
As does the preceding couplet, every time.
Fred
Whoa, didn't notice that! Pretty cool
Nobody exudes more childlike joy at maths than James Grime.
2:40 James slightly singing "601 529" made me instantly think about the new emergency number from The IT Crowd
0118999881999119725
3
i am from argentina, really proud of our future!!
I didn't realize until the other sequence at the end of the video that the hypothetical "predictive" version of their constant was almost identical or that they were completely on the right track for it. I thought that the next new biggest prime found would throw their number way off. Bravo to them for doing this, it makes it so much more impressive with that knowledge.
This number is so cool. Now someone has to find a way calculating it without using primes. Then it would be really a prime predicting number.
"ahhh constant! We love a number" will be printed on my tombstone.
I like constants, we need more of those in these times of uncertainty
Ordo ab chao.
This is such a crazy improvement to classical “get primes” functions you can write today on computers.
This is a certified James Prime (James Grime) moment.
He definitely should make a typical ad of "I am James Grime and I approve of this constant" :)
A Prime Grime moment
Yesss!!! Dr. James Grime after a long time ig!!
We missed James!
Hey intel, how are you
Dr. Grime is the best. I love how enthusiastic he is.
As you point out, this method of compressing the sequence of primes into a real constant depends on the sequence being increasing and p_n < 2 p_{n-1}. If you wanted to compress a sequence of positive integers which doesn't necessarily have those properties, make your sequence's terms a_0, a_1,... the terms in a real constant x's continued fraction
x = a_0 + 1/(a_1 + 1/(a_2 + 1/... ...))
My new favorite constant is social anxiety.
The constant with which you never find your prime :P
@@CLBellamey HELPPPJSJSJF
when james dropped the second instance of the constant my brain just popped
Numberphile hasn't changed in years and I love it.
9:58 - And this proof is left as an exercise for the reader
Papa flammy's fan?
Thank you so much James Grime for the great number!
Dr James Grime es una gran inspiración por la alegría y el entusiasmo que transmite en cada conocimiento, me hace sentir un apasionado por los números aunque no sea la ciencia a la cual me dedico. Todo mi respeto desde Argentina a los amigos de numberphile
Fvjoid, freufjo donfn eefj donicv onjf fon juowf ijvjie vif. Mej cei dcim foqr frij ecj cic, cehj eijc eomc mefok fij. Efj jfo jfi vjn rvhr ckj. Numberphile veoj ejv eovj bej ewjfie James Grime.
It's lovely to see James back. This feels like what numberphile used to be all about
11:20 that series looks like how the musical scale is built when pulsing a string. half the notes are the note that the string is tuned, then come the thirds, the fifths and so on following prime number proportions. looks related
Yes, it does look like that. I think because the constant is in a sense a geometric average of all primes, which are the harmonics of a monotonic increasing sequence - the PNT, analogous to RH zeroes
This is my new favorite constant! So happy to see Dr James Grime back at it again!
It's so cool that it doesn't even skip twin primes since they're so close together
Well, it's made so it doesn't skip those. After learning about how they made the constat, the spell kind of disappears.
Loving the framed signed Graham's Number brown sheet. RIP Ron.
this guy is so damn cool
He's proof the nerds won :-)
A cute bonus fact which I discovered after fiddling around with this for about ten minutes: try starting this same procedure, but instead of starting with the constant in the video, start with the number e.
The result may surprise you...
Bravo.
In Excel, with A1=exp()
A2=INT(A1)*(1+A1-INT(A1))
and continuing down, it pukes out at 18, though I haven't analyzed it with regard to floating point imprecision.
@@Bill_Woo I actually worked this out backwards, I thought to myself "how could I, rather than generating the sequence of prime numbers, generate all the positive integers?" So then I went to the formula for doing this, calculated the first few terms, and realised it was the same as the series expansion for e.
It's not a coincidence, it would carry on forever if not for floating points.
@@alexpotts6520 Shrewd, working backwards. Great accomplishments come from that at times.
Then again, working for managers in today's short sighted large corporate myopia, it has almost always seemed that my employment framework is always to be given (or wink, wink, implied) the answer, and asked to build the solution. In other words, I believe that a sadly vast number of the programs that I have written were under the ominous umbrella that I was asked to do it in order to justify a premise, rather than actually seek "the answer." Ergo, working backwards has ironically formed the launching point for a frighteningly large amount of my career's work :( However,
In my defense, I have been something of a PITA maverick rather than corruptly playing along, when appropriate. And certainly many times I've worked backwards and actually "disproved" the premise - that no plausible set of "forward" inputs could support the end result that was first presumed.
The big thing I suppose is that working backwards is more common than one might think. And it's the fastest way to solve some problems. For example, recognizing that "the answer must be both nonnegative and less than the U.S. population" often initiates a "backwards-oriented" approach that eliminates inefficient false paths.
Great teachers produce great minds.
This guy is the happiest man alive.
Me at the start "Hm, what's the catch?"............"Ah!"
Inspired viewers becoming scientists. This story proves the channel is a success. Great job Brady 😃
BREAKING: the constant is actually equal to (pi + e) / 2
this isn't actually true but it's pretty close
What if you just found the actual constant that works for all primes? Did ya test it?
@@amir3515 yeah i tested it, it only works up to 7 though :(
(15pi + 9pi^2)/(12 + 11pi) works up to 31
That was so cool how the average of the sequence was the very number of the video. Amazing!
James is baaaack!
About "the smallest prime that doesn't divide n", every term defining the constant has an intuitive meaning:
Let p(n) be n-th prime. Take multiples of p(1)*p(2)*...*p(n-1). Out of every p(n) such multiples, p(n)-1 are not divisible by p(n), and therefore their corresponding value in the integer sequence is p(n). Their value of p(n) cancels out the p(n) in denominator when you take their average, leaving you with p(n)-1 over p(1)*p(2)*...*p(n-1).
Since you're always multiplying by "1.(some junk)" does that mean the next prime is never above double the value of the previous?
9:36
Actually, Bertrand's postulate decrees that the prime after a prime p is always less than 2p-2.
@@k-gstenborg3847 damn thanks, I was just listening to the first few minutes while on break
I'm really bad at maths, I had no idea why I used to watch these videos as I don't understand anything about them.
Then I saw James Grime and rememebered that I draw happiness from his passion! I've missed this guy!
when numberphile posts
math nerds:
*the return of the king*
When it even stars James Grime-
It's funny because I was just reading up on arithmetic encoding. It's a method to encode a sequence of symbols (e.g., characters of a message), as a recursive series of fractions, given a known probability for each symbol. That is: take a given range, and partition it into a series of bins; whichever bin the number's integer part lands in, that's your symbol. Subtract the offset of that bin, and divide by its width: now you have the next number, which falls in the same range, so you compare it to the bins and get another symbol off, etc. This exact process requires some refinement to deal with carrying (for practical purposes, we don't want an infinite-length fraction -- we only want to have to deal with, say, bytes at a time, or a bit stream).
This works very similarly, with the trick that, whereas arithmetic encoding works over a fixed domain (e.g. a finite field), this has to expand the scale every time, hence using multiplication instead, taking the residue times the previous term to recover an ever-increasing sequence.
Ironically, a property that's useful for storing information, is counterproductive for most mathematical purposes: if the terms of the infinite series are similar to each other (i.e., lots of common symbols present), presumably very little information is stored per term, i.e., the compression ratio is high (arithmetic encoding can very closely approach the Shannon entropy of the sequence; if a given symbol is very common indeed, it might end up with less than 1 bit per symbol). Which is equivalent to asking: how many primes do we correctly recover, from a given precision approximation of this constant? However, when we want to calculate that approximation in the first place, we want very sparse terms, so that the series converges quickly!
Has... has anyone used this, mechanistically? -- that a slowly-converging series has low entropy or something like that?
I see Dr James Grime. I click instantly
It's muscle memory! :-)
We're in tenth year of Numberphile with James. How time flies...
I love the way Brady says "pretty important junk!"
This is amazing. Thanks sir. You have made me gather courage and confidence to start my channel.
"We love a number," yes, James, that's kind of the thing
Proving that the Buenos Aires constant is trancendental would imply the riemann hypotesis. The proof for this is a bit complex.
*A question*
How many decimals would you need to accurately generate the first N primes?
If it is 1000 decimals for lets say 1,000,000 primes, if someone could then compute 1000 decimals of this constant, then other people could use this constant to quickly generate primes, without needing to download huge amounts of data.
I'm working on a project now, and need to generate the first trillion primes. I can't download them anywhere, and generating them myself using conventional methods takes way too long. If I could copy a pre-computed constant like this one with way fewer digits, I could quickly generate primes that way.
I was thinking the same
@@sjoerdiscool1999 Try using a prime (eratosthenes) sieve for generating the primes, 1 billion primes should be generated in a few seconds with it. Took me 10 seconds for 2 billion with one I made once. I'm also almost 100% certain this constant does not store prime information more efficiently than just a sequence of primes.
@@sjoerdiscool1999 use prime sieve with optimizations (bitmasks instead of lookup tables, skipping evens etc); even with basic (erathosthenes) sieve, it only takes about half a second on an average machine to generate primes up to a billion; there are a lot of improved, hyper-optimized versions out there, which can achieve amazing runtimes
With just some very quick testing it looks like the number of significant digits in the constant is equal to the number of correct primes generated before the sequence fails with a composite number.
Dr James never seems to age.
maths isn't done until we find a function
p: ℕ → ℙ, n ↦ p(n)
where p(n) is the n-th prime number
There is one, just not a closed algebraic form
how are you defining function? In a mathematical sense, and computational sense, this function exists, defined by how you just described it
But you've just described it 🤔
@@25thturtle48 but i didn't describe the algorithm. i want an algorithm which has time and space complexity
@@toniokettner4821 there isn't one.
Simon Plouffe has something similar along this line of investigation in his newest paper "A Set of Formula for Generating Primes". It's on the Arxiv. If you're not familiar with the name, he's the "P" in the "BPP formula" for the digits of Pi.
Isn't this just an "encoding" of the primes? I feel you could create infinitely many "constants" from which you can extract the primes again.
Yes, it is an encoding of the primes, that is what they mention towards the end. But it is not obvious you can encode them so that you can extract them in such a neat way.
The averaging process of "least prime that doesn't divide n" is an interesting way to encode such a constant though. But yeah it can't, to our knowledge, be used to predict new primes, which would set this apart as something revolutionary rather than something neat.
There is a typo about Bertrand’s postulate in the cited paper: the authors wrote p_n < 2p_{n-1}-1, but it should be
A shame that the paper is paywalled. Would've liked to read some more about their findings.
There's a version on the arXiv as well.
You could probably find it on scihub, lol
SciHub is your friend
Not much information but I thought you'd be interested. I tried it out in Java and unless I made mistakes, it was only accurate to about 37 then started deviating greatly. I also tried the generator algorithm and got a similar result.
@@saudfata6236 Did you run out of precision? This sort of algorithm only works as far as you have deeper and deeper digits to feed it.
the question is tho, if this is transcendental? It can be described in short terms so it's likely not. Also tells us something about information theory. A set of numbers (for example primes). Can be encoded in a single constant given the right function. However you can also define the set of primes in three logical predicates as well. The question here is: can you do this for any number of sets? For example one that has duplicates?
Another question would be if it's possible to do a function that crosses the Y=0 line at every single prime.
Wonderful. As usual with Mr. Grime, the non-ageing mathematician :-)
This is one of the coolest videos that inspires me to keep looking into math :) I have been trying to get back to college for years, and this is one of those videos that makes me believe I can still do big things in my field
The only thing i want to say is that i wish they tought maths in school with this excitement and these problems. Many more people would like maths.
The problem is they don't have the time for, frankly, unnessesary maths like this. The curriculum is very strict and time sensitive, even for the normal stuff, which you might actually have a chance of using irl. The teachers are doing their best to squeeze all they have to teach into the few classes you have in a school year. Stuff like this is reserved for either recreational mathmaticians or university level number theory courses (and even in those, most of the stuff is watered down).
I just realized that if this works forever then we can prove that the next prime is always going to be less than 2p, where p is the previous prime. O.o
We already know that, that's what Bertrand's postulate is.
Also, the authors used Bertrand's postulate to prove the constant exists, so such a proof would be circular.
9:41
Chebyshev said it and I'll say it again,
There's always a prime between 2n and n.
Isn't that what the video said?
@@thomasi.4981 -- Nope, the video says there is a prime between n and 2n where n is a prime. Shankar Sivirajan is quoting Chebyshev, who apparently said there is a prime between n and 2n for *any* n, not just prime n.
@@fudgesauce Oh, okay.
@@fudgesauce That, and it's a mildly amusing rhyming couplet.
If you play F1-F4 (2.920050, 3.840101, 5.520305, and 7.601529) as raw frequencies for a speaker, the speaker moves in the rhythm of a human heartbeat.
I saw this video in my recommend 2.920050977316 seconds after it was posted.
James Grime and getting excited about a number, the classic Numberphile video.
So, what do we need to replace the "1" in the construction with so that the constant ends up being e=2.718281828459... instead?
Talk English not math
The extra footage actually answers that question!
@@refrashed No, it answers the question about the sequence generated by e but still with 1.
Great question! Probably around 0.9 or 0.8. But what do we need to get 3.14159..? A little bit above 1. It would be mad if the answer is 1.14159.. !
@jj zun to get the full replacement constant for the 1 we would need all the digits of e and all the primes, but you can get the replacement constant to a specific number of decimal places with just a finite number of digits of e and a finite number of primes
This video makes me so happy.
James Prime back at it again
Contending with the parker square
By a viewer?! Dang. I can be inspired now to find a constant or invent a new arithmetic. :P On a smaller scale, I've made contributions to OEIS, and I probably wouldn't have known about it without Numberphile.
I suppose it'd be silly to ask for a closed form of the equation...
This channel is really great
The key to the value of this constant or one like it is if it can reliably approximate only the very next prime beyond the last one used to define it, or if it fails, even sometimes, immediately after the last prime used to define it.
According to another comment, it actually fails *before* the largest prime used to calculate it.
Hey, idea: how many ways are there to paint a cube with 6 different colours with repetition... BUT taking into account the rotational symmetries
Look up Polya's Enumeration Theorem and Burnside's Lemma. They use group symmetries to answer questions like these! Both are super nifty and useful.
@@poissonsumac7922 many thanks! I'll definitely take a look on that!
@@CarlosToscanoOchoa No problemo!
I am no kind of mathematician (in fact I’m a freelance philosopher and esotericist, so that should illustrate how useful my knowledge is LOL), but numberphile always helps to keep me honest. Keep up the good work.
This makes me wonder if it's possible to create a similar function and constant that generates *any* number sequence.
Just using the same formula and different starting constants, you can generate any monotonically increasing integer sequence, so long as the next term is always less than twice the previous one. (Which is something about the primes which has been known for a very long time.)
Inituitively yes, but only if the property fn < fn+1 < 2*fn holds for all n.
Yeah.. I guess you can use this for those limited sequences -- but can you do it with any sequence in any order without the x2 limit?
Yes, with certain conditions on how the sequence grows (different conditions could be obtained if one futzes with the recurrence formula: e.g. you could probably make it super-flexible by adding a tan function in there). I suggest thinking of this number as more an "encoding" of the sequence of primes rather than "generating" it (this is just a semantic distinction in the end). In that sense there's nothing too magical about it: it must exist as a constant because the sequence of primes is constant. Looking at its properties is certainly interesting though.
I am a Civil Engineer approaching 40; videos like this make me want to go back into maths as a field of work.
Is it possible that this constant could be calculated to an arbitrary number of decimal places without the use of primes, or are we definitely limited by the amount of primes we know?
It's entirely unclear how you'd get it without knowing the primes to build it, but it has not been proven to be impossible
Possibly. If it was, it'd be kind of a big deal
@@maxkolbl1527 Kind of a big deal is a liiiiiitle bit of an understatement. It would probably be the most important mathematical discovery to date.
@@romajimamulo The bit he talks about at the end, where the other place the number arises means you can deduce the percentage of 2s, 3s 5s, etc that average out to make the number, makes me think that you could use a method like that to get the constant to a particular number of decimal place, then churn out at least a few more primes than you needed to know to start with.
@@MrDannyDetail I suspect that in order to work out the proportion of numbers with each value, you need to know the prime numbers (as the values are all, by definition, primes). So again, to get more precision, you need more primes.
I love the framed Graham’s number brownpaper. That along with magic circles video are my two favorite Numberphile entries.
That framed paper with Ron Graham's signature on, unexpected feels ... R.I.P. Ron Graham
I believe I've found a more generalizable way of generating a constant like this, and it's no more complex. It allows for encoding any sequence of positive integers, including those with repeats, or with decreasing values, subject to the constraint that 1
Yes! James Grimes! Long time waiting for a video with him
Wonder if there’s some interesting data encoding properties here. Being able to encode a very precise floating value as a series of integers
Oh, interesting. Most people wondered about the other way around. With regard to storing an arbitrarily large series of integers as a single floating point number, it's basically at best barely more efficient because the computational time of computing offsets the memory compactness benefits.
For your idea though, I feel it could be valid. However, the restriction I believe is that any following number in the series can't be more than 2x as large as the previous, for such a thing to work. I'm not smart enough to confirm and test anything though, I've only grasped this a bit better by some comments.
@@thomasi.4981 the reason I mention is that storing floating point value is notoriously difficult. Rational numbers can be stored as a pair of integers, but irrationals almost always end up with some rounding error no matter what base you use. I know expansion formulae are used for calculating very precise values of pi, e, etc, but I’m not sure if those techniques are general purpose. For applications where processing time is cheap but memory is expensive, and storing values using some technique like binary-coded decimal is therefore infeasible, I think this could be interesting. Obviously there’s no way to just cheat your way out of storing the same amount of information, it’s all about space versus time trade-offs
@@rlamacraft I was feeling that a series of integers would take more space than an arbitrarily large floating point number, but maybe I'm incorrect. Either way, a given system could keep whichever form it has an easier time with.
This is what we do every day. You can encode the fractional part of pi as the sequence 1, 4, 1, 5, 9, 2, 6, 5... this is literally what calculating a decimal expansion is.
On the other hand, this is much more interesting when the sequence has a rule to generate it, of course. Rational numbers have trivial rules (1/2 and 1/3 can be encoded by 5, 0, 0, 0... and 3, 3, 3, 3..., both of which are very obvious to write down in closed form), but some irrational and even transcendental numbers can easily be encoded this way. There are many interesting ways of encoding irrational numbers as integer sequences other than decimal expansions (for instance, √2 and e both have a very nice encoding as a continued fraction), too.
Amazing, the constant and also the relation of that constant with the average of prime numbers that doesn't divide the integer n number anymore in an integer. The average of all outside boundaries still doesn't tell you the next boundary without processing the boundaries.