2.920050977316 - Numberphile 

It's always a treat to see Dr James Grime know every constant to 10+ decimal places

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!

James is always telling constants they're his favourite but he keeps dumping them for newer, hotter constants

“I’ve got a new favourite constant” (with a beaming face of joy). This is the purest form of numberphile and I love it 😍

This title is so classic Numberphile

Dr. James Grime still looks like the age when we used to solve puzzles on his channel

You've seen elf on the shelf, now get ready for James Grime on primes

His smile never gets old

"Pretty important junk" "We need this junk" - Haran & Grimes, 2020

The sobering real-life side of research: ... "Received 16 Sep 2017, Accepted 29 May 2018, Published online: 30 Jan 2019"

“I used the primes to calculate the primes”

I can't imagine being smart enough to see a maths video on RU-vid and go "you know what, I can do better than that" and then find a new, seemingly very useful, constant.

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!

Me to Mill's constant after watching this video: I don't want to play with you anymore.

OMG ....It's James Grime💚💚💚💚💚....It's soo good to see him back making videos with Brady.... Numberphile you are my favourite channel 💚💚

Engineers: Three take it or leave it

A disadvantage for Numberphile is that nobody will write that number in the search bar even by mistake and find this video

Math teachers in primary school: prime numbers have no pattern. Every mathematician ever: You’re wrong but I have no proof *yet*

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.)

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!

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

On US Thanksgiving Day and I wake up to a new video from James. Now that's something to be verythankful for!

Anybody else instantly click when they see James grime?

Framed demonstration of Graham's number, by Graham himself, on the wall. My jealousy knows no bounds.

"ahhh constant! We love a number" will be printed on my tombstone.

We missed James!

Yesss!!! Dr. James Grime after a long time ig!!

I like constants, we need more of those in these times of uncertainty

when james dropped the second instance of the constant my brain just popped

This is a certified James Prime (James Grime) moment.

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.

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

Dr. Grime is the best. I love how enthusiastic he is.

i am from argentina, really proud of our future!!

This is such a crazy improvement to classical “get primes” functions you can write today on computers.

Loving the framed signed Graham's Number brown sheet. RIP Ron.

My new favorite constant is social anxiety.

Thank you so much James Grime for the great number!

It's so cool that it doesn't even skip twin primes since they're so close together

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/... ...))

Numberphile hasn't changed in years and I love it.

this guy is so damn cool

Me at the start "Hm, what's the catch?"............"Ah!"

9:58 - And this proof is left as an exercise for the reader

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

James is baaaack!

We're in tenth year of Numberphile with James. How time flies...

Great teachers produce great minds.

It's lovely to see James back. This feels like what numberphile used to be all about

Since you're always multiplying by "1.(some junk)" does that mean the next prime is never above double the value of the previous?

That was so cool how the average of the sequence was the very number of the video. Amazing!

I love the way Brady says "pretty important junk!"

Dr James never seems to age.

I see Dr James Grime. I click instantly

This guy is the happiest man alive.

*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.

This is my new favorite constant! So happy to see Dr James Grime back at it again!

James Prime back at it again

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?

This channel is really great

Proving that the Buenos Aires constant is trancendental would imply the riemann hypotesis. The proof for this is a bit complex.

A shame that the paper is paywalled. Would've liked to read some more about their findings.

James Grime and getting excited about a number, the classic Numberphile video.

when numberphile posts math nerds: *the return of the king*

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...

"We love a number," yes, James, that's kind of the thing

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).

So, what do we need to replace the "1" in the construction with so that the constant ends up being e=2.718281828459... instead?

Inspired viewers becoming scientists. This story proves the channel is a success. Great job Brady 😃

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.

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.

Wonderful. As usual with Mr. Grime, the non-ageing mathematician :-)

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...

I saw this video in my recommend 2.920050977316 seconds after it was posted.

This is amazing. Thanks sir. You have made me gather courage and confidence to start my channel.

9:41 Chebyshev said it and I'll say it again, There's always a prime between 2n and n.

CONGRATULATIONS GUYS !!!! GOOD JOB !!!!!

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.

This video makes me so happy.

BREAKING: the constant is actually equal to (pi + e) / 2 this isn't actually true but it's pretty close

I love the framed Graham’s number brownpaper. That along with magic circles video are my two favorite Numberphile entries.

maths isn't done until we find a function p: ℕ → ℙ, n ↦ p(n) where p(n) is the n-th prime number

I just noticed, for the first time, the signed explanation of Graham's Number on brown paper, framed and hung on the wall. I love it.

That framed paper with Ron Graham's signature on, unexpected feels ... R.I.P. Ron Graham

Dr. Grime is always my favorite. Impressive kids!

This makes me wonder if it's possible to create a similar function and constant that generates *any* number sequence.

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!

-What is your favourite number? +Yes.

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.

Hey, idea: how many ways are there to paint a cube with 6 different colours with repetition... BUT taking into account the rotational symmetries

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

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?

This is great -- I am impressed! I did not feel the same about Mill's Constant, since it very quickly became too large to confirm the primeness. As I recall, after only a few iterations we go into multiple digit numbers. :( At least for this one, we can check them! And all we need is one failure to know that it doesn't work.

Wonder if there’s some interesting data encoding properties here. Being able to encode a very precise floating value as a series of integers

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.

If the Riemann hypothesis is true, would that give a way to compute that constant without having to know the primes?

Sadly my version of Excel fails - I get primes 2,3,5,7,11,13,17,19,23,29,31,37,41,41,68,128. (I am using the same Excel as the Test and Trace one).

VAAAAAMOS ARGENTINA !!! 🇦🇷🇦🇷🇦🇷🇦🇷