Awesome Prime Number Constant (Mills' Constant) - Numberphile
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- Опубліковано 17 лип 2013
- Have you ever heard of Mills' Constant? Featuring James Grime. bit.ly/grimevideos
More links & stuff in full description below ↓↓↓
Several people have pointed out the n=4 prime is 2521008887 (we missed an 8)
More on prime numbers: bit.ly/primevids
This video features Dr James Grime - / jamesgrime
The Mills Proof is at: bit.ly/MillsProof
Video supported by (& free book): www.audible.com/numberphile
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Hey! (This is James *waves*)
singingbanana hi James!
*🌊*
आप हिंदी में नहीं बोल पाव हिंदी में दो ना नंबर हिंदी में इंडियन इंडियन
आई लव यू
🤝
Do a video on " what if the Riemann hypothesis is wrong"
Phonzo Cisne heck that’s going to be a great watch
Yes please! +++
* whispers * It's probably true..
I agree. That would be the best video ever. _EVEN IF_ Riemann hypotesis is true
That would be a long video.
"The rockstars of math are..."
In any other field they would have named people, but he names numbers.
oh, there are human rockstars too in math too.
Ryan N In chemistry, you usually name elements.
So the rockstars would be... maybe Hydrogen and the noble gases?
no wouldn't it be carbon?
Muzik Bike Carbon for organic chemistry definitely, along with Hydrogen
"Wanna know what my conclusion is? That number is awesome!"
"I know! I know!"
I didn't know I there were people like me.
niklas schüller same
"I KNOW! I-I I completely agree!"
Made me happy
What if i disprove the Riemann Hypothesis ? I'd be the most hated man in maths ever. I wouldn't even get the Fields medal. They just couldn't give it to me. "You ? You destroyed everything."
+Enkel Magnus get a fast computer, use the calculated value for mill's constant and check to see if the successive numbers are actually primes. If you find 1 that isn't a prime, then there is a proof that RH is false. The proof will be in the form:
If RH is true, then mill's constant is x. (James in the video said that they are calculating it on the assumption that RH is true)
x is not a mill's constant. (a number in the form MC^3^n is not a prime)
Therefore, RH is not true.
You KNOW that you are working with very precise numbers here. You would never use a simple 'double' primitive type or anything like that. You can be extra careful and calculate extra digits of Mills Constant and be sure that it is not an issue. The bigger issue is that you can't find a counterexample and that proves nothing (although evidence that RH is true, it is not a proof).
+YK L Can the RH be used to calculate constants for other formats, such as floor(k^(n^3)) being prime or floor(k^(2^n)) prime? I expect such ones exist.
Actually, it's such a famous hypothesis in maths (so many other branches on maths have rested on this unproved hypothesis for millenniums) that proving or disproving it would make you famous. MyThey would definitely still give you the medal. Mathematicians are concerned with the pursuit of truth, not what they want something to be.
+Irene Pretty sure he was joking.
For those wanting to see Mills' proof, Dr Grime sent me a link and I have put it on the video description!
Clicked over expecting some very long, in depth thing. Guys, it's only a single page long. You can print it on one side of one sheet of paper.
3:15 I love his reaction :DD
It's even funnier in slow motion (speed 0.5). :)
+Guepardo Guepárdez They sound like little kids! So cute!
+Patrick Wienhöft A true mathematician there
Actually the whole video is hilarious at 0.5 speed.
in the video there was a typo with Mills' Constant to the power of 81. he said it was 252,100,887 but it is actually 2,521,008,887.
Glad you mentioned this, as I was about to point out that 252100887 = 3 * 84033629.
We want more videos with James Grime!
Go on @singingbanana
James Grimes is my favorite of all. He always seems excited to share what he knows!
I would love to see a video explaining both how this guy figured this formula out, how he calculated theta, and what the proof is that this will always work. Because this number is AWESOME!
They did that in this video. You start with a prime, and work backwards. They told you that nobody knows the constant, that you have to calculate it from known primes.
@@stargazer7644 yes but how do you know which primes to choose? I think it obvious that the user meant that
@@hakarraji5723 You use the ones that appear to fit the progression. You have to calculate the primes using other methods, as mentioned in the video. I suspect how to figure out the original formula would be apparent from an explanation of the proof. There's a deeper level here than just plugging exponents into a constant.
@@stargazer7644 i think there is a misunderstanding:) You cant just calculate the number without knowing the primes and you cant just know the primes without knowing the number. So the user wanted to know where it all began. either there is another way to calculate those primes or another way to calcucate the number.
@@hakarraji5723 Start with an approximation. Calculate the power and if it's not a prime adjust the number until it is the nearest prime. Rinse & repeat. The amount the exponent increases guarantees that the fidly bits you add & adjust in the end won't affect the smaller exponents. If you listen James carefully from the start he implies that there are more such numbers than just the Mill's constant. Probably is. You can try to find the k that guarantees that k^(4^n) gives you primes with integers 1 to infinity.
I think I see what's going on. Basically, Mills proved that a factor 3 in the exponent is enough of a difference that you can always fine-tune θ to find a prime p2 without the *previous* value going out of the range [p1, p1+1). Tiny changes to θ only affect primes past a particular n, see, so once you've locked onto a prime for one n, you then move to smaller changes to find n+1's prime, and so on. This proves that θ exists; finding θ then means tuning it more and more finely to find the primes.
You've nailed it. It's claimed that Mills's constant generates primes. The point is that this claim is bogus. You need to generate some primes in a different way in the process of fine-tuning the constant, and as you correctly say, this is possible.
I love how excited he gets when Brady says "That number's awesome!"
His face at 3:19 is priceless
3:18
I know!! It's genuinely adorable how excited he gets :)
5:21
well it is numberphile, ya know ;)
Charlotte Cole - Hossain u
lost
You gotta love James' mischievous grin. It's a shame he rarely features in Numberphile's videos nowadays. By the way, I'm not saying the rest of the people making the videos are boring or don't bring interesting content to the channel, but James has this unique way of explaining things, that drives your interest even if you're not into math that much.
I love the way James says "we don't know" in these videos.
0:31
I've also seen "math.floor".
+cyndie26 It's the floor function, yea.
3:15 James was expecting some serious conclusion but became overjoyed with Brady's comment haha
I love James' enthusiasm towards Mathematics, he makes these videos addictive.
can we just appreciate how beautiful this equation looks
Its beauty is in its simplicity.
"Dear Excellent Translator,
Your translation in Portuguese (Portugal) (Português (Portugal)) for the video "Awesome Prime Number Constant - Numberphile" has been approved!
It should appear on UA-cam very soon at: *[this video]*
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:D *.* ♥
Thanks for the vid Brady. Great job. Im always happy and greatful for my weekly dose of numberphile. What's more is that its a viddy with dr James Grime, arguably the most genuine quirky and interesting person on the channel.
agreed, I just love how excited he gets. you can really tell how much he loves this stuff.
6:04 turn on the captions
"it's related to crimes"
lol
I was gonna ask if Ø was transcendental, but we don't even know if it's irrational. Guess I gotta start working up a proof
Grass Observator You cannot be sure of it without a proof.
I've always been interested in maths. But since I watch Numberphile, especially the videos with James Grime, I absolutely love maths. He has so much passion! You can really feel the awesomeness of numbers in his voice. Keep on the great work!
I"ve loved the last two prime number videos on finding their locations.. very interesting.
6:06 TURN ON CAPTIONS
related to *crimes* and how they’re distributed
That is captions for you.
Its not even the autogenerated one hahhhahahahahahaah
Property history
How exactly would you prove that this works?
William1234567890123 Cook there are many ways to test for primes.
But testing the formula's output doesn't prove that it holds in general
Cheeki Breeki Nice profile pic
The way I see it the primes are encoded in the digits of theta. Think about, you calculate additional digits of theta using primes you found from another source and fitting the additional digits of theta in such a way as to produce these primes. If your process of calculating the additional digits is rigorous that is the proof that it will always produce primes. It's a circular argument and not really impressive. I have a constant alpha which constitutes a picture of infinitely many cats, you just find a picture of a cat and encode it as additional digits of alpha, there you got a constant with infinitely many pictures of cats.
I wonder why this wouldn't be true for all integers (not just 3) then.
The last sequence of videos on prime numbers are simply fantastic, I always kind of assumed that prime numbers are random and unpredictable in distribution. Keep up the great work Brady!
Gosh this is really awesome. Amazing how people can come up with this stuff
Dear Dr. Grime:
I remember hearing, back when I was in high school, of some sort of prize (à la the x-prize) for the first and/or best formula for finding primes; do you know anything about this?
Your friend and fan,
Brad
if there are an infinite number of primes.. then theta must be irrational.. otherwise.. if theta were rational.. then theta would terminate eventually.. not allowing you to generate further primes as guaranteed by the proof that Mill's formula works for n ≥ 1
No. 3.3333 never terminates but is rational. Although it is probably irrational you can't prove it without knowing it's formula.
So it would seem
Thanks NUMBERPHILE for these interesting videos.
Knowledgeable video by James as always he did.
Really enjoying this channel, thanks for your work. Also, I love the Foundation series. I was not introduced to it till later in life, but once I read it I was hooked. Good to see you promoting Asimov's work. :)
From wikipedia:
"Currently, the largest known Mills prime (under the Riemann hypothesis) is (((((((((2^3+3)^3+30)^3+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220,
which is 20,562 digits long."
+Mart Mists That's what I was wondering, if you could cube one Mills prime and add something to get the next, but those differences seem pretty random.
you mean like cubing the same number over and over again until it is a palindrome? might make a script on that :) sounds like a new type of numbers is coming... gates's numbers :P
+Mart Mists 11^3 = 1331 is already a palindrome, my point was that you already wrote the derivation of a Mills prime and showed that you have to add a bit to the cube each time and the amount you have to add isn't too predictable. Never mind palindromes anyway, write 'em in another base and they aren't.
What are the differences between the cube of one Mills prime and the next, from what you wrote?
3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220, .....
894 is a multiple of the prime 149 and 97220 is a multiple of the prime 4861. Mills's constant sounds like a transcendental irrational number.
+Mart Mists It sounds like such a constant does exist for k^(2^n).
One day i will find a general formula to predict all primes.
I like this channel because it has English subtitles for those who aren't English native speaker like me! So I can understand them more easily.
please do more videos. They're just so interesting and I cannot get enough.
I guess this implies there are infinitely many primes, since n has no upper bound.
It's very easy to prove there are infinitely many primes. Imagine there were a finite number of primes. If there were, you could multiply them all together and add one to get a new number that is greater than all primes. If a number is greater than all primes, it can't be a prime number itself. But the new number would not be divisible by any prime, so it must *be* a prime number. Because we've reached a contradiction, we know our original assumption (finite primes) is false. And so there are infinitely many primes.
Another proof (using the zeta function) is this:
Zeta(1) is the harmonic series, which diverges to infinity.
However, zeta values greater than or equal to 1 can be written as a product of primes.
The only way to multiply finite numbers to get infinity is to multiply numbers.
Therefore, there are an infinite number of primes.
interesting work
having infinitely many primes is one of the most basic proofs in number theory
Actually that new number could be composed of primes that weren't on the list used to generate it. Either way, it results in other primes that weren't on your finite list, and therefore the number of primes must go on forever.
I consider it an 'inefficient' proof because the number of primes it generates is actually very small compared to the actual number of primes out there. For example 2*3*5*7 + 1 = 211. The proof generated just one extra prime on the number line to 211 yet we know there are a lot more than that. So even these 'proof generating primes' is a small subset to the total number of primes out there, it's still an infinite list.
So we know that there are other numbers with this same property. My question is, are there other numbers that always give primes for (theta^(x^n)) for all n, where x isn't 3?
I was wondering this too! The proof is fairly short (linked in the description, only one page) and seems reasonably straightforward to understand even without deep knowledge of mathematics if you give it enough time, though I'll admit I still don't get it after reading it over a few times. I can't see a step where x=3 is required, but since the paper doesn't claim the general case, I'm guessing there's a reason I'm not seeing.
If it's true for the case where _x_ is 3, it clearly must be true for the case where _x_ is any positive integer power of 3.
jimpozcaner True.
remuladgryta well, it seems that they calculate it by taking the inverse function and seeing if it matches up. So if you wanted to, you could program/excel a spreadsheet where you take a list of primes and apply that function to many constant values and see if any strings start showing up.
Thanks for the book recommendation! Will take a look.
thanks, your channel has created such a big interest about numbers in me!
I wish you would cover where did that Theta constant come from in first place.
by using different values of n and finding primes close there and retroactively fitting a curve
That is not something you can discuss in a video. These proofs are very complicated.
What if n is a fractional value?
n = 3.4
n = 5.25
Could this somehow be a way fill in all the "in between" primes?
PrivateEyeYiYi that was what i was thinking
PrivateEyeYiYi it has to be integer
adi paramartha
Care to explain why?
Its on the paper
Deboogs But that would give you a fractional power.
James! Always my favorite Numberphile guy
Dr Grime's face when Brady said the constant was awesome.
Inspired.
Hi, Im looking for a beautiful mathematical proof or conjecture or solution to a problem that would fit on an A3 piece of paper. I'm renovating my room, love maths and I want to frame the proof. Any suggestions for a proof that looks beautiful, is super elegant and links seemingly unrelated areas of mathematics? (last one is that surprise factor some proofs have.)
Pretty cool idea :)
Moisés Prado Yeah, i wanna have something with Integrals. Its so annoying I've seen lots of great proofs but I cant remember what they're called or how they go.
danobot Perhaps use a proof of Euler's identity. If you must have integrals, you could use the comparison of Gabriel's horn's surface area to its volume. My personal favorite "math thing" is the derivation of the quadratic formula from standard form.
this seems provable easily enough by induction? i'm too lazy to do it though
Proof of infinitude of primes:
Let N be composite. Then N(N+1) has more prime factors. Q.E.D
This is probably the best proof I've ever seen.
I think it's non-transcendental but irrational.
Well, statistically speaking, it's probably transcendental, as are most real numbers.
+KasabianFan44 Using a huge Mills prime to approximate theta is (massive prime) ^ (super tiny power). These approximations are algebraic irrationals, but the limit should be transcendental.
Foundation - a great recommendation !!
You guys rock !!
Another great video. I love this channel and computerphile.
It is really interesting but for n=4, the value 252100887 is divisible by 3. It might be because Theta is irrational and it needs its precise value to the hundreds of its digits to be raised to 3^n and then the floor function applied, for the resultant number to be prime.
Actually it's 3 8's at the end. He only wrote 2...
2 521 008 887
They wrote in the upper right corner in very small letters that it's actually 2 521 008 887. I didn't notice those letters at first.
I believe they should make the text slightly bigger to avoid downvotes.
Question: If we are only now coming up with a way to calculate Theta, then how did MIlls originally come up with this number? Did he get some kind of revelation from heaven or something?
Mills only proved that there exist such a number, he did not calculate it
They showed you in the video how he came up with as much of it as he came up with. He started with known primes.
I love how MrGrime is so enthusiastic and has always been (:
damn, i don't know anything about Prime Numbers but Dr. James Grime makes me learn about it further.
Make a video on the Riemann Hypothesis.
If the Riemann Hypothesis is true and we can calculate huge primes easier than we can now, is this going to affect cryptography? If so, how is it going to affect it?
+Rodrigo Camacho It probably won't. The mills primes get exponentially spaced out as the numbers involved get bigger. There will only be a handful of mills primes in the 2^2048 range (which is what we're using currently). If mills primes are used in cryptography, and it's easy to calculate them, then someone guessing someone else's key will be pretty easy. Because of this, we're probably just going to stick to the usual methods of calculating large primes.
+Rodrigo Camacho P vs NP (another millennial problem) on the other hand...
Him getting all excited over a cool number completely made my night.
Mill's constant and Foundation in one video? I'm sold.
5:47 "I'm not so impressed by it" ... this was my gut reaction the moment I saw him put the dots at the end of the constant. I became even less impressed as I saw that it produced far fewer primes than it had digits.
Still, it was fun to watch this video.
If theta supposed to give infinity of primes how on earth can it be rational ?
+Taqu3 Well, it's not that simple. For example: 1.5^3^n also goes to infinity as n gets larger, but nevertheless, 1.5 is rational.
+Daniel Șuteu I think the argument is that since primes are distributed more or less randomly, having a constant that generates them all should contain infinite information. A constant that contains infinite information would not only be irrational, but transcendental as well.
+DaffyDaffyDaffy33322 Well, it doesn't generate them all, just some of them.
Yes but it would give an infinite SUBSET of them
It will contain infinite information if you ASSUME that primes are distributed randomly.
Fabulous, good to see all is well and you are up to speed right there.
Great video! Please explain the Riemann Hypothesis, and perhaps the generality of the Millenium Prize Problems.
What if n=0?
If n=0, then the mill's prime would be theta (mill's constant) ^3^0. Anything to the power of zero is one because anything divided by itself is one. Therefore, the mill's prime would be 1.306...^1 which equals 1.306... . If you round this down, as it says to do in this video, you get one and one isn't a prime. Therefore, if n=0, it wouldn't be a mill's prime because it's not a prime.
Isn't the name of rounding down "floor"ing?
+Brandon Boyer yeah, it's called the floor of a number
+Brandon Boyer Same thing as a greatest integer function
@@chrisandtrenton5808 no, floor is end rounding. greatest unteger is intermediate rounding
Foundation! So good. Love it.
Love the prime number content! And +1 on doing a Riemann Hypothesis video. Heck, do a video for each millennium prize problem!
How is theta calculated then?
Commented before I watched the whole video, that seems like the main reason we cant use it to find primes.
I think it's calculated by the cube root of Pn
Right, but you have to know theta to find these numbers (Pn) effectively... Damn ^^
Recursively - P_{n+1} is always the next prime after P_n^3 and Θ=lim P_n^{3^{-n}}. Something on the density of primes (Bertrand's postulate or something sharper?) is needed to get a handle on the growth of P_n^{3^{-n}} so that the rounding down isn't ruined for earlier values of n. Essentially you need to ensure that the intervals [ P_n^{3^{-n}} , (1+P_n)^{3^{-n}} ] are nested.
Bertrand's postulate doesn't cut it, I don't think. It needs P_n^3 < P_{n+1} < P_n^3 + 3P_n^2 + 3P_n, so we need a version that says for any x, there is a prime between x^3 and (x+1)^3.
Very close to Legendre's conjecture! I think it's true, but can't find a quick reference.
If Legendre's conjecture is true, then there is a number Θ such that ⌊Θ^{2^n}⌋ is always prime.
Well, I think that your Theta actually is not a constant. It is a number that is being constructed according to the quantities of primes that you want to represent by the algorithm. As n increases, more decimal you need to include in theta, and that additional decimals should be such that they have not effect on the previuos calculation (and rounding) for n-1. The construction of theta is a simple mechanic, only a bit boring.
Conclusion:Theta in your algorithm is not a constant, in the sense thar are e, pi , 2 , i or others.
No, but, there's been proven to be such a constant. As in, the theorem goes, 'there exists some constant theta such that floor(theta^(3^n)) is prime. And that's been proven. The value of that constant is unimportant for theoretical purposes.
Your comment is about as useful as saying that irrationals don't exist because you can't write them down. In other words, your comment is ridiculous.
Actually, e is such an algorithmic constant too! "e" is constructed by defining an infinite power series whose derivatives are itself, I.e. an invariant under the differential operator. Since differentiation amounts to the limit of a computation- the delta of f(x) divided by the delta in x- it is effectively defined by an algorithm. You compute e and pi both to arbitrary accuracy by truncating some infinite sum or product to a finite # of terms,
This constant is far less useful or widely known because it offers no new insight or data into the primes- it skips most of them and provides no new primality test. It would be interesting to see if a lot of results like the paper defining this constant still held up if the Riemann Hypothesis was falsified. I guess it would depend on whether the disproof was merely an explicit counter example or if there was more of a theorem behind it.
Great video, I enjoyed a lot!
Starting my day with a dose of James Grime... This day cannot go wrong now :D
If we don't know how to get to this constant, how did we even figure it out in the first place?
It helps if you watch the video to the end, you know. He named TWO different ways there...
+SayNOtoGreens gg
just test it
Experimentation in boredom.
rather simple: many mathematical proofs, most actually, don't use numbers directly. this one was probably shown by showing that there are constants that have that property. finding such a number can then be done numerically. for example starting with cube root(2), and then adjusting it numerically, by iteration using higher prime numbers.
so this prove that there's an infinite amount of prime numbers
Gamer placE yeah true
cul
Gamer placE Except that it was already proven more than 1000 years ago.
panescudumitru about 2300 years ago
+Diego Rojas La Luz 2300 is still more than 1000. The statement by panescudumitru is not wrong.
Simon Plouffe has a new (well, 2022) paper out revisiting Mills and Wright. He produces some new formulae, and discusses the order of growth of the Mills constant functions, and creates some new constants and prime generating functions with slower growth rates. You should really check it out! I believe he set some records for the longest prime generating polynomial in the paper. The paper is titled "A set of Formula for Primes" and is on the Arxiv. By the way, if you aren't familiar with Simon Plouffe, he's one of the co creators of the Bailey-Borwein-Plouffe formula (BBP formula) , which is is an explicit formula for the digits of Pi.
@numberphile Great video. Thanks for linking the proof also.
If you liked Foundation (one of my favourite books), try The Last and First Men by Olaf Stapledon.
I think its irrational.
How Mill found this number?
test
Mill: "Which number is awesome..?"
.
.
.
"Let's take 1.306...!!!!"
For fun. He basically realized that since the value grows so rapidly, he only has to make sure it passes through a few known primes in the beginning, then tweak the value to guarantee that it passes through other primes as n goes up. It's not really that impressive of a feat because it's just basically creating a number that satisfies your arbitrary rule.
Amazed. Every day I think I know "a bit" about maths, Brady comes along with something like this. I don't know the tiniest slice of the whole thing. If we have a field of study that is itself infinite, it its probably mathematics
This is a very interesting case of algorithmic hashing! The amount of information stored in the decimal encoding of the constant itself can completely account for the information that each answer contains, because there will be a prime that "lines up" and adjusts the trajectory. If this is true, then there are other constants that can be encoded to represent other types of information, which is effectively hashing.
thetha=1.3...
put n=0
you get thetha which 1.3.., round it you get 1. that proves 1 is prime.
n is a natural number.
If one was considered prime, then we could allow n=0. n > 0 is explicit stated to not generate 1.
prathamesh sawant XD what a beautiful and flawless piece if logic
Y U tryin 2 break math?
prathamesh sawant 1 is prime but it doesn’t behave like other primes so people leave it off the list now because they don’t want to have to keep saying “primes except 1”
Truly fascinating video.
That is actually incredible, I wish I could meet him
Thank you! I've already seen it, but after watching this vid I understood it a little better. Thanks for the recommendation :)
A fun side note is that this is fundamental to most computer algorithms. We force data into data-structures, that allows for accessing, deletion and insertion in a logarithmic bound. So what would have taken ages to search for linearly, is now lightning fast. Instead of doing a million computations, you do ~19.
Excellent video
That's a very cool constant indeed. It's interesting that something like this exists. Now we need to find a closed formula for it.
*Any math problem:* *exists
*Riemann Hypothesis, pi or e randomly appearing from nowhere:* bonjour
I would love to see something about Aleph numbers and stuff up that alley. I don't know whether or not there is much more to it that what you have already mentioned in one of the previous infinity videos.
That "rounding down" notation is called the "floor function" (alternatively rounding up is called the ceiling function and is the same symbols but flipped so the horizontal lines are on top... Rounding the normal way is just denoted like this [3.4] (or sometimes double brackets [[3.4]])
This is pretty cool, I can't believe I had never heard about the Mills' Constant before.
Reading the closed captions: "Riemann's hypothesis is a very important hypothesis in mathematics that hasn't been proven yet, that is related to crimes and how they are distributed."
Wow, the Riemann hypothesis really does appear in all sorts of apparently unrelated phenomena, doesn't it?
I have a hunch that theta is transcendental. That's because it contains an infinite number of primes encoded in it and everything that we've ever managed to use for describing primes were infinite series that couldn't be written down otherwise.
There are an uncountable amount of transcendental numbers (I've proved it today, consider the set of all polynomials with rational coefficients and prove that this set is countable) so theta probably is transcendental. Numbers like gamma(Euler's constant) and most values of the Riemann zeta function probably are too.
+icemaster523 So this means that the probability of a random irrational number being transcendental is >99.9999% but
CORRECTION: 252100887 is not prime. 2521008887 however, is. (sorry for the pedantry but I just wrote a program to check for primes and couldn't help but run 252100887 through it)
You should make updates whenever sth new is coming up. 5 years have passed and new observations about the mills constant have been made.
oh what wonderful enthusiasm!
also i totally agree on foundation being a must read. i finished the first trilogy some time ago and just got the last two books. such a mathematical book :)
Good book pick!
Awesome, keep it up.
03:18 - 03:20 my response when someone voiced an unpopular opinion that I never dare say myself
are there more constants (infinite constants?) that produce primes in similar ways?