JAMES GRIME: So I’ve got

another prime number generating formula for you. So this one is more modern. It was 1947, a mathematician

called Mills, and he found this formula. He said there exist numbers–

which we’re going to call theta, the Greek

letter theta– where you raise it to the

power 3 to the power n. And you actually then have

to round it down. That gives you fractions, so you

actually round it down to the nearest whole number, which

you write like this. That’s the symbol for

rounding it down. There exists a number like this

that will always give you primes for every value of n. So you might have n equals 1,

n equals 2, n equals 3, and every value is a prime. The smallest value for theta

where this will work is called Mills’ Constant. I’ll write it out for you. It’s called theta, and

Mills’ Constant is 1.306377883863080690, something,

something, something, something. So you can see it’s not

a whole number. So the value you get when you

raise it, it’s not going to be a whole number, and you

have to round it down. Let me just try the first few

to show you what I mean. Let’s do n equals 1. Let’s use my constant. You put it in. So then it’s 3 to the power

1, so it’s just cubed. So what I’m going to do is, for

each time, I’m going to take this constant and then

raise it to the power 3 to the power n. And I’ll do that n equals 1,

n equals 2, and so on. So, n equals 1. 3 to the power n is just 3. So that’s going to

be theta cubed. That’s the first prime. And what is the answer? What is theta cubed? Theta cubed is 2.229, something, something, something. You round it down, so

the prime is 2. Hey! We’ve got that is prime. In fact, every number

should be prime. Let’s try the next one. Let’s do n equals 2. So this is going to

be 3 squared. That’s theta to the power 9. So take the constant to the

power 9, and that is going to be 11.082, something,

something. Round it down, you get 11. So you get gaps. It’s not consecutive primes,

but every one is a prime. You get 11, that’s

the next one. Let’s do n equals 3. So I now need theta 3

to the power 3, 27. Theta to the power 27 is

1,361.000, something, something, something. 1,361 is a prime as well. Let’s do n equals 4. Theta to the power 81. You get 252,100,887– BRADY HARAN: Oh, yeah. That’s clearly a prime. JAMES GRIME: Which is clearly a

prime, which is a a prime– Point something, something,

something, but you round it down. But you’re actually guaranteed

to get a prime every time. BRADY HARAN: Do you know

what my conclusion is? That number’s awesome! JAMES GRIME: I know. I completely agree. That number is completely

awesome. So the rock stars of math, so

I have pi, and e, and these golden ratios, and

things like that. It’s not nearly as famous. It’s a constant that

gives you primes. Brilliant! I love it. BRADY HARAN: Is it proven? I mean, there’s a lot

of n’s you could put into that equation. Is this a proven thing? JAMES GRIME: Yep. This is proven. This is proven. So you might think then,

what’s the big deal? We’ve got a formula to find

guaranteed primes. How amazing is that? And the problem is, did

you notice how big the powers were getting? The powers very quickly become

so huge that even computers can’t deal with the problem. So the next one, n equals 5,

is theta to the power 243. And this is about 1.602

times 10 to the 28. Once you start getting bigger

than that, you can’t do it. The computers can’t

cope with that. The other problem is you need

to know this constant to a great accuracy. Because, what I’ve written

out for you– let’s say if I terminated

it there– isn’t accurate enough to tell

me what the next prime is. The next prime actually

is something huge. We’ll put it on the screen,

I think, instead of writing it out. But this was not accurate

enough to work it out. That’s a bit of a problem. You need to know theta very,

very accurately. BRADY HARAN: Is theta

rational? Does theta have an end? JAMES GRIME: We don’t know. We don’t know. How amazing is that? We don’t know if theta

is irrational or not. That’s a cool question

as well. We don’t know. One of the other problems with

theta is, at the moment, we don’t know any way to really

work it out, apart from taking one of the Mill primes– You take one of the Mill primes,

cube root it, and theta is approximately that. So I’m afraid it’s a bit

of a circular argument. We haven’t got a good way

of working out theta. You have to know the

primes in the first place to work out theta. BRADY HARAN: Oh, well I’m not

so impressed by it anymore. JAMES GRIME: I know. So you can see why it’s

not so practical. One of the things that we might

be able to do is, if Riemann’s hypothesis is true– Riemann’s hypothesis is a very

important hypothesis in mathematics that hasn’t been

proven yet, that is a Millennium prize, and it’s

related to crimes and how they’re distributed. If that is true– (WHISPERING)

it probably is true– If it’s true, then we

have another way to find Mill primes. Using that method, we can now

calculate what theta is probably going to be. It’s been calculated up to about

7,000 digits, but it is relying on if the Riemann

hypothesis is true, which hasn’t been proven.

(WHISPERING) It’s probably true. We have worked out larger Mill

primes, and we have worked out theta to a large number

of places. BRADY HARAN: Our thanks to

audible.com for supporting this video. They have a huge range of audio

books you can listen to. Great for putting on your

handheld device or listening to in the car. And you can download

a free book at audible.com/numberphile. This is the part where we get

to recommend a book, which I always enjoy. And today I’d like to recommend

“Foundation” by Isaac Asimov. The first in a brilliant series

of books, probably my favorite series of books ever. So go to

audible.com/numberphile, get a free book, and why not check out

anything by Isaac Asimov, but especially “Foundation,”

or any of its sequels and prequels.

hey! im reading Foundation

I love constants

This guy is a massive nerd – very sexy, ðŸ˜€

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.

Geogebra 5 can only calculate it up to n = 7

Then the question arises: How did Mill work out Theta in the first place?

Actually this number is not that impressive after thinking of it for a moment.

You pick a prime. Thats your first prime for n=1. (in this case "2"). Acording to the fact that you have to round it down to get 2 Î¸Â³ can be [2 and 3[ (between 2 and 3, but excluding 3), that means Î¸ has to be between the cube root of 2 and the cube root of 3. Than you raise your boundaries 2 and 3 to the power of 3 ( as Î¸^nÂ³ = Î¸^((n-1)Â³)Â³ , which means Î¸(n+1) = Î¸(n)^3 ) and it gives you the interval you can search for the next prime.

Pick a prime between Your new boundaries 8 and 27. In this case 11. As you round it down Î¸ is anything between 6th root of 11 and 12. Raise those to the power of 3 and you get new boundaries to search for primes, in this case 11^3=1331 and 12^3=1728.

You can pick a prime between those boundaries, state Î¸ more precise acording to the nÂ³th root of the prime you have picked and the nÂ³th root of the prime+1 and do that again and again and again… As long as there is at least one prime within the boundaries of Î¸(n)Â³ and (Î¸(n)+1)Â³ this can go forever and for every prime you pick. The interval of Î¸(n)Â³ and (Î¸(n)+1)Â³ is 3Î¸(n)Â²+3Î¸(n)+1 long. All you have to proove is that there is always a prime within that interval (and that interval is pretty large, so it should not be that difficult to prove. See the huge interval of 1331<->1728 it already creates for n=3) and you have the prove that this number exists, you have the prove that this number goes on forever and so on… And you have proven, that there are endless constants always according to the prime you have picked… Mill's constant is just the constant you get when you are always choosing the lowest prime within that interval

Thats why i said it is not THAT impressive after thinking of it for 5min… You just always pick a prime within the interval of Î¸(n)Â³ and (Î¸(n)+1)Â³ and state Î¸ more and more precise acording to your picked primes…

Hey James, I have a conjecture having to do with primes. In my equation, if you plug in a prime number as the value of the variable, the answer is always divisible by 3, along with other things about this specific equation. It is very simple and not complicated. Please reply so I know ur active…I'm hoping to have this featured in one of ur videos… ðŸ™‚

It seems to me that we can infer one of two possible scenarios if Mill's Constant is proven to give us Primes.

1. That theta MUST be irrational

OR

2. That theta is rational and there is an as yet undiscovered algorithm for predicting prime numbers of any size once we have determined when it repeats.

The first seems more likely to me.

Theta seem not only irrational but possibly transcendental?

1-(1/2)

~~(1/6)~~(1/15)…2 eliminates half the possible future numbers from being prime.Â

3 eliminates a further sixth of possible future numbers from being prime.Â

5 eliminates a further fifteenth of possible future numbers from being prime.Â

This series??? gives a maximum value of primes past a given value and should converge to n/ln(n). Can you make a video on this or direct me to someone that has?

Citizens of the world while working on a problem I found the following to which I conjecture:

The set {2,3,A,B} contains all prime numbers in the natural set.

Where:

A={6n-1: n is natural}/{36xy-6y+6x-1: (x,y) are natural}

B={6n-5: n is natural}/[{1} U { 36xy-6y-6x+1} U { 36xy+6y+6x+1}: (x,y) are natural]

MC. Makhetha

I don't think theta is rational. How would it give you infinitely many primes? Doesn't sound right

6:06 TURN ON CAPTIONS

What comes to mind is to ask if all primes are associated to a generator number this way? And if so are they

uniquelyassociated to those numbers?Hah, I feel that theta may not be so asesome, sorry.Theta can be forced to give only primes, it's constructed starting from the primes.For example if you know that 1.3…something gives say 4 consequtive primes but not 5, you can add to it a correction that forces it to give the 5:th consequtive prime without the correction affecting the 4 earlier primes. That's because the decimals in the beginning are those that affect the rounding most.

this is not done in the correct way, when you raise a power by a power you multiply the powers together not raise them. Thats so simple mathematics i cant believe they did it wrong

theta is just suited to have some primes

So combining this knowledge and the biggest (atm) known prime would result in the most accurate theta?

Curious, Does this have any relation to the 3x + 1 problem?

I'm glad to see my basic math wasn't wrong, as the digits of 252,100,887 totalled 33, implying it was divisible by 3.

Funny that they joked it was "clearly prime". Granted it would have been more obvious it wasn't prime if the number ended in 0, 2, 4, 5, 6 or 8, and some patterns (like all the digits being the same and not equaling one, or the same sequence of numbers repeating for the whole length of the number) make it obvious a number isn't prime, but adding the digits and dividing by 3 is the easiest test that can be done in your head that isn't downright trivial.

"This is an amazing number! But it isn't great. Cya later."

1:58 dark side

Things are getting interesting day by day but actually what they are in don't know yet????

Thanks a lot????

Can we assume that we have an infinite number of prime generating constants?

Basically if you take a subset of primes P1, P2, P3 up to infinity, you can always apply operations that take you from P1 to P2, from P2 to P3, and so on. Thoughts?

I wonder if James Grimes is working on to prove Riemann Hypothesis. I meant an extremely serious approach to it, like Andrew Wiles did on Fermat Last Theorem.

Numbers? So there is more than one mill number?

Am I correct that it was proved that the mills' constant exist, but we don't know the value and the later digits are found by trial and error until it results in a prime?

Why is it Theta^3^n rather than x^n where x=Theta^3 ?

Simpler to write yet its the same thing.

Brilliant

Excellent video

We want more videos with James Grime!

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)

Proof of Riemann's Hypothesis:

Abstract : It's Probably True

Proof : It's Probably True

Conclusion : It's Probably True

Its amazing to think that we know for certain something like theta^3^grahams number is prime, although that number is unimagineably large…

Why 3?

What if n=0?

Is this nummer Just adjusted

Are Therme multiple options.

If you go bigger you nee 10 digits more for one n more and propably multiple options.

Or is the nummer more then a generated one.

But propably IT fits the way you cant take different follow ups without failing to reach a prime

I habe to Look how fast the Nummer growth and how man prime are possible for next digits But many

"I KNOW! I-I I completely agree!"

Made me happy

Do a video on " what if the Riemann hypothesis is wrong"

More like Awesome Grimes Numberphile Constant

N = 0 gives 1. Guess he forgot to specify that n > 0

But how did Mill discover his constant??

Doesn't the mills number need to be irrational?

So you can just make an itrational consant, which is defined by some sum including primes, so you still need primes to even compute it to a decent number of digits and then you just make a formula, which these primes just digs out by some reverse cycle and you're done, you have a formula for primes. Then just tell everyone, that the ancient gods are giving you its digits when you pray at night.

One day i will find a general formula to predict all primes.

How does the Riemann hypothesis relate to primes?

So far the largest Mills Prime confirmed is for n=11.

The sixth one is: 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499

The proof in the description is improperly typeset.

In several places the pdf shows something like P^(3-n). The correct form is P^3^(-n).

I would recommend reading this page instead: https://proofwiki.org/wiki/Mills%27_Theorem

Not 'owe' it is ZERO (exclamation)

You should make updates whenever sth new is coming up. 5 years have passed and new observations about the mills constant have been made.

03:18 – 03:20 my response when someone voiced an unpopular opinion that I never dare say myself

7777=1! know that

FUN FACT: 252,100,887 is NOT in fact a prime. 3 * 84,033,629 = 252,100,887. But Mills Constant still works, he just forgot an 8.

(Mills Constant)^81 = 2,521,008,887 (which is prime)

3:15 to 3:23 is adorable.

Prime numbers are interesting because they are not completely chaotic but not completely orderly either. They have both chaos and patterns.

If Riemann's hypothesis hasn't been definitively proven to be true, and Mill's constant is dependent upon the former being true in order for it also to be true, then how can one definitively conclude that the latter is proven?

Millsâ€™ constant^3^giantass number= new biggest prime

0:14 "which we're gonna call feta â€“ the Greek cheese fetaâ€¦" SCNR

More seriously, the statement that computers have trouble working out large exponentials is rather absurd. As long as the precision does not exceed what can reasonably be stored in RAM, this can be done (there are even numerous easy to use "bignum" libraries that can do

fairlyefficient [1] large integer or floating-point exponentiation). The main problem is that "only" about 7000 digits of Mill's constant are known, as you later state.[1] Specifically, algorithms exist that can take a floating-point number with n bits of precision to the power of a positive integer with m bits in O(m * n * log n * log log n) time (at least, assuming that n > m).

I love the way James says "we don't know" in these videos.

This number doesn't make sense to me, you can totally do this in reverse, because you say there's not enough decimal places, if the number generated is not a prime you can most likely adjust the digits until you get a prime, and the change most likely also won't affect previous numbers because the power is so high

If theta is only calculated up to 7,000th digit there is no way to calculate more than maybe 8 or 9 primes with it

252100887 is not a prime, it is divisible by 3

The seventh mill prime is 6.95838… x 10^253

6:05 (subtitles): …. mathematics that hasnâ€˜t been proven yet, that is a Millenium prize and itâ€˜s related to crimes and how theyâ€˜re distributed.

So RH will probably never been proved, cuz then you would know the distribution of crimes! ðŸ˜‚ðŸ¤£

So what happens when you take 7000 digits of theta, take it to some ridiculous power of 3, and see if you get a prime? Considering that most of the digits are based on the Riemann hypothesis, you could conclude "it's probably true"

Computers CAN deal with big numbers like these. There's a library for various programming languages called big nums that handles numbers bigger than 32-bits numbers.

So it doesn't give you all the primes. Is there a transcendental number out there that will generate every single prime number when you put it through a given formula?

How is this not just BS. Couldn't I just make my own constant and keep refining the digits out to an insane degree to make sure each one hits a prime

gajab!!!!

Raising a power to a power, you need to multiply the powers, so (5^2)^2 is the same as 5^4. While you squared the three..

252100887 is NOT prime!

What about Theta^G64?

circular expression! that number is not really helping. the constant itself is defined by the same fact, that primes have to be found in the first place to get it accuracy higher. meaningless formula. I can write my own constant, that will contain every single prime found so far in it's digits and than write you a magical formula that will extract single primes from that magical constant each time! it's more like a database for primes we already know, nothing usefull.

Do the values of n have to be whole numbers?

cant you just plop in a number that is massive even bigger than the largest prime known and get an even bigger one? if computers were able to do it?

If theta is rational then we can do this bigger

Isn't it weird that 3 appears in the exponent ….I mean has anyone tried it for 2^n or 5^n or does it only work for 3?

teach me math little british turtle man!

3:18.

6:09 Turn on the captions. Reimann Hypothesis is related to crimesðŸ˜‚ðŸ˜‚ðŸ˜‚

Another awesome thing is that the value gets extremely close to the prime numbers as 'n' gets larger. For example, n=7 gives 69583804376962741608539276573538592864835917395924759602456009555710439562534603942132717654085619871657656850305900008136962235482689306913393638227620908148480337200487348871845278976469184329953753965251639715257390268487534179757699110378097045955949.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000061546…

Wth is going on

Plot twist:- theta proves the reamans hypothesis

There's an hypothesis which is probably true

Hypothesis:- i am a mathematician

Its probably true

I was gonna ask if Ã˜ was transcendental, but we don't even know if it's irrational. Guess I gotta start working up a proof

2019 ???

Grime ASMR

I think this would be an excellent tattoo for me ðŸ™‚

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.

What about n = 0?

It so funny how 1 passes all these prime tests but isnâ€™t prime because of some arbitrary rule

So, if this is proven to always give a prime, then there must be infinitely many primes.

could someone explain why this formula could not be used to constantly come up with new world record primes?

Maybe its just a quirk of how base 10 works?

What does this number look like in a different base

are there more constants (infinite constants?) that produce primes in similar ways?

That is actually incredible, I wish I could meet him

Donâ€™t you think that there could be multiple values for pheta and multiple formulas for a different series of prime numbers. Considering the number of digits after decimal and the magnitude of difference in the powers to which pheta is raised, it seems that pheta need not be a unique number.

0:32 IT be like: floor for integers, int and trunc for positive integers, ceil for negative integers.

I let n = 0, which is 1. I guess 1 is was a prime number after all. ðŸ˜ˆ