DR. JAMES GRIME: So another

prime-generating formula is actually the most important

one of all. It’s called the prime

number theorem. So let me write it out. I think it deserves

being written out. Really important result

in mathematics. It’s so important that you

can refer to by its TLA. You can refer to it by its

three-letter acronym. Just call it PNT. People know what PNT is. It’s the prime number theorem. It’s that important. And the prime number theorem is

about how many primes are there, less than a number n. So how many primes are there

less than a billion? Something like that. And so the formula says

that the number of primes less than n– well, the symbol for

that is pi of n. Now, don’t be confused. This pi is not 3.14. It’s nothing to do with pi. We’ve reused the symbol

for something else. So it’s P for prime. So the number of primes

less than n. And this is approximately n

divided by natural log of n. A couple of things to say,

because some of you won’t be familiar with that

natural log of n. Some of you will. But for the ones who aren’t

familiar, I’ll try and quickly show you. It’s just a function. You could call it–

it’s just a thing. Any number– you

pick a number. Any number, like n, can be

written this way, as e to the power x. e is another constant. It’s 2 point– I never remember what it is. 718, something, something,

something. So any number can be written

in that way. It’s 2.718 to the power

of something. And the log of n is this bit. It’s x. It’s the power. So that’s what a log is. If I plot log of x, it

looks like this. This is actually growing,

but it grows really, really slowly. I think the log of

1 billion is 21. So you put in a big number, and

you’re actually getting a very small number out. It’s always getting bigger. It’s actually tending

to infinity. One more thing to say– this twiddle is a new

symbol to you. Tilde– or twiddles, as I like

to call it– it’s a new symbol. What it means is if you divide

those two things– so I’m going to divide

them, like that. If you divide those two things,

that’s tending to 1 as n gets bigger. That’s what it means. It’s slightly technical. You can say approximately

equal to, if you’re happy with that. But that has a proper meaning. So what it’s told us

is it tells us how many, on average– it gives an approximation, at

least– of how many primes we can expect less than

a number– 1 billion, 10 billion,

a googol. But we can then get some

results from that. That means that the proportion

of prime numbers is– well, if we divide by n, the proportion

of primes less than n is 1 over log n. So if we’re talking 1 billion,

I said the log of 1 billion was 21. So the proportion of

primes is 1/21. It’s like 5%. 5% of numbers less than

1 billion are prime. You can call it a probability

if you want. If I picked a number at random,

what’s the probability that it’s going to be

a prime number? It’s 5%. What does it mean? It does mean that the prime

numbers are getting rarer, more spaced out, as the

numbers get bigger. So this proportion is

getting smaller. The gaps– we can write the

gaps, as well. The average gap between primes

is now equal to log of n. So if n was 1 billion again–

let’s stick with that– it turns out the average gap

between the primes is 21. Some have bigger gaps. Some have smaller gaps. Some are twin primes. But the average gap is 21. BRADY HARAN: Let me give

you a number, and you give me all the stats. I’ll give you a number. Give me the state of the nation

for the first 5 and 1/2 billion numbers. DR. JAMES GRIME: 5

and 1/2 billion– 5 and 1/2 billion is this. So 1 billion has nine 0’s. So this is 5 and 1/2 billion. We’re going to take the log

of that, the natural log. What have we got? 22. BRADY HARAN: 22.4. DR. JAMES GRIME: 22.4. BRADY HARAN: So what

do you now know? DR. JAMES GRIME: So I know that

the proportion of primes less than 5 and 1/2

billion is 1/22. Again, it’s just like– well, let’s see. Is it 4%? Let’s see if we can do it. Yeah, so about 4% of numbers

less than 5 and 1/2 billion are prime. The average gap is about

22.4, 22, 23. So that’s a lot of

information. And then one more thing, then. One more thing we can

learn from this– our prime number formula. The n-th prime twiddles,

or is approximately, n lots of log n. So you might see where

this comes from. If the average gap is log n, and

you want to know the n-th prime, you can multiply by n. You add the gaps all together. The n-th prime is approximately

n log n. Which means 5 and 1/2 billion,

the 5 billionth, 500 million da-da-da-da-da prime is– it actually gets better

results the bigger the number is. So it helps you find– but it is an approximation,

rather than those formulas, the prime number formula

giving ones. One more thing I wanted

to say about this– this average gap we’ve

discovered, which is log n– and I said it was an average. So sometimes you get

bigger than that. Sometimes you get

a twin prime. You make an average of

it, you get log n. The gaps do have

bounds on them. You can’t go as big

as you like. There’s actually a little– it’s called Bertrand’s

postulate. And it says if you pick any

number– let’s pick the number n– he said that there will be

a prime that is bigger than n and smaller than 2n. So you can’t have a gap

that is too big. There’s a little

rhyme for this. The rhyme is, “Chebyshev said

it, and I’ll say it again, there’s always a prime between n

and 2n.” And if you put n as a prime instead– let’s take it as the

n-th prime– then the next prime you’re

looking for has to be sandwiched between prime

and 2 times the prime. Let’s pick a prime. Prime’s the nice version

to do it in. Well, pick a prime. BRADY HARAN: 11? DR. JAMES GRIME: 11’s fine. Right, so let’s have 11. All it’s telling you is that

you’re guaranteed to have a prime between 11 and

22, which is true. So your gaps can’t

get too big. For primes over 100, you

can do better than 2. That can actually shrink

down to 1.2. So that gives you quite

a narrow gap. So that just means that the

primes can’t have very, very, very long gaps between them. So they’re quite regular. Primes are always turning up. They’re fairly regular. And there’s infinitely

many of them. But they’re like weeds. They’re always popping up. You can’t get rid of them. There’s another prime. You don’t go for too long

without another prime popping up. BRADY HARAN: Surely when we

get up into sort of the Graham’s numbers of the

world, those weeds become a bit more sparse. DR. JAMES GRIME: They

become spaced out. But proportionally, compared

to the size of numbers that you’re using, the proportion is

actually very, very tiny. In absolute terms, they’re

really spaced out. But as a proportion to these

huge numbers we’re talking about, they’re close. This prime number formula is

actually the easy version of the prime number formula. There’s more sophisticated

versions of this. They look the same, but they’ve

added on extra terms. And so you can see that the

error is sort of zigzagging around 0-ish.

forsenPrime DONT SMASH IT forsenPrime

Because(2,3) it's seed spread further grow more weed, for example : 5=5/2+1/2=3, grow 5 prime by 2 and EULER PRODUCT (2-1)/2, 11=11/3+1/2-5/6+2/3+1=5, from 3,(1*2)/(2*3), 29=29*4/15+1/2-5/6-9/10+29/30+2/3-14/15+4/5+2=10, from 5 and 4/15, it keep growing never stop, like Euclid's infinite prime number.

Speaking of weeds and primes, 419 and 421 are twin primes.

I love your primes Dr Grimes but your acronyms need polishing. Acro = initial, nym = word.

PNT is not a word (unless, perhaps, you pronounce it punt). But in any case there's a much more elegant way to say "three-letter [non]acronym". The word is trigram. It even sounds mathematical.

Smoke primes Everyday

4.20 + 0.80 Prime it

20 years later, the last video of dr james will be entitled : prime numbers are really son of bi…

Why are primes or the number of them even imortant?

what if I found a way to make a primality test that does not involve module operations or probability? 🙂

Legalise maths

Why primes are so important , still just numbers like any other natural numbers.

How do you plot twin prime gaps.

I have better formula than this and will give a correct approximation,but I don't know how to show it to you.

Damn he's so charming

dr james grime always has the best thumbnails

(@6:23): …looks to me like this postulate could be used as a proof for why the number 1 isn't prime.

Primes. The higher they get, the more spaced out they get.

What's that got to do with weed?

Anyway, weeds have roots. Natural roots.

And they don't have logs.

So if ln(x) tends towards infinity and the approximation approaches being exactly correct doesn’t that mean that prime numbers make up 0% of all whole numbers

I'm a simple man. I see Grime, I see prime, I click.

A milliard has nine, a billion has twelve

Nice video, one thing thoug; I'm afraid you calculated 5.5 milliard, instead of 5.5 billion. Not a real bit deal, the US-Americans have that messed up permanently, and for that reason, English actually allows this mistake now. I thought I would point it out though.

Thank you for the video.

I was doing some math and found that (2n)+(n^2)-1 created primes very well if n is even. Example: (2 x 99922222222220)+(99922222222220^2)-1 is prime. I also saw that up to 200 being n (leaving out odd numbers) it spit out a prime 42% of the time.

there's a proof that you can have gaps as large as you want between primes

4:20 is a prime

PNT, and dynamite

No one can say this isn't cool…

123,354,076,611 isn't a prime number, but 123,354,076,613 and 123,354,076,609 are. Close!

E = 2.7 1828 1828 45 90 45 2 3 5 360

2.7,then 1828 twice,then the angles in an eqilateral triagle,then the first three primes,then the sum of the oofs in a psqare.

But what if primes were divisible by more numbers than 2? And I'm totally doing a lateral swipe to imaginary numbers but it could be useful

I'm proud of California, for legalizing primes 😎How to quickly check if a number N is prime: check all prime factors under √N, if they all aren't factor N is prime

But why is this true? You don't say why or that you don't know why.

Wow I actually could follow that . Yay..

I think that an equation for nth prime that is more accurate is (ln(n) + ln(ln(n) * n))/2*n. It is more accurate because the nth prime is a larger number than n and so has bigger spaces between the primes that n. So, n *ln(n) will always underestimate the value of the nth prime.

Yo

Has anyone worked out a magic table of products of very large primes?

I'm trying to steal Satoshi's bitcoins

Only positives reee

math.log(GrahamsNumber)

There actually exists gaps between primes that are infinite in size.

You can have arbitrarily large gaps between primes. n! + (any number 1 through n) will not be prime, giving a gap of n.

This is why marijuana was made legal in a lot of the planet places.

Weed!!!

Proof that there is infinite primes

Composites are just a combination of primes, so that means composites are also just a combination of the primes I smoke everyday

y

No wonder why I feel addicted to then.

The log of a number is roughly the number of its digits (times 2.3).

hi. i just noticed. what proof do you have that x/lnx as x approaches infinity is equal to 1???? because if u try to use calculator and increase the value of x little by little, you will get infinity too. assume x/lnx has a limit, then you can use lhopitals rule right?

because inf/inf… then =1/(1/x) =x = inf not 1. pls help

Dont give away your secrets….someone might steal the prize