Lecture: Terence Tao: Structure and Randomness in the Prime Numbers, UCLA

Transcript

0:01

Joseph Rudnick I'm the interim dean of physical sciences and it is by a

0:06

privilege my pleasure to introduce the speaker at today's UCLA science faculty

0:13

research colloquium this ongoing lecture series showcases the exciting work being

0:19

done by our faculty across the campus you know it's sponsored by the division

0:24

of life sciences and the division of physical sciences in the our UCLA's College of Letters in science with the

0:30

participation of the schools of medicine and engineering and it's made possible by the generosity of SL and Betty Huang

0:36

okay so now now the introduction actually a very smart man who is

0:43

coincidentally my father used to say that the secret to giving a standing-room-only talk is to arrange to

0:49

have it in a small room actually in this case it's a even though we chose a much

0:57

larger venue for this talk you can see what the turnout has been and that's

1:02

actually no surprise because the speaker today professor Terence Tao is the

1:09

equivalent of a rock star in mathematics his talent emerged early on in life and

1:14

that he became an taking college courses at the age of nine and was earning medals in international competitions at

1:21

the age 11 immediately after earning his PhD at the ripe old age of 20 he came to

1:27

UCLA and four years later was promoted to full professor Terry's work has earned numerous honors

1:33

and I'm only listing a few he's been a Sloan fellow a packard fellow he's one of fellowship from the clay mathematics

1:40

Institute he was just awarded a MacArthur Fellowship you know the

1:46

so-called genius grants and of course this year he was awarded a Fields Medal

1:52

colloquially known as the Nobel Prize of mathematics the citation for that medal

1:57

is for his contributions to partial differential equations combinatorics

2:03

harmonic analysis and additive number theory for distinct fields of

2:08

mathematics he's been called the Mozart of mathematics but to my my

2:13

it's actually more useful to refer to Matt's own prodigies in terms of the

2:20

number the range and the impact of his contributions terrorize comparison to

2:25

Carl Friedrich Gauss and Leonard Euler he is in short the genuine article today

2:32

Terry will be telling us about structure and randomness in the prime numbers

2:47

very much that's an election joke that's a good - actually speaking my homeland

2:53

I'm University for a change even if the weather wasn't quite what I remember to me but anyway so I'm going to give a

3:01

talk not so much on my own work actually but I'm I feel that I work in number theory wonderful fields you'll mention

3:08

so number theory and specifically in the study of prime numbers which is a very significant slice of your number theory

3:15

and this is a very old topic you know 2,000 years of research are starting to

3:21

list this topic and I'm not going to talk about all of this in one hour so just a whirlwind tour and so the year

3:28

the equivalent of going through you know Paris interesting the Eiffel Tower in the after tree or so forth but not

3:33

really giving it is impossible to get more than just a taste of the type of things that are going on but I just want

3:39

to convey that this is a lot of work is going into this we know a lot more than we did 2,000 years ago but there's still

3:46

a lot more to be done and and it's still very exciting field but lots of discoveries every year basically so

3:55

we're talking entirely about the prime numbers so you're not a prime number is or just review the definition a prime

4:01

number is a natural number larger than one cannot be split up as the product of

4:06

two smaller numbers so 2 & 3 & 5 are prime before is not supposed to have to unite first 20 or 30 prime numbers here

4:14

and so there they are so why do they care about these numbers well one way to

4:22

think about them is that this will be analog of atomic elements in chemistry they're sort of the the fun little building blocks of multiplication in the

4:30

natural numbers and this comes from one of those theorems about primes dating

4:35

back to Euclid 300 BC which is the fundamental theorem of arithmetic that any actual number bigger than 1 can be

4:42

expressed as the product of one or more primes so it's just like every molecule can be expressed in terms of atoms every

4:49

natural number chemically successor primes and furthermore there's really only one way to express it no matter the

4:54

primes other than the trivial thing of be able to rearrange the other products of example number 50 you can split up

5:01

into the 5 2 times 5 times 5 or 5 times 2 times 2 plus 2 times 5 but there's no other than this trivial rearrangement

5:07

there's no other way to express some 15 types of primes other the reason to this

5:12

view is so important this is this is the reason why we don't consider one a price I mean one copy split into smaller numbers either but if you've made one of

5:19

primes in the fundamental theorem of the FinTech wouldn't be true and we really want to see a free trip so we we do not

5:26

go out one membership in the primes okay so this is the first reason why times

5:32

are important and after this theorem the first really interesting theorem about

5:39

the primes is that there's a lot for them so first theorem is Youkilis theorem so do do you good that there are

5:46

infinitely many prime numbers so unlike atomic elements of what you only have on your twenty six of them or so we

5:51

actually have infinitely many Prime's and it's and which is in sharp contrast of say addition if you look at addition

5:59

instead of multiplication the only prime so this is one because every number came is every number is just a double whole

6:05

bunch of ones but that's not the case for multiplication so I'm actually going

6:10

to give you a proof of this theorem right now this is a non-technical audience and I don't want to I'm not going to play any real mathematics or

6:16

any real detail mathematics on you but this this one proof is so beautiful that I think it is just and quite simple it

6:24

has to be shown so it introduces one of the one of the best tools in improving a

6:34

very honored to have choice in mathematics I don't reductio ad absurdum proof by contradiction you want to prove

6:40

something well pretend is force and then show that that can't happen in the net indirectly shows that is true JJ Hardy

6:47

is the famous British repetition or once said that this is the final gambit than any any chess gambit or any gambit

6:55

navigate with so that area a chess player may offer you of a brook or queen

7:00

you know in order to win the game but the math addition offers the game itself and still with so okay so here's how the

7:09

proof works so you want to prove that there definitely many primes well suppose it were suppose that when we find any many Prime's then you could

7:15

have them they'll be you could call them P 1 P 2 up to P n so suppose for example 2 3 & 5 for some help the only Prime's

7:21

in existence then what you do is you take the dense crimes and you might have them together and you add or subtract

7:27

one so you take two three to five sake you multi them together sturdy you add a strike one you get 29 or 30 one so that

7:34

gives you two two new numbers at a distance two apart which I'll come back to later and so these numbers these pair

7:41

of numbers they're both larger than one but they can't be Ducati divided by any of the prime numbers that you really

7:46

have okay because 2 times 2 times 5 is already divisible by 2 3 & 5 so if you add subtract 1

7:52

they can't be divided by 2 3 or 5 and those things that are out there so you constructed some numbers which can't be

7:58

divided by any prime number at all and that's a that contradicts the fundamental theorem of arithmetic to the

8:04

degree that ties it every number came still in the price giving you some numbers like copies they're from the primes so hence can't happen finally / how many

8:14

Prime's must have in that name so it's a very short proof but it's very

8:20

it's an indirect proof it tells you the Infini many Prime's but it doesn't tell you where they are

8:26

it just tells you it small is it really is only talking that there can't be finitely many plans so they might be

8:32

thinkin too many and so there are some more direct routes which more explicitly

8:38

try to run for Prime's but the nowhere near as short or elegant is this one so we know abstractly the intially many

8:45

Prime's don't actually have a good way of I can't show you into many Prime's and in fact you know we only know I'm

8:52

finitely many primes and just to give you the world record right now the

8:58

largest prime that's actually known is to you know like we can write down explicitly is 232 million five hundred

9:04

eighty two thousand six hundred fifty seven minus one so that beast is almost 10 million digits long

9:10

and it required a distributed internet project to show that is prime so that's the biggest primary goal so even though

9:16

we always infinitely many that's sweet that's the biggest one reconnection right down okay so in Euclid's proof I

9:28

took a bunch of primes mop-up together and add a subtract one and I got these new numbers which which differ by a

9:34

factor of two a difference of two so that suggests its own defining foreign

9:40

concept we divide a pair of twin primes to be a pair of numbers which are both prime and this is two apart so a number

9:46

P its prime in the number P P plus two which is also prime so for example with

9:52

three and five twin and seven eleven thirteen forty one forty three so forth

9:57

so given that you could prove keeps generating all these twins which are not divisible with what any prime that you

10:03

really have it then becomes natural although he never actually stated this term in his work but it still becomes

10:12

natural to to conjecture that just like it instant many primes may be there in

10:17

ten minute windpipes um so and because the proof for four of you Kasim was so

10:22

simple maybe this should also have a very simple proof but the amazing thing

10:28

is that even after two thousand years of research into primes we still don't know the answers was conjectured we do not

10:34

know whether the infinite mini train tracks Euclid's arguing suggests maybe we

10:40

should start multiplying some Prime's together editor of subtract one but it doesn't work example if you add the first four Prime's together two three

10:46

five and seven and you subtract one you get two hundred nine it turns out not to be prime it's not divisible by two three

10:52

five or seven but unfortunate of is what it is divisible by 11 and 19 so the

10:59

largest known protein primes is that thing written down there to two billion three million six hundred sixty

11:05

three thousand six hundred thirteen times to the margin 95 thousand plus or minus one those twins 58 thousand digits

11:13

long I know discovered there this Monday I read about of the very bottom term

11:19

even that yesterday actually so I update my slides so just the previous records 51 thousand as long so they beat the previous record

11:25

by about 7,000 pages so anyway so that's

11:32

that okay so why are the prime so hard

11:38

to understand basically the things we don't understand the sequence of primes all that well here's the sequence of prime 2 3 5 7 11

11:45

they behave it I mean it was increasing but other than that it behaved in a very sort of random manner much more random

11:52

than other sequences so here's another sequence for comparison with square numbers 1 4 9 16 that sequence is much

11:59

more orderly because given any I can tell you what the well I can tell you what they 1,000 elemental of what the

12:05

1,000 square numbers we get will I take without any thoughts 1 million at the end square number is is N squared but I

12:12

can't tell you what the NF prime number is not easily okay I can't tell you the 1,000th prime number without a lot of

12:18

work so the nth prime numbers often called PN but we don't have a good

12:24

formula for PN what I mean either P in itself is not very good for but we're

12:29

not a useful formula for the nth pine-sol and this is so the prime to do

12:36

is that they're trying I mean that there's only one prime sequence then what is it it's deterministic it's not random but somehow it seems to behave as

12:42

if it was random in many ways so there's a famous quote from poor Kurdish or the great mathematicians of the 20th century

12:49

that referring to an early quote of Einstein Einstein they did not believe

12:56

in the interpretation of car mechanics given by his colleagues because it seemed to imply that the universe was

13:02

had some randomness in it he said that God did not play dice with the universe and then Paul sir graphing on that said

13:07

well God will not play with ties to the universe but certainly something fun is going on the prime numbers okay so we

13:15

don't as I said we don't have a good exact formula for the nth prime right I do I cannot give you a nice form for

13:21

what what P sub n is but we do have an amazingly good in exact formula if you

13:26

just willing to know what the end time is approximately I can tell you with very high accuracy what the nth prime is

13:32

approximately and the big theorem here it's uncle the prime number theorem

13:39

which was proven over a hundred years ago that says that the enzyme PN is

13:44

approximately n Times log n and this is log not log base 10 which would be kind

13:49

of weird since you know the natural numbers don't care that we have 10 fingers but it's a it's log base 10 after the natural log e 2.71828 okay so

13:59

PN is approximately n log N and what what approximately means is the precise meaning here but I won't talk about that

14:05

so this is it's a funny theorem it's really it's widely considered basically

14:11

the essentially the best theorem the best achievement in a number theory there a single achievement in number

14:17

theory however it's the the proof is much more advanced then it uses much

14:24

more advanced mathematics the nucleus proof you can say give you a complete proof and it's a much more it's a much

14:30

deeper result and the proof is me it's really quite amazing and I'm going to give you a somewhat cheating simplified

14:37

version of it but just to give you a taste of just how mordant it is compared

14:43

to what you just saw so I'm going to use lots of imprecise terms so what you do

14:51

is that you you analyze the primes as follows so you have to create well I

14:56

caught a sound wave like at the precise name is the bond my goal function but create a sound which at prime times is

15:03

noisy and at them at other times it's quiet so so a time one is silent a time

15:09

to make a noise time sync noise silent five make a noise six silent so forth okay so and the volume low noise is a

15:16

formula for that but so this is this is funny sound wave that you can make it as if it's a function and you take this

15:22

function and you listen to it okay so the precise or all this and means is

15:28

that you take a Fourier transform actually you don't quite do that you take my photo Mellin transform which is almost the same as a Fourier transform

15:34

so you you listen to the sound wave and you will hear certain notes and you work all the notes that you hear and the

15:41

notes have a name then the notes are called the zeroes of the Riemann zeta function which I'm not talk about you

15:46

colloquially the sometimes referred to as the music of the primes and each note with refers to a certain each note will

15:54

give you some hidden pattern in the pride and the oscillating at a certain frequency then they will should then

15:59

that will show up as one of these notes and in principle the entire description

16:05

of primes can be recovered from knowing what these notes are and then what you

16:10

do is that you show that of all the possible notes that could possibly come show up in the in this sort of musical

16:16

the primes a whole bunch of them actually don't appear and that's a sport

16:22

fast tricky to state but also tricky to prove that's the hard bit if you want to talk about here but once you can exclude

16:28

certain types of notes in this music that tells you a lot about what that music does and it using that and either

16:35

Fourier analysis or or complex analysis which is very related you can then prove oculus prime number theorem so that's of

16:42

course so that was very sketchy but that just so shows you just how modern is

16:47

proof is that there are more elementary ways to prove the prime number theorem but but those proofs are longer and also

16:53

I'm not very intuitive in fact in many people yet they don't the the the elementary proofs are not considered

16:59

anywhere near as elegant or as informative as it's much more modern proofs these non elementary proofs so

17:08

just to give you some example of the partner with you in action the 1,000th prime is seven thousand nine

17:13

or nine whereas this n log n prediction is roughly six thousand and seven that's an error over 13% if you can look at the

17:21

millionth prime or the building of the prime or be trillions prime the error slowly gets better but it's not it's not

17:28

all that great this this this approximation you know it's only good to one significant figure but but nevertheless what the problem

17:37

with even says is that as you keep as you keep going out in the prime sequence the the error will eventually go to school will eventually go to zero okay

17:46

so in mathematics you know it's great to prove initial results but but often the

17:53

the techniques used to prepare those results are even more valuable than the result themselves because they can be

17:59

then use to prove other things so for example the techniques used to be the prime number theorem can be used to

18:04

prove many other many other statements and I just give you two examples so one

18:09

is that if you take a large prime any prime bigger than five actually and yogas last digit you've added base 10

18:15

because last digit it's not hard to see that they're all large primes the only possible last stage is they can have is

18:21

1 3 7 or 9 there are no crimes n + 6 or 8 or zero and only one fun instant in

18:28

five or two so all large Prime's in in 1 3 7 or 9 that's easy but what's not so

18:34

obvious is that is that those digits are Co equally likely that there are just as

18:39

many Prime's ending in 1 as there are ending in 3 or 5 or sorry about not 5 is 1 3 7 or 9 23 25% of all large Prime's

18:47

in 125 in int 3 and so forth that is not an obvious fact and that took quite

18:53

quite a few proof and there's a similar statement for other bases in base 10

18:58

here's another famous theorem I've been in garros theorem from 1937 but all

19:04

large odd numbers any one number that is large enough you can you can write as a sum of three primes

19:11

so this this this tomb as a matter of history it was conjectured by a go back with an amateur mathematician in the

19:18

18th century so he in fact conjectured that that all odd numbers larger than 5 are the sum of 3 all primes three pipes

19:25

so become 7 is is that 3 + 2 + 2 and so on and so forth so ok so he conjectured

19:33

that all our four numbers are some of these crimes and Vinogradov show that all large odd numbers ultimately France

19:40

well how large is large well that barrier for for large means has been lowered over over the years the the best

19:46

result right now in 2002 all numbers larger than 10 for 1346 all odd numbers

19:52

larger than 10 to 1000 336 are known to be somewhat for your tracks we also know

19:58

that all numbers less than 10 to the 20 for your crimes by a very different method so that's the odd go back inject

20:05

yourself presumably as the this gap will close eventually but you know even with the best computers right now list

20:12

there's there's no charge so we can get up to 10 to 1 10 to 1,000 or so that's more than minimal atoms in the universe

20:18

so that cannot be tested by group or search there is an even go back injector

20:24

which is stronger than the odd Goldbach conjecture it's congested at all even numbers larger than 2 I was sum of two

20:30

primes and this is still open in fact it's it's basically as hard between prime conjecture so that's still

20:37

an open question ok so the prime number

20:43

theorem is tells you approximable the nth prime is it's n log n so the chose

20:48

out to be a more accurate formula or formulary belief in more to be more accurate which is given by the Riemann

20:55

hypothesis which I won't state exactly but one of its equivalent formulations is that it tells you a more precise

21:02

formula but a more complicated formula for the nth prime at the end prior has to obey this funny equation which I want

21:07

to discuss more this is the this hypothesis is 150 years old and one of

21:13

the most important tons of questions in number theory so much so that the clay mathematics

21:19

Institute in Boston has listlessness one of its seven clay price of clay millennium prize problems and there's a

21:26

1 million dollar prize it adjective someone can actually prove this ok and it this one will not mean anything you

21:35

don't like you know what the halters is is but it's saying that this musical the primes is actually arranged like a chord in some sense but it's a very cryptic

21:43

comment at this point so just to illustrate how accurate this prediction

21:49

is here again is the thousandth prime building a prime building for Prime Minister Prime and on the right hand

21:55

side this is the prediction that that the Riemann hypothesis gives except for

22:00

this error that was over there's a low error term in the Riemann hypothesis so it the Riemann hypothesis even that

22:08

doesn't predict what the M prime is exactly but it does give an astoundingly good approximation so

22:13

so the thousandth Prime is only accurate to like two percent but once it gets to

22:19

say the trillions of Prime it's exactly at point zero zero zero two percent the prediction for that formula so it's a

22:25

it's a very good prediction for what the primes out so yeah so the Riemann

22:33

hypothesis will go back a little bit so it's it's it's not an exact formula it's got this funny big old thing and I won't

22:39

say what Big O means but this is just some some weird error term that we don't understand and this error term is

22:46

actually the same type of error that you would get if the primes were somehow just routed randomly if if you know if

22:53

somehow you know God has measured shuffle the numbers and deal them deal out the primes in some random fashion

22:59

in some sense you would expect that the Riemann hypothesis type of prediction

23:06

there to hold exactly this error due to some encoded or large numbers which one will talk about here so what the Riemann

23:14

hypothesis is saying somehow is that the primes behave as if they were they would they were they were dealt randomly

23:20

somehow among the images so the precise well we so we call starting

23:26

pseudo-random they're not actually random but they behave as if a random it's like the random number generators

23:32

that has used our using computers they're not actually random they come from some complicated algorithm but they behaved randomly at least we believe so

23:39

so we would very much like to prove this Riemann hypothesis which is the statement of primes behave randomly but

23:45

we are not very good at proving that things being randomly the problem is Adam you know maybe the primes look like

23:51

they gave randomly but maybe secretly after the trillions primate what they all decide to to maybe they take a big

23:57

break there's a big gap where there's no crimes and available bunch together and we don't really know that the primes you know sort of conspire to do something

24:04

after a while that there we we can't see yet and so how do you show that conspiracies don't happen you know I

24:10

mean that's this but it's more philosophical question but it is then we'd only have good tools to to prevent

24:18

this from happening okay so this is all pure mathematics but

24:25

this whole business about showing crimes and things relate to the primes are pseudo-random this is not a purely

24:30

academic concern there are very many important things in which underlie

24:38

internet security or banking security these days which rely on a belief that many things with little Prime's do

24:45

behave as randomly as the Madison's predict them too and so in many ways the

24:51

you know trillions of dollars that hang on on on on certain beliefs while the

24:56

primes actually and so to give one example this is some some aquatic diffie-hellman key exchange protocol

25:02

this is a protocol invented in 76 and it's what it does is that if you have

25:09

two strangers which are traditionally called Alice and Bob and I was supposed to tell Bob something secretly but but

25:17

she can't but she can only communicate with Bob over an insecure Network like

25:22

the new place you can only send emails through the internet saying and and the this network is completely open if

25:28

everyone can read again can read every communication if if there's no secrecy how can you still communicate and some

25:35

secret code for example or some code code number G you know maybe the number of your bank account how do you how do

25:41

you communicate this number from A to B without and without anyone in the middle

25:47

any eavesdropper learning what that secret is even though the network is completely open so that's that's not

25:52

obvious how to do it if they were strangers if it had previously met they could arrange some secret code but but

25:58

they're not met before like it they can only talk to each other in a complete open ways if they arrange a code

26:03

everyone hears them so you can't just sort of cut it set up a code that way

26:09

there is a this is a digital problem but you can think of a physical analogy

26:15

imagine that Alice wants to mail a physical map you know an envelope a

26:21

message to devolve but but he knows

26:26

about that she knows that all that all all her messages can be opened and read by people in the middle so how can you

26:31

still send secret messages when everyone can open all your mail so you have to do a trick and in the physical

26:38

problem here's the trick it's a very cute it's a very cute algorithm what you do for

26:44

Alice does instead you take your message and you put it in a box and you put a padlock on the box again you keep the

26:50

key and you mail the lock box off to bottom okay now so anyone anyone opening

26:55

the mouth this is a lock box and don't they don't have a key that can open of course Bob doesn't the key either so he

27:01

can't open it either but what she or Bob does is that he puts his own lock on the box too so now it says it now has two

27:07

locks okay and knows it back to Alice okay so so now it has a double lock box

27:12

coming back I was can't unlock Bob's lock but she can unlock unlock take it off okay now

27:18

only bops lock is left mail it back and unlocks the lock again and gets back the message okay so so the

27:27

use dropper never sees an unlock box and the Allisyn Bob never have to communicate with each other but they can

27:32

simply say they can transmit the message so that's how you solve a physical problem and the digit problem is done

27:40

analogously math is all about Alice and analogies and so roughly speaking if you

27:47

want to send a secret number G if as opposed to send us a secret emoji to Bob what you do that you first agree on a

27:53

large prime you could pick any prime that's that big primer I mentioned before you could use so Alice picks a

27:59

key a just ain't what her favorite number a and she takes this message actually and she locks it and the way

28:05

you lock it mathematically is that you raise to the power a so you take usually a and then you have to take the remainder modulo P so but never mind

28:13

that so you lock it you take usually a and you get a new number and you send that you send that sort of Mach number

28:20

over to Bob Bob cannot can't he call that but he picks his own number B raises raises the GCA again to be and

28:28

giclee a racially beast seems G to the a times B so now now is doubly locked

28:33

sends it back to Alice Alice takes the eighth root of the number she got and there's a way to do that modulo P which

28:40

I want to talk about and so that gets you back from GTA V back to G with G B and then she sends it back to

28:46

and then she takes a bit of that he takes the beetroot of that and that gives back the G it's the same algorithm

28:51

that I said described physically but it is this is the this is the sort of the mathematical analog okay and this is

29:00

believed to work we don't actually have a proof that this is secure and if the each property sees all these secret

29:06

numbers going back and forth it might be possible that this the each droplet could actually crack the code in some

29:14

reason or other time we don't know that's true we believe it's we believe it's secure but we have no proof that's

29:21

related to another one of the seven clay millennium prize problems six if the P equals NP problem if P equals NP was not

29:27

if P was a sin alpha one four but this is that is the sum equipped circle NP

29:32

and the belief it different if they're equal then we could do many things in particular crack this code in fact we

29:38

could crack any code really but in particular this one we believe that P is not equal to NP and

29:43

we also believe that this code cannot be correct we don't know that's true but we

29:50

do have some data there is a recent result I'm quite fond of our my friend John Gorgan who shows that the data that

29:59

they use chocolate intercepts by this protocol right so so the deer a new trouble will see a whole bunch of

30:04

numbers going to whizzing past usually a gdb to do to a B these numbers what can

30:11

be shown is that if you take the most significant digits of the numbers that you see they are what's called uniformly distributed which I would define here

30:18

but it means that you can't distinguish them from ramp they look like random lots in a certain sense and you can't

30:25

decode anything random noise a lot of the more random noise so that that suggests that that the code is secure

30:30

but it doesn't there's only the most significant digits it doesn't the if if

30:36

the issuer gets all the digits that maybe maybe he or she can can decode the code one result because it uses one of

30:43

one of my own results from a few years back that's the story

30:48

okay Oh some disclaimers what I described is not actually how to be

30:54

human works its clothes oversimplified but that says that's a congregate only

31:03

the book for experts that's okay they did this is a slight difficulty generate

31:08

the secret but the other disclaimer is that I mentioned two types of Oz - do I

31:15

know - so the Riemann hypothesis is one type of pseudorandomness probably a concerning the primes and the

31:21

pseudo randomness that that is believed to make diffie-hellman secure is a slightly different type of pseudo

31:26

randomness and they are in some sense similar in spirit but they're not the same sometimes it's inaccurately

31:32

reported that if you saw the Riemann hypothesis you can start cracking codes and that's not actually true it but I

31:41

came anyway severe enough of publicist true so blends believer factor these

31:46

codes are uncrackable but but there's no direct connection that is of spiritually

31:51

connected relevant then sort of directly connected okay so my next topic is

32:00

another is another part of number theory primary theory course if TV so I've

32:06

talked about how the primes are behave somewhat randomly but they're not completely random and they're asking

32:12

obvious patterns in the primes right for instance almost all the patterns are hard with one exception

32:17

almost all the prime numbers are adjacent to a multiple six as a with two

32:23

exceptions this time that's a nice little result and so on so forth so so there are some obvious things that the

32:28

parents do I mean basically the primes cannot be divided by two or divided by three unless they are two or three and

32:34

so the doctor which it tells you something and there's a way to capture

32:40

that very efficiently and it's a tool for safe theory and it's one of our basic tools to our designing the primes

32:46

so I just want to mention this because it connects with talk about so Sif's the

32:53

the idea of SIVs is in some sense you don't try to look at each prime separately because that has proven to be very hard you sort of

32:59

view them in aggregate it's like if you want to count all the how many grains of

33:04

sand around a box one grain at a time that's very painful and and not very fun

33:09

but what you can do is that you can just take a look take a box set and weigh it and then divide by the average weight of

33:15

a bubble of a grain and that will give you at least a good approximation to how many grains there are in your box and so

33:21

you can see about in aggregate rather than individually you can you can do things more efficiently and so the idea

33:29

is to try to uncover the primes as a set in aggregate the primes you can think of a sort of sculpture buried inside the

33:35

images so the the natural numbers want you to provide you can pick up some sort of a block of granite and what you're

33:42

doing is you're trying to carve out the the the primes out of this block of granite so what you do is that you can

33:48

take this big mallet and you want you you're going to back off big chunks of all this of this kind of sculpture and

33:54

then after a while you take away your map mallet and put you take a chisel and you just often with some smaller pieces and then you think you pour your chisel

34:00

and then you would take a needle and you think you've lost a little little tiny fine adjustments and then eventually you'll get a point and this is sort of

34:07

how sibs work the most famous sieve and the one that you may have learned in

34:13

high school is the sieve of Eratosthenes which also dates back to 200 BC and what

34:21

it does is that given any n like a million you can get all the primes up to n watch it all the practice between square root end and end but doing all

34:27

the primes up to n as follows so you take all the numbers from square root n to ends like all the numbers from a

34:33

thousand to a million it's like this big block of granite okay and then you just chop away half of it you feel all the

34:38

even numbers like Holger and then and then you throw it all the order box or three you can cross them

34:45

off and then and then mobile to five to seven or so probably 11 and so forth you just keep void it's a city key keeps

34:52

avoiding all you can do you think of air--so sifting out the primes or I like to think of itself as doing a sculpture

34:59

and so you keep you keep making these changes again and again and again all the way up to a thousand so I think the

35:05

largest prime is 1997 so finally you take the remote the most of 1997 for the route and then at last

35:12

after all that everybody that's device is a prime and this is get all the primes from 1,000 to 1 million okay so

35:19

this is the standard sieve for various reasons it's not a very efficient sieve there are more advanced tips we use

35:25

nowadays which are much more complicated it's it's more like another it's much

35:31

like a student application form for now you you and you you assign numbers

35:36

various scores or wastes depending on various things that if you give up of two you lose some way to develop to you

35:42

you gain some weight and they're always weights that you add or subtract depending on various statistics and then

35:48

you finally have the score and then the idea above a certain threshold or something you gain emission and then

35:53

this dissipated and these are your primes but okay but that's there's a much more technical technical detail so

36:01

as I said these these these sips they cut off very very simple like it's it's easy to understand what happens if I

36:07

take a whole bunch of numbers and I throw away all the even numbers and not what I want have less the odd numbers okay I have to throw most of - and also

36:14

three okay what I have left is just everybody that's next to move all six but but then after a while it gets

36:20

really complicated and we don't really and we've never really managed to use sieves just by themselves transient

36:26

primes but what you can do with a sieve is that it can take you as the sitting

36:34

process and just stop it halfway and you don't quite and if you see this opposite of halfway then you don't throw out

36:40

enough stuff you get the primes but you know you also get some some extra stuff attached to the prime is called it

36:46

almost primes almost plants are numbers which are not prime which may not be planned themselves but very few prime

36:51

factors may be numbers that only have two or three prime factors and what SIVs

36:56

have done for us is that basically roughly speaking while we don't understand the prime soil we do

37:02

understand the almost primes quite well so it's just a very typical result here and go chains theorem which is them so

37:11

you remember the twin prime conjecture there should be infinitely many Prime's P where people's who's also crime we don't know that yet dearly love to do

37:18

to see that proven but this is the newest the newest we've got so far which is that they're infinite is P

37:24

where P plus two is either a prime or product of two primes so sort of it's

37:31

almost the twin prime conjecture but that's the best we can do it visit the arm and that's basically the limit of

37:38

surf theory in many ways it's we know in some sense of sift we cannot do much better than this ago okay so finally I

37:47

can talk about my own work so as I said we even after 2,000 years we can't

37:54

detect many patterns in the primes I do not know how to find watson also twins in the primes there are lots of other

38:00

things I don't know how to do or no one knows how to do no we don't know how to write even larger even than versus assemble two to two primes but there was

38:07

one type of pattern that we can actually do something of because we there's a cheat available for for one type of

38:12

pattern ethnic progression an ethnic progression is the sequence of numbers which are equally spaced everything

38:18

every number is the same spacing from the previous one and my feeling would been that I proved two years ago Ben

38:25

green is that the primes inside the primes somewhere you can exist opening

38:32

progressions of arbitrary length so somewhere in the primes there are progressions of length 1 million some

38:38

enterprises if there is a progression of length 1 billion I can't tell exactly where they are but I can tell you that just like you Chris theorem I can't tell

38:45

you exactly where they are but I can tell you that they're they can get as long as you please and also this many of

38:51

them so as a consequence given any K not

38:57

only can you find a progressional length k but in fact you can keep an infinitely many progressions in landscape okay so

39:04

this builds on a number of early results for example it was known before that you could find progressions of transforms

39:10

length 3 but now we can find progressions of primes of any land fertile so just to give you some

39:18

progressions of primes here are the first few suppliers of each length so the first progression of length 1 is 2

39:25

the first progression of F 2 is 2 & 3 the first progression of primes of them three is three five and seven and so

39:31

forth so it's the first few progressions look look very small and you think that

39:37

it's easy to find the next few tricks in sequence thank you already finding the first progression of

39:42

them seven is not easy and I haven't written it down here what's the longest progression that of primes that we know

39:49

of we only the only explicit the longest explicit progression of primes that we know of that's 23 okay we we have an

39:58

example of a progression arithmetic location Prime's over 23 just 23 that's all but already these Prime's out is it

40:05

what is 56 trillion and the spacing is 44 trillion it's some you know yeah of

40:11

course you need a computer to do computer trying to find these we don't know of an explicit progression of them

40:17

to 24 but nevertheless we have shown that some way out there there is a quotient of 24 and 25 26 there's

40:26

absolutely no chance I can tell you about how this is proven except well so

40:33

I'll give something which maybe it's not so enlightening but but the the one

40:40

second sound bite one minute Samba is some okay so severe

40:46

that much before that allows you to tell it doesn't tell you that the primes contain long progressions but what I can

40:52

tell you eventually is that the almost primes get a long progression so you can find lots and lots of really long progressions which are not prime but

40:58

they're almost prime so what does I talk about the primes well the primes are subset of the almost

41:04

crime so there's somewhere in the primes and the almost primes but we don't know how the distributed maybe maybe they're distributed in a very uniform spread out

41:11

manner maybe this there or distribute in a very in a very structured sort of matter we we don't know we still don't

41:18

know we're working on a but we don't know exactly how they're distributed but

41:23

what we managed to do what what been winning I'd like to do is we met by two different arguments we could show that

41:28

regardless of whether the primes are structured or random inside the almost primes it doesn't matter which one regardless of how they distributed in

41:35

either case they must necessarily contain altima group they have they have to contain a fraction of the

41:40

progressions that they pair I'd set the almost prime student as well the almost the either question systems

41:46

of hereditary property that at once set has called us lots of progressions then then lots of subsets will also have

41:52

blossoms and progressions as there's some weird hereditary property just special to think progressions and it's

41:57

not it's not troopers a Twins this method does not give the twin prime conjecture but for some reason

42:04

progressions have a special property that that you can pass the subsets and you still keep a lot of progressions but

42:09

that's a whole long story anyway so just to finish up we have a lot of

42:18

work still left to do our as I said our theorem doesn't tell you exactly where

42:23

to find say the first progression of length 1 million we do it does give you

42:28

a bound as it turns out I can we can tell you that if you want to find the first progression of primes of them over the K we can guarantee you that that

42:36

progression will exist and not only that that all these entries to be less than 2 to the to the 2 to the 2 to the 2 to the

42:41

2 to the 2 100 K that's what that's what our argument gives us that's not the true should be and the the conjectured

42:48

is more like K to the K so one exponential there's the seven Exponential's there's several inefficiencies in our

42:54

proof that we have to resolve but just to connect with a previous thing this

42:59

this Riemann hypothesis that we see we can't so if we can if that's true we can get rid one of the Exponential's no bad

43:06

but not not the rest unfortunately anyway that's a good place to start

43:11

thank you very much

43:21

I personally don't the all the records that I listed up there what we're doing by computers but I work on more the

43:28

theoretical side of things I do know myself is computers well I was that of course - yes so for example I have a

43:43

paper from what a couple years ago that there's an analogous theorem for the Gaussian Gaussian prime so the gaseum

43:51

fans are arranged in a grid rather than on a line and the analogous theorem is not only contained on progressions but

43:58

they contain constellations of any of any given shape there was a movie called

44:03

a beautiful mind where at one point the protagonist who serve Russell Crowe I want to impress his girlfriend so she

44:11

can just scope into him to name a shape at the key an umbrella and she and he looked up the night sky

44:16

and after what he finds a bunch of stars arranged like my brother so they got

44:22

some prizes the same property if you give me a shape like an umbrella eventually somewhere in the program there's some arrangement or Gaussian Prime's that is shaped like that

44:28

umbrella it's a they use the same method actually

44:39

yeah so that particular number is you may observe that it's um it's almost a

44:46

power of two it's what's going to ascend Prime and for four numbers that are

44:54

close to two numbers that are close to a pure power like a power - there was a special way to test for parity to just

45:01

present is primal law and that that's what yeah so most numbers that are nine

45:07

nine million digits long even the fastest computers cannot tell whether the prime but that but the special numbers say

45:14

special methods no fewer sales and so forth I just is not no subject that I

45:20

know that much about possibly so there

45:32

are seven seven clay price problems in and one is more team proven right now

45:38

the point way conjecture even policies I think would be somewhere on the second

45:43

to last week movement I think P equals MV we the last I think but yeah it's

45:51

possible but I think some of us a stumble across something unexpected you

45:57

can't solve it just by using all the known technology something you have to discover something that hasn't been discovered before yeah it might happen I

46:06

mean we didn't expect the point wedge we can do it to be proven I do it was open

46:20

has their own favorite well you know there's there's a sofa you want your

46:28

problems that are well compelling but also have such a sweeping soul so there's you know I mean I myself like

46:38

you know I have fun a problems that just look barely out of reach just he's I need to do a little bit of stretching to

46:44

to reach them and and I'm I mean these these these big really famous problems

46:50

they're you know you can deacon they're very frustrating to work on directly and

46:56

while they could escape know about this you know you shouldn't spend spend too

47:01

much time on dividing tally of logs actually you know these aren't bit with a more Within Reach so no there are many many emitter there

47:09

are thousands one so problems listed in the literature and we don't just work on

47:15

all the big ones pecking noise never worn a big once actually see cotton there's one finally

47:27

called the dean's reputation at work it's really certificate of appreciation for coming everybody I'd like to thank

47:36

you