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

Tao is a very good communicator. Modest, fluent, responsive, considered, honest, and humorous. Very good person, a great scholar and a gentleman to the core.

He stutters and says “um” a lot but still managed to give a concise, interesting, and clearly understandable presentation. So called “public speaking skills” only matter when you really don’t have anything of substance to say.

I have so much respect for all of you mathematicians. It is truly unbelievable what you all do. Cheers!

he is as beautiful as the mathematics he talks about. The key the learning mathematics is to listen to someone who speaks with such passion. That passion is contagious, wakes up the natural curiosity that is dormant inside of us. Just beautiful.

I was at office hours for one of my math classes here at ucla, and I turn around and just this guy (Terence Tao) just casually walking around. It was pretty crazy to think about all that he's accomplished and to go to a university where people like him work at!

It's insane how we can listen to this gold while just sitting at home ! Thank you Tao, UCLA, Internet, electricity, etc...

I don't know anything about maths; but I find this strangely therapeutic. It reminds me of how stupid and insignificant my life is, and it is very humbling.

Very thankful to have found a recording of this lecture. Deeply gracious, UCLA.

"How does one disprove a conspiracy?" The question with the greatest resonance in 2020!

Thanks for putting into simple wording the method of double-lock information transfer

One of the most beautiful lectures. Love it.

This presentation really opened my eyes and mind to prime numbers. I now wish I had taken a life path in mathematics. There is SO much in the pattern of primes that I wish I knew how to express mathematically.

"You want problems that are compelling but also have some chance to be solved" (True Intellect begets humility)

As someone who has been a Math guy, I love the great ones such as Terence who are so humorous. We love this subject and it brings joy that cannot be described. Number theory is deceptively simple in the sense of making statements, but notoriously difficult to prove. The proof of Fermat's last theorem is a canonical example. Taniyama, Shimura (also Weil) conjecture... What the hell have Modular forms have to do with a seemingly simple statement about some integer power sums. They do! Another example is from my thesis adviser's thesis adviser (Feit) ... Groups of Odd Order are Solvable (Feit - Thompson). This is beautiful. Once the commutator descends, one can apply induction and prove all kinds of things. This why the whole subject is so spiritual. Love it!

He's such a great explainer for both technical and non-technical audiences. A mathematician not afraid to use metaphors and analogies in explanation? Yes please. So humble, so brilliant. Love Professor Tao.

25:09 about how to send secret info over a public line, very cool. i never understood public key before.

I feel wonderful to hear this great Mathematician explain math...

Probably one of the best lectures I've ever watched, it's amazing how he can state the elegance of pure mathematics to his target audience with ease. Thank you. Things that I will continue to explore/learn/appreciate more into the future: *The Reimann Hypothesis *Goldbach conjecture *All primes except 2 are adjacent to a multiple of 6 *trillions of dollars is kept safe by prime number randomness. *There are musical "chords" of primes. *Sieve theory is like a sculpture that you carve out prime numbers from. *The arithmetic progressions of the primes and almost primes. *There are many unsolved prime number problems and most mathematicians work on the problems that are closest to being solved. This has encouraged me to try to study number theory as a hobby... Why not?

Even though the topic is very deep and difficult to grasp but he simplified it for a large audience. Amazing lecture easy to understand.

Beautiful, very clear lecture.

Indeed, we can only speculate! :) There were some parts in the video where he would find something funny but no one else would, it seems most people were laughing at things which relate to normal life, whereas he found mathematical formulas funny. A difference in humour, and humour is all about unsaid truths.

A memorable lecture that shows the "lumpyness" of the Number Line via the concept of Prime occurring in the unit cycle of "counting", which is a particular identification of ePi connection multiples.. And if you stare at the blank page or black board, the context in which we visualize Numberness condensed from Mathematical Imagination and Intuition, the omnidirectional-dimensional Origin of Superspin preceeds all function cause-effect of form, In-form-ation formulae. So the reference to the logarithmic relationships with Prime Number, the fact of the inherent-implied processes of Differentiation and Integration "Interfere" as numberness in potential possibility, of time-duration = calculation manifestation, substantiates the Mathematical holistic conditions of Polar-Cartesian Coordination and Geometry in Spacetime, by which observation in Methodology self-defining difference in Principle produce these Proof/Disproof circumstances. This lecture also promoted the idea of pseudo random fields of Uncertainty in otherwise specific resonance interference positioning Image condensation and certain vertices in vortices of e-Pi-i interference positioning resonance of resonances.., ie "simple", and messy, complexity. Mathematicians may prefer Spinfoam Totality of Superspin, a Modulation conception, to "resonances"? Terminology is the real problem of further Education, in my worthless opinion. The antilog "Chord of Superspin" Modulation Totality, (could be an opening to a "Music of Prime", Eternity-now Interval of Vector, Temporal Opera?). There's another well defined concept, pseudo random conglomerations of primes and cofactors, either in a lump of Granite conglomerate, or box of unsifted Sand, the idea is reducible to mathematical expressions. Same applies to "fluid" fields of Uncertainty in otherwise specific resonance interference positioning that becomes turbulence and the Navier-Stokes Equations.. The intuitive reason for the properties of numbers is the temporal vectors of time duration timing modulation, by the naturally occurring superimposed probability, in potential possibilities conception, of whole number base logarithms, in the Superspin-calculations=> antilog-> self-definition numberness context of e-Pi-i continuous creation connection QM-TIMESPACE Principle. Time is ONE at zero-infinity, and Timing is the hyperfluid state of zero-infinity sum-of-all-histories wave-package probability ONE, from which all phenomena self-define. (P=nP doesn't seem to define itself from the logarithmic fluid state, intuitively) Put simply, the time duration timing modulation functional probability of Primes and multiples of Primes=> cofactors, are the cause-effect embodiment of the QM-TIMESPACE Principle Universe.., this Mathematical Eternity-now Interval/format. The Kieth Newman book, "God is a Mathematician", is consequently a degree of fact, because "Mathematics has Substance" in Actuality.

Thank you so much for this video. It cleared up important questions I had about randomness.

By the way, beautifully clear presentation. Thanks.

It's somehow so hard to imagine him being a mathematical prodigy (in addition, also actually, a child prodigy, if more precise). He is one and it has been proven that he is a genius. But he looks so normal. Like a normal Asian young adult, just like any one of us (except for the fact that not all of us are Asian, lol). What makes him what he is? His physical appearance is also very pleasing and he seems like a sympathetic, honest and a polite person. I really do find him likeable. I watched this lecture some days ago already so I just thought I'd drop by and say what I just said. If I had to say something about this particular lecture, I do not have any strong opinions in one way or another. It was a nice basic presentation without any technical detail. I also stumbled upon Terry's personal blog and saw his mathematical powers in action. I'm a mathematical noob and would like to learn A LOT but, hah, I might never reach the level to even understand any of Terry's mathematical stuff.

Did he just unload about 80% of Number Theory in this lecture? ...

I used to help this guy with his homework

Great job Dr. Tao and thanks UCLA!

and keep note he said we do not grant the number 1 the property of primality, as the primes as a group use the number 1 in axiomizing them as a subset of the integers, thus using the number 1 is a self reference hence circular arguement, ie treating 1 as a prime is equally useful as saying 1/1 = 1, which is the required property of an identity element under the operation of division!

the encryption analogy kind of blew my mind

A beautiful man with a beautiful mind. All my consideration! Very interesting presentation - I liked so much.

In your wisdom may you find a path of light for us all to follow my friend.

Thank you, UCLA, for posting this video. Terrence Tao and his work has further inspired me to pursue mathematics, I someday wish to come to UCLA to take courses or go to lectures under Mr. Tao.

Brilliant, Terry has the guts to say "we don't know", very unusual. A brilliant lecture from a young genius

Nice voice to listen to and the knowledge is fascinating

Complex stuff made simple, very well explained.

Such a nice guy and a good lecturer!

This shows that as a number grows faster the density of prime gets lesser?.what one can infer or what about the significance is ,one can draw from this statement

Love his humility!

I HUMBLY LOVE YOU PROFESSOR TAO. INFACT YOU SAVED MY LIFE.

U can feel his mind going too fast his speech having a hard time keeping up

26:43 A neat trick and analogy! This should be listed in every schoolchild's book of puzzlers like the "get the people across the river" game.

Because I am also a programmer, I wrote 2 applications to demonstrate how Diffie-Hellman works and how RSA works. I wrote my own large integer math library in C to aid with the effort. I also used a large prime generator based on a Rabin Miller test. It can handle 100 digits easily.

Yeah mod functions can change things a lot. It's a simple concept though, which is used a lot in maths but not taught in schools. All you do for mod P is divide by P and keep the remainder. For example, our clocks use mod 12, so if 25 hours have passed, we divide by 12 and keep only the remainder. Thus, the time shows 1 hour later.

Honestly just by looking at this dude he seems like a super genius.

I...... never knew that about prime numbers. That took, what, one minute to say why primes are interesting. No one in my entire 12 years in school could be bothered to say it, or anyone else. Astonishing. Once you know the trick, just astonishing. Pretty neat is what I'm trying to say.

Just wondering: How many transfinite primes (if any) might there be between w1 and 2w1 ? P.S. I’m using w to stand for Omega.

I listen to this to help me fall asleep every night

I love this stuff. Pure genius. The great ones are very humble. I was fortunate to meet some of them as a young child (Weil, Borel, Atiyah, Bombieri, Godel … all at IAS (Princeton) ) where my dad spent some years.

There is actually a much more satisfying reason for not including 1 as a prime. If you define a prime as being a number with exactly four unique divisors, 1 and itself and minus 1 and minus itself, then the number 1 does not fill that bill. As far as I know no one uses this definition, but it seems to me that it would eliminate a common student confusion about the role of 1, and the role of 2 for that matter.

Brilliant ! Thanks

I like the analogies. Easy to understand

Constance Reid (sister of the brilliant Mathematician who died so tragically young and author of (among other books on popular maths) "Mathematical People") wrote: People have spent more time think-ing about prime numbers than war. A really cool talk. Also, there is an *excellent* popular maths book on the Poincaré conjecture. The part that Professor Tau (great name) left out, was that so many people were try-ing to claim the prize (for make-ing what were (it seems) minor corrections to his 3-part proof) that he refused to have anything to do with the money and returned to teaching and research. This is parallel to Richard Feynman's comment about the Nobel Prize he shared that the reward was to work on the problems and make some headway to *understanding* the universe. thanks, for the post -r.

I guarantee this man could fix the budget and point our country in the right direction. He is the type of man that would tackle any problem with cold calculating logic, and be smart enough to recruit any assistance he needs to fix any problems he encounters.

An elegant explanation of private and public keys in computer encryption :)

Thank you for this very much for this video. The video shows that pi is approximately 22 / 7. This value is approximately 3.14. Using the properties of this value we can compute prime numbers in sequence, which is based on the existing computing capability. The formula was an algorithm, that was developed by a well known mathematician. Using his formula and the method that I discovered, I can compute prime numbers in sequence using 22 / 7 .

What a charming great scientist. It's amazing that one can reach him on his blog, for advice or to read his thoughts.

19:05 all primes (except numbers 2& 5) contain a last digit of 1, 3, 7 or 9 or to put it in a different way: is n*10 + {-3, -1, +1, +3} !!!

Third slide and I lost it. I salute you for your intelligence.

I started a project where I'm took the base video and started cutting out ums, uhs, and repeated words. Already got it down to 42:25. Fascinating talk, though.

[-well-known result-] p.s. what does it sound like when you play the "music of the primes"... play a sample for us...

He's such a great teacher in a way that, despite his problems with the idiom (also he speaks too fast) and his obvious shyness, he can make very difficult math problems sound easier than they really are, also he makes you want to know more...

26.43: Why doesn't Alice simply subsequently mail the key and save a step?

He is full of awe at his own presence

id really like to study under this guy he is part Australian so I followed his career really inspirational

42:30 k^k Would be the inefficient bound given large k, as lim (trash^k)/(k^k) as k->∞ is 0. The trash^k is still interesting despite being terrible for small values of k.

Wow, what a great person. I wish I was his disciple.

I spent years searching for the essence of truth through poetry and fiction. Became a lawyer but was miserable every minute I practiced. As I continued searching for that essence, I was led to study mathematics. I feel that math has provided all the solace I had been searching for. If you're reading this and have wondered if mathematics is worth your time and effort, trust me it is. Math is an inexhaustible well of discovery and enlightenment. You will never regret it.

Terence Tao is the greatest living Mathematician.

19:27 aw come on man, why you gotta creepy zoom

**ROUGHSHOD CAPTIONS throughout e.g. [**22:27**] should be "What the prime[s are]" etc.**

I think I was channeling a little Douglas Adams with that thought. But I'm not even exaggerating: math is beyond comprehension in it's beauty and complexity. There's symmetry, fractals, recurring themes and hidden spaces everywhere. It's like an infinite cathedral built at the height of the late medieval. My apologies if you already know this; I'm taking some classes on modular forms and I'm just beginning to wrap my head around the symmetries and relationships.

he should write a book. "The Tao of Mathematics" :D HAHAHHAHAHAHA!

Tao comes in at 2:40

Really interesting!

very nice video!

Why right ear!

God delivered this extraordinary human being. It is not just his outstanding talent, but because of his impressive character!

Terrence, a brilliant mathematician. 7:51, Euclid proved (by contradiction) that 2, 3, 5 were not the only prime by multiplying then adding or subtracting one (2x3x5+1 or -1) thus creating 31 and 29 which are two prime numbers since if 31 and 29 WERE NOT PRIME, then by the fundamental theorem of arithmetic which states that every positive integer can be expressed as a product of at least one prime then 31 and 29 should have been, but they could not. That is, if 31 and 29 were not prime then they could be expressed as a product using 2 or 3 or 5 or DIVISIBLE BY 2,3, OR 5. Since they can't then the fundamental theorem of arithmetic would have been violated. Therefore 2, 3, and 5 were not the only prime.

It was a though sorry. For missing ? at end. Great presentation very passionate and addictive. Thank you.

Distribution of primes is explicitly defined (without randomness of any kind) by the following statement (MATRIX DEFINITION OF PRIME NUMBERS): There are two 2-dimensional arrays: |5 10 15 20 ..| 6i^2-1+(6i-1)(j-1)= |23 34 45 56...| |53 70 87 104...| |95 118 141 164...| |149 178 207 236...| |... ... ... ... | | 5 12 19 26 ..| 6i^2-1+(6i+1)(j-1)= |23 36 49 62...| |53 72 91 110...| |95 120 145 170...| |149 180 211 242...| |... ... ... ... | Positive integers not contained in these arrays are indexes p of all prime numbers in the sequence S1(p)=6p+5, i.e. p=0, 1, 2, 3, 4, , 6, 7, 8, 9, , 11, , 13, 14, , 16, 17, 18, , , 21, 22, , 24, , , 27, 28, 29, ... and primes are: 5, 11, 17. 23, 29, , 41, 47, 53, 59, , 71, , 83, 89, , 101, 107, 113, , , 131, 137, , 149, , , 167, 173, 179, .... There are two 2-dimensional arrays: |3 8 13 18 ..| 6i^2-1-2i+(6i-1)(j-1)= |19 30 41 52...| |47 64 81 98...| |87 110 133 156...| |139 168 197 226...| |... ... ... ... | | 7 14 21 28 ..| 6i^2-1+2i+(6i+1)(j-1)= |27 40 53 66...| |59 78 97 116..| |103 128 153 178...| |159 190 221 252...| | ... ... ... ... | Positive integers not contained in these arrays are indexes p of all prime numbers in the sequence S2(p)=6p+7, i.e. p=0, 1, 2, , 4, 5, 6, , , 9, 10, 11, 12, , , 15 , 16, 17, , , 20, , 22, , 24, 25 , 26, , , 29, ... and primes are: 7, 13, 19. , 31, 37, 43, , , 61, 67, 73, 79, , , 97, 103,109, , , 127, , 139, , 151, 157, 163, , , 181 .... So, in order to find primes we need not to perform any calculations, we simply have to find positive integers which do not appear in these four arrays. See: ijmcr.in/index.php/current-issue/86-title-matrix-sieve-new-algorithm-for-finding-prime-numbers www.planet-source-code.com/vb/scripts/ShowCode.asp?txtCodeId=13752&lngWId=3

Terry Tao is the real real Will Hunting.

That's true! We now have so many people in the world, that there will be enough people to do all the important and interesting things in life.

To say that there is randomness in the primes sounds so strange and yet it's so wonderful to know.

And I thought I was smart...

FORMULA OF PRIME NUMBERS Positive integers which are not members of arithmetic progressions: Q(ik)=a(i)+b(i)*k a(i)=6*i^2-1; b(i)=6*i-1 R(ik)=a(i)+c(i)*k c(i)=6*i+1 Are indexes P of all prime numbers in the sequence S1(P)=6*P+5; i=1, 2, 3,…; P=0, 1, 2,…, k=0, 1,2,… Positive integers which are not members of arithmetic progressions: T(ik)=d(i)+b(i)*k d(i)=a(i)-2*i; U(ik)=f(i)+c(i)*k f(i)=a(i)+2*i; Are indexes P of all prime numbers in the sequence S2(P)=6*P+7; i=1, 2, 3,…; P=0, 1, 2,…, k=0, 1,2,…

I am addicted to this topic.

OMG, this guy is a fields medal winner. He deserved it.

This mans IQ is so high, he isn't considered smart, proficient, genius, or even extreme genius. He is in a category all his known named unmeasurable genius (around 230).

I like the conspiracy analogy.

You can find interesting facts and puzzles about Prime Numbers and Magic Squares, Smith Numbers, and Arithmetic and Palindromic Primes on Glenn Westmore's blog.

At the trivial zero x = -2 of Riemann's zeta function are all the second twin prime numbers existing?

of course there are only finite amount of primes, just watch revenge of the fallen :p

Pn is approx nLog(n) Entropy = kLog(n) Interesting.....

I thought that it is about keeping the classroom at a certain level of concentration. I've seen so many people do it, and I never understood why. I guess you might be right, but we can't know for sure.

Can you help me with this... Other Twin prime conjecture If (P) is a prime number and (P + 2) is also a prime number, where P and (P+2) are twin prime numbers, then the result of (P mod 3) will be equal to 2. The remainder of dividing by 3 to the first of the twin prime numbers will always result in 2. So the only prime numbers that can be started with a pair of "twin prime numbers" are those whose residue with respect to 3 equals 2.

I smoked some pot and as always meditated upon reality. I could actually grasp the pattern of all the prime numbers but I can not remember nor explain; but at the time it was clear and very easy to understand how and why all prime numbers seems to appear randonmly and how they form everything there is in the material world up to some levels above where it is yet not spiritual but is mid way, for example the part and level of the universe where dreams occur.. I am positive that the insight I had about prime numbers is the final answer to mathematics, but I unfortunately cant recall the main ideas.. I can recall something that was fundamental for this understanding and it has to do with the real nature of numbers itself. I remember realizing the existance of only 3 fundamentals for creating all numbers, something like binary systems where the idea of 1 (one) is to the masculine principle and the idea of 0 (zero) the negative or feminine. the intercourse of both creates the plurality of all existing numbers.. And I also remember I could understand how and in what degree each of this 2 (1 and 0 ) would relate to each other and create "pockets" where each of this "pockets" would be enclosed by a positive prime in the beggining and a negative prime in the end enclosing this pocket.. for example in between the 2 and the 3 is one individual pocket the 3 and the 5 is another pocket and so on for eternity revealing all primes.. I can recall the theoretical part of all but I cant grasp the insight that made me recite primes for almost one hour amazed becouse I didnt know primes.. I actually could tell primes forever without calculating anything, thus I know it was not a pot trip, it was real ... It was really scary and I remember felling shivers all over my spine that felt like jolts of real electricit and my body was somehow numb.. It really felt like an extraordinary degree of inteligence where in the realms of numbers I could understand absolutly everything yet it was something really simple and fundamental.. I wish I could think of this again.. But the more I try when I get high the harder it gets to recall .. I guess it something sponteneous;; But if anyone into mathematics, try to think of the numbers as only 2 "beings" inteligent and living beings habitating higher dimentions of our personal universe. Try to imagine them "having sex" yeas its funny but I remember imagining numbers having sex and really "having a swing party" thats how I can describe my thoughts at the moment, than it was unfolding into understanding and seem where each prime would "give birth " to the created numbers.. It was in other words like looking into a lineage of numbers where the first ones to conceive would first conceive primes and the primes would conceive "lesser beings" . I know it sounds crazy, but at the moment I was really concerned about my sanity upon this thoughts.. 

Reductio ad absurdum, not absurdam. But it's ok, Latin isn't a prerequisite these days for being a genius in maths.

3 reasons why I like this guy: Genius, interesting and quick at math. Not faster than me though. If I asked him what 2+2-1 is, he would have to hear it first. I already knew the answer from the start.

Actually, 10, ten, is very special in number theory-there are constructions that work only for radixes 2 and 5 whence 10, e.g. dragon curves by 90°, (radix 3 works at 60°),... (P.S. I tested radixes 2 and 5 but not radices 2 and 5 together..., and radix 4 closed up...)