Terence Tao on Prime Numbers

[@subodhjam]

A great mathematician can be gauged by his inability to make eye contact. This guy is a boss.

[@swollpenispok8172]

Is more better or less?

[@jocabulous]

follow up question, how can you tell how much/little eye contact he makes from this video?

[@happy1288]

Wth lol

[@andrewolivetreemixing]

Lol

[@I_discovered_civilization]

Many geniuses are on the spectrum.. hence the lack of direct eye contact.

[@SalesforceUSA]

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...

[@hamburgeryumyum7491]

When this guy was 7 years old he could do math stuff at the level of a 24 year old

[@cloud9847]

"I don't even know everything going on" - fear strikes the crowd

[@mattmahoney890]

Feeeeaaaarrrrr

[@gris186]

Could listen to him explain math all day.

[@แม่เล็กแดงสว่าง]

gris186

[@bustownbc2787]

Shits boring and common sense

[@MixMastaCopyCat]

@@bustownbc2787 This is a terrible attitude for exploring math

[@YashSingh-mr9dz]

@@MixMastaCopyCat After some Math they're gonna say Shits boring and explodes head

[@jellyslapper2872]

Yup I could listen all day and still not understand math haha.

[@lionpersia]

Terence Tao is a top mathematician; the mathematics of 21st century will be remembered with his name. I've read his PhD thesis. Normally, a PhD thesis must be about 170 pages but his was roughly 40 pages and accepted. He's a genius harmonic analyst, which let him prove, along with Ben Green, that any residue class of any modulus has infinitely many primes. Also, he's a chief editor of one of the journals of the AMS. Oh, by the way, his annual worth is 463 000 $.

[@pranitgandhi6832]

If this is true, that's crazy!

[@zerosugarmatcha7348]

@Anderson Jeffrey He's not paid for writing on blackboard dude, he's paid for advancing the knowledge for humanity. He's well underpaid comparing the celebrities, athletes and politicians.

[@allall8695]

@Anderson Jeffrey That club of high rollers have gatekeeping mechanisms (*cough* income taxes *cough) that prevent individuals even with the fattest paychecks from getting in or sustaining their position there. It's a different pecking order entirely.

[@luigy0648]

@Anderson Jeffrey as @Zero says, this guy is quite underpaid compared to his advances and all he is given to human knowledge. Is not just writing on a blackboard. You could say stupid things like that about sports for example.

[@luigy0648]

@Anderson Jeffrey totally agree that there are people out there doing great stuff, for example, a lot of scientists with pretty mediocre salaries due to bad politics. Tao's work is great and I think he deserves that, as I also think there are a lot of people just getting to much

[@Light-vu6ws]

3:55 I didn't know that Euclid was attending Terence's lecture.

[@Light-vu6ws]

@Anderson Jeffrey I was joking

[@ruslannuriyev]

@Anderson Jeffrey The guy on the right looks like Euclid. That's what he meant.

[@procheck9220]

@Anderson Jeffrey Bruh.. is this your first day on the internet? the guy means there is a person in the audience that looks like Euclid...

[@amir3515]

@Anderson Jeffrey r/whoooosh

[@khoavo5758]

@Anderson JeffreyPretty sure everyone got the joke (beside you ofc)

[@johnchessant3012]

"prove something is true by proving that it is not false" So obvious, yet so useful

[@AbhishekSachans]

Yeah, Could listen him all day! His comprehensive expression of mathematics is very beautiful plus useful.

[@samanthawylie893]

I have no idea how I got here, but this is my third video in a row of him I've watched; and I'm beyond intrigued! What a beautiful mind.

[@Chataine91]

Surprisingly insightful. I could follow him quite easily and I'm not a mathematician.

[@stickyrice2141]

So, you're a good listener... LOL

[@PoliticallyCorrect]

@@stickyrice2141 shh

[@DivineMaunze-e9u]

Mr tao is my inspiration and indeed my fav mathematician,I listen to him very much

[@lakiboiBB4L]

its an honor to even be learning from him on youtube

[@Longshlong99]

I am wondering, 11 years later, if you would reply to this comment, how crazy would that be

[@raph8057]

it'd be even crazier if you replied to this one

[@Sutapa-qj1ir]

More crazier if you reply to this one

[@cosmicsapientia2447]

siuuuuuu

[@yangerrai1605]

This is the only video of him where I understood his lecture just because he talked about basic of real numbers also in a beautiful way

[@Qritiqal]

"It takes a while to get used to this type of argument." -- says the guy who understood it at the age of 4.

[@loggins2182001]

I have no idea what he is talking about, but I continue to watch anyway.

[@JimmyBoosterCrate]

This man is from another planet. Plain and simple.

[@normaljohn6035]

Learned about this guy doing research for my math history project (I’m a math major) literally yesterday. This guy is awesome

[@Nikkikkikkiz]

KZbin or Google collected your data

[@adrianusraditya8329]

I don't mind him being my maths teacher.. he's just so passionate

[@shucklesors]

oh you 'don't mind' him... wow what an honour it would be for him to not be minded by you to teach

[@petehenry7878]

@@shucklesors Why must you be an ass? Obviously Adrianus meant, I "WOULDN'T" mind him being my math teacher.

[@michelberden3717]

@@shucklesors lol

[@eurko111]

@@petehenry7878 you do realize how entitled it sounds?, to be the one to "not mind" have a renowned mathematician as your tutor?

[@petehenry7878]

@@eurko111 BTW sweetheart, Tao is a professor, a professor is a teacher. Either way he teaches more than one student at a time. Where as a tutor is a private teacher, so if anyone is making any kind of entitled comment, it's you.

[@dsbmgrey9504]

Euclid was a real genius.

[@MsRyanstone]

Yes he really was a towering genius

[@Runtime_dragon]

3:04 the most excellent reasoning.

[@winstonsabellona2204]

5:04 when you thought Terence will talk about something too complex and advanced(y I hesitated playing this vid) yet end up listening about basic number theory.

[@niemand262]

It's fascinating to hear that Euclid was "rejecting the null hypothesis" so long ago. This is a fundamental tool in science even today.

[@JM-us3fr]

I wouldn't exactly think of it this way. Rejecting the null hypothesis just tells us the null hypothesis doesn't fit the data as well as the alternative hypothesis (with high confidence), whereas a contradiction proof says we can't even assume the contrary without arriving at a paradox. One is incompatible with the data we happened to sample, while the other is incompatible with logic itself.

[@niemand262]

@@JM-us3fr It's fundamentally the same process. We bisect a distribution of possibilities, we demonstrate that one of the possibilities can't be true, so the other must be true.

[@nomarxistspls90]

@@niemand262 you are clearly not a pure math major. That’s ok. But they are NOT “fundamentally the same concept”…🤦🏻‍♂️

[@mauisstepsis5524]

This feels like a primer to primes for elementary schoolers not college students and professors.

[@hennyhalim3786]

he speaks so fast like his brain also thinks like that fast.. Cool!

[@keshavl1089]

I have seen so many dumbs speaking very fast

[@bipensubba4709]

Fool... You clearly are a hypocrite just of what you said. I suggest that you exercise your flawed logic.

[@felipebrunetta2106]

Considering tao has one of the highest IQs in human history he should have a hard time putting all of that in words

[@robertveith6383]

He speaks too fast.

[@ndk4]

He's like roger federer of math

[@Savage-ws7sy]

lol

[@bedroom7653]

ndk4 they both computers

[@jmiquelmb]

You mean Federer is the Terence Tao of tennis

[@rodrigo100kk]

Actually the current Nobel Prize mathematician is Arthur Ávila.

[@procrastinateurreformateur5968]

more Nadal :-)

[@XhanetMP]

I'm actually currently learning this in class. I love it !!

[@TravelWorld1]

Terence Tao is the greatest living Mathematician.

[@jenniferlawrence944]

ever heard of gregory perelman?

[@TravelWorld1]

@@jenniferlawrence944 no

[@jacoboribilik3253]

@@TravelWorld1 how can you not know who grigory perelmen is. He proved Poincare conjecture. And don't swallow everything Numberphile says.

[@tuberaxx]

Perelman is great, but I think Terence Tao is more versatile like Gauss and more collaborative like Erdös.

[@nomarxistspls90]

@@jenniferlawrence944 yeah the guy who turned down 1million and lives in his mums basement?

[@chaijackleng4486]

He is Bruce Lee of math

[@maxwellsequation4887]

He is too great to be compared to some dancing boi

[@nodeathingames2701]

tao of math.

[@AstroSully]

@@maxwellsequation4887 😴

[@joeyboyztng6400]

How about optimus prime that came to invade our world

[@xXxBladeStormxXx]

Optimus Prime didn't come to invade our world moron, he was trying to save it.

[@xeno4162]

yo surely are a moron

[@EDEsouth]

1059 vieuws ? this guy is a legend ! guys spread this and have it as favorite ! so we promote it ! and give it a 5 star

[@vlogsbyrow]

I read Simon Singh's "The Simpsons and Their Mathematical Secrets" which mentioned this exact proof, but I find it odd that he didn't mention one thing. There are two parts of Euclid's discovery. The first is what Tao mentioned which is that if you multiplied all the primes and add 1 it could result in another prime that wasn't part of the original set. You know it wasn't part of the original set because it is much bigger than all of the numbers in the set (for example 31 is much larger than 2, 3, and 5 since you're multiplying them to produce a new number). LONG STORY SHORT: Tao mentioned the first part of the theorem. What he missed was also amazing. Euclid said that if the number produced by multiplying all the numbers in the set and add one to produce a COMPOSITE number (i.e. not a prime number), then you can come up with even more primes. Let's say you have the set 2, 3, 5, 7, 11, and 13. If you multiply them and add 1, you get 30031. That is a composite number meaning it has factors besides 1 and itself. It turns out its other factors are 59 and 509, which are 2 new primes that were not included in the set. Why does this always produce new prime numbers? If you try to divide 30031 by any of the numbers in our set 2, 3, 5, 7, 11 and 13, then the remainder will always be 1 (which makes sense). Therefore, if a composite number is formed by multiplying all the primes and adding 1, it will always produce at least 2 new primes. I see that a lot of the comments are either saying that Tao's fast talking/stuttering is due to his fast mind or that they didn't understand anything, so I don't think this comment really belongs here. Respect the man's content.

[@98danielray]

that is an addendum if anything, since the "first part" already proves the theorem by LEM.

[@98danielray]

oh I see what you mean, you werent talking about expliciting them. the thing is this proof is generally given in such a way that the second step is considered obvious when I agree it should not be.

[@KeithRowley418]

Wonderful!

[@SuperRigo90]

I only watched the prime number parts of the video starting at 0:02

[@ryanchiang9587]

prime numbers pure elements

[@matthewkemp5343]

One of my academic heros

[@learnershome1251]

Me too. I love Prof. Tao

[@watherby29]

"This is abc fora" hits me like a sleep twitch.

[@areyouarobotz]

I did lol irl

[@ComputerCurry]

Lol

[@willcrawford7896]

5:20 what an interesting way of coming to a conclusion. I find that so creative!

[@kamon9339]

Will Crawford thats basically how most math problems get solved: by stating the opposite and proving that this isnt possible after

[@soondooboo1]

True, but there are many forms of proofs. There are direct proofs and induction is effective when dealing with sums.

[@nuc1eu52]

Millennium problems such as reimann hypothesis which is claimed to be proven uses proof by contradiction

[@nuc1eu52]

There are lots of other theorms which are proved this way, cause in mathematics you have infinitely large number to prove such thing lots of mathematics use this

[@guilhermefurquim8179]

@@nuc1eu52 Riemann Hypotheshis wasn't proved lol

[@Vandfeducky]

Terence Tao is one of the most smartest people in the world and yet still gets nervous talking to the audience.

[@TheOriester]

But (2 x 3 x 5 x 7 x 11 x 13 x 17) + 1 is not prime because You can divide it by 19

[@phillipchien]

Had to read that over a couple of times

[@adelarscheidt]

I'm glad youtube offers the option to slow down 0.75x

[@userma_r.cr123]

Adelar Scheidt loool

[@umarjanbhat3819]

😂

[@intelligence6743]

😂😂😂😂

[@gerjaison]

He does sound so much better, and understandable. You're a "practical" genius

[@IrregularPineapples]

Full talk somewhere? Link in description doesn't work.

[@abghaneedulla5049]

I wonder if Terence is able to calculate as fast if not faster than Ramanujam.. coz both of them are masters in number theory

[@ulilulable]

From the description: "To view the full talk visit [broken link]" Any chance this will be fixed?

[@barmouthbridge8772]

This guy and Dr James Maynard intonate the same when they say the word "Prime" .

[@ashutoshkumarjha41]

Awesome set induction of how an element or compound is composed of atoms by using concept of prime or fundamental theorem of arithmetic.

[@inocente106]

i was lost from 0:05 to 5:29 .. the rest i understood

[@FitnessFreak]

haha what is left lol

[@Anya_Boo]

Terence Tao defenitely needs a beard

[@piousseph6219]

Bro looks like he gonna live till 120

[@_glitchy]

Uploaded on my birthday

[@abrahamsikazwe7538]

The moment I noticed his habit of constantly touching his chin unconsciously I knew this man is a Genius.

[@prasadraos]

What a cool guy!

[@Phymacss]

He’s simply the best mathematician

[@СергейПавленко-х5я]

Perelman

[@daviddoch4872]

so much head happening in these comments... must be a rock star+

[@mahikannakiham2477]

What I wonder is why do we consider natural numbers as the product of primes instead of the sum of 1s? For example, instead of considering 8 as 2 x 2 x 2, why don't we consider it as 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 instead? By doing it this way, there would be no need for prime numbers, a sum of 1s is all we would need. Just a thought btw, because to me it seems that the rule "only divisible by 1 and itself and not equal to 1" is arbitrary.

[@SmileyMPV]

Mahikan Nakiham The two most important structures on natural numbers are addition and multiplication. While 1 might be the additive building block for all natural numbers, the prime numbers are the multiplicative building blocks for all natural numbers. This fact is used all over number theory and even other fields of mathematics. For instance, finding the greatest common divisor of two numbers is equivalent to finding their common multiplicative building blocks. In English: you can find the greatest common divisor of two numbers by looking at their prime factorizations and finding their common factors. Note that this is not the fastest way to determine the greatest common divisor, it is just an example of the usage of prime numbers

[@mahikannakiham2477]

Thanks for the explanation. I understand that primes are the multiplicative building blocks but isn't multiplication just a series of additions? For exemple, 2 x 2 is just 2 added 2 times. So to me, multiplication just seems like a concept we invented to facilitate calculations but doesn't seem to be part of the real world.

[@98danielray]

the natural numbers are in fact mainly constructed by successors

[@ZeroKelvin]

I'm clearly missing and important piece of information. It seems like we can generate new largest primes by just multiplying all of the prime numbers up to the largest then adding one. Or is that outside of current computational abilities?

[@Empyreangg]

If we had a list of all the prime numbers up to a certain point, then yes we could do that. The issue is that you can't be missing any primes up to the largest one you know about. Suppose you knew 2 and 7 are prime, but didn't know that 3 or 5 are prime. Then multiplying all the primes you know about (2 and 7), then adding one you would get 2*7+1=14+1=15, but 15 is not prime. The largest prime number known currently is 2^74207281 − 1, which is 22,338,618 digits long. We could find a larger prime if we knew all of the primes smaller than this one, but it would take more effort to find all the missing primes than trying to compute a bigger one by other methods.

[@thiantromp6607]

Martin Derige the number that you get from multiplying all the primes is not guaranteed to be a prime number, just to have a previously unknown prime factor.

[@divisionzero715]

There are a coupe of problems. One is, that primes tend to be more or less randomly distributed. Using this method on its own may leave gaps. The second is, as you mentioned, computation. Integer multiplication is a very fast operation, however, any machine would choke up for months trying to multiply 10^25 numbers for example. It's a good way to start, but it's not feasible in the long term.

[@powerdriller4124]

@@thiantromp6607 :: Right. It means that none of the known primes is a factor of that product-plus-one number, so it is either a prime, or has a prime factor larger than the largest known prime (and of course, smaller than that product-plus-one number.)

[@syahnazmi29]

if Terence Tao is a rapper, Eminem would be the pizza delivery dude while Nicki Minaj would be working in McDonald's

[@robinwu7333]

He speaks too damn cast.

[@aditya234567]

Wish Ramanujan's work got recognised similarly.

[@jonmoore9015]

Aditya N Ramanujan's work is very well recognized. The novelty of his story has actually eclipsed the work of more prolific mathematicians of the same era.

[@username17234]

More people (specifically mathematicians) know and revere Ramanujan than Terence Tao.

[@jmiquelmb]

I don't think I know the name of more than 20-30 mathematicians. I know who Ramanujan was

[@cobaltbomba4310]

The one who knew ''infinity'.

[@98danielray]

you indian nationalists are everywhere. everybody already knows about ramanujan

[@pronounjow]

Whoa, I get Euclid's proof now! That remainder of 1 is the key!

[@davithov]

I didn't understand the proof of the theorem that there are infinite numbers of primes, because you took as an example {2, 3, 5} set and then said that 2*3*5 + 1 = 31 is prime => the initial assumption that there are finite number of primes is wrong. But we took here {2, 3, 5} as an example and that 31 is prime and which contradicts our assumption just means that {2, 3, 5} is NOT the finite set (if it exists). So, maybe {2, 3, 5, p1...pk} is that set.

[@moosapatrawala1554]

he is agenius with iq 230 it is totally normal for him to speak like that

[@TheKrazyLobster]

I love this man

[@GreyWind]

Dude is overclocked

[@alexanderealley9992]

Answer to Riemann The answer to the Riemann Hypothesis is Infinity. Infinity times infinity equals infinity to the power of infinity. Infinity squared equals infinity to the power of infinity. If 2 is a prime then so is infinity. You are all welcome. All numbers are comprised of Primes but not all numbers are comprised of non-Primes. Primes make up the building blocks of infinity. They are telling the other numbers what to do. People are looking at numbers and infinity incorrectly. Infinity is Prime so case closed on the Hypothesis.

[@andik70]

The argument is very subtle. If you take all the primes until some number N, the construct P again as the product of all those +1, then this number is *not* always a prime. (I believed that for too long)

[@WilcoBrouwer]

of course the number of primes until N has to be an uneven number, since each prime is uneven, and even numbers cannot be prime (beside 2)

[@jasminebe1630]

He wouldn't mind saving me from IP class and doing my homework would he?

[@davidbolf2442]

nice.^ and where are you from? I'm 17.

[@intelligence6743]

Where can i find this complete video

[@xuanhuong1810]

thankssss you

[@dr.rahulgupta7573]

Sir factors of 1 are( cost + i sint ) and ( cos t-- i sint ) . Where i^2 = --1

[@thegoonist]

how do u prove the fundamental theorem of arithmetic?

[@alonamaloh]

It's a bit involved, but you can start here: en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic#Proof

[@GammaFZ]

video after terence tao teaching something: *this is abc fora* me: *nobody cares*

[@sreenivasramana5466]

GREATEST OF MATH TAO

[@Leeengold]

I don't get the proof. The fundamental theorem of arithmetic doesn't state that in the factorisation every prime can appear at most 1 time - they can appear more often. So why is the sum of all known primes + 1 not dividable by any other combination of primes? Lets say we know 1000 primes and the sum of all + 1 is dividable by 1000*p2 + 54*p3 + p99.

[@spherica1438]

1000*p2 + 54*p3 + p99 is not a product of primes (what this is would be called a 'linear combination'). Basically, any number x = p_1^(i_1)p_2^(i_2)...p_n^(i_n) Where i_1...i_n can be 0, or a positive integer. So for the proof: P = p_1...p_n + 1. If this isn't prime, it should be representable as a product of primes from p_1 to p_n (repeats allowed). This means some p_i is a factor of P. So, P/p_i should be an integer. But P/p_i = p_1...p_(i-1)p_(i+1)...p_n + 1/p_i p_1...p_(i-1)p_(i+1)...p_n is an integer. But 1/p_i is not, because the smallest prime is 2, so 1/p_i is either 1/2 or smaller. So P/p_i is not an integer.

[@Leeengold]

Spherica I failed with the linear combination :D But thank you for the further proof.

[@MsRyanstone]

He is not taking the sum of all known primes, but the product of all known primes

[@mujtabanasir2970]

5:02 i literally screamed out "FUCK YOU!!" to all the people yawning. I'm sorry

[@rohinandgaonkar9013]

Great man he can solve each and every sum and problems just because of his mind and memory.

[@hellopleychess3190]

the "memory" is not a thing, it is a matter of how you are

[@Chris_L034]

wth kinda of double talk was that?

[@davidbolf2442]

Perfect! :) How are you old?

[@GoldenAge3210]

I am a student Please anybody suggest me some genural Where I can get maths research paper on number theory

[@stolenlaptop]

Fun drinking game, take a shot every time he says "umm"...

[@ligesh5520]

im ur 1000th subscriber

[@ih8mcfly]

It’s amazing how some people seem to be just naturally talented at maths like him. And then there’s people like me who aren’t.

[@johnpaularango8627]

I am sure there is something amazing about you. ✊

[@ih8mcfly]

@@johnpaularango8627 my stupidity is off the charts

[@johnpaularango8627]

@@ih8mcfly live your truth King 🤴🤣

[@short-eu7bs]

I doubt he's naturally as smart as he is. Nobody who is super smart is naturally that smart. They might have an easier time learning, but you get to a point where you have to put in a ton of time and effort to be as smart as him. Being a master at something (especially as complicated as math) will never come easy to anyone.

[@mattsmith1039]

It's important to put into perspective that it's not a good idea to compare yourself to someone like T. Tao. He often spent hours after school reading mathematics textbooks, along with his naturally attributed genius. Many times people struggle with one aspect of mathematics, but due to the nature of the education system, this small error isn't patched up or fixed, so after getting say a 70% or so on a test the teacher moves on to the next subject. When the student moves forward, they find their journey in math more difficult because there are more blank spots in their knowledge. And this leads to them thinking they just aren't good at math, when this isn't the case. Sal Khan makes a great ted talk on this.

[@pigcake101]

Mr math himself explains some math

[@stephenlaw8451]

5:01 Woman on the upper left with the FAAAT yawn. LOL. Awesome video though, Terence Tao is an amazing mathematician

[@allstarmark12345]

2:48 my man Verdi!

[@cpxie3061]

Proof. Suppose there’re finite number of primes, Multiply them all plus 1=p. Clearly, none of those primes divides p since none divides 1. Now, we have an p that’s not a product of any prime and bigger than 1, by definition itself must be a prime, yet p is not within that total prime list we assumed. Hence, it’s a contradiction, and the original statement “there’s only this finite number of primes” must be wrong, and primes must be infinite.

[@Eyenn_n]

Where is the rest??

[@Gupitor]

live in Connecticut u.s.a where are u from?

[@piyushuniyal6129]

what a god...

[@patrickodoherty1334]

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.

[@si_quest]

I understood Everything But also understand Nothing. It's kind of odd.

[@jbman890]

+Super ATP Synthase Schrödinger disagrees

[@222222225574]

You got AIDS for sure!

[@pranavmalik6242]

If it's odd it may be prime.

[@u.v.s.5583]

Mathematically speaking, there exist even prime numbers, too.

[@bedroom7653]

U.V. S. Two*

[@lionpersia]

I am a Number Theorician. I've just received my PhD.

[@adnanashraf7621]

Hey man! Can you contact with me i want to ask about Mathematics?

[@Bikerider919-23]

@@adnanashraf7621 hello friend

[@SMDz]

THIS IS ABC FORA

[@ingzimmerman899]

Exactly what I have always said.

[@mantiilove]

Why does he like prime numbers so much 🤔

[@mehmetaliuzuner1858]

why there are chaotic gaps between prime numbers? why there aren't have any rules?

[@K92E]

Im counting the number of time his finger comes into contact with his chin and ... I got a prime number :D

[@howitworks404]

Beautiful presentation but would that proof be necessary as if there’s an infinite number f numbers then there has to be an infinite number of primes no matter how low the chance is of one showing up?

[@DrakePitts]

this argument does not work. you have to show why specifically the property of being a prime number cannot be limited to a finite set of numbers, which is what the proof here shows. it's not enough to have a hunch like this.

[@howitworks404]

@@DrakePitts ah yeah my bad I was sleepy when I wrote the comment thanks for correcting me

[@sickman8372]

He's talking faster than how Eminem raps

[@short-eu7bs]

Wait so any math guys in the comments want to explain to me why if you multiply all the "finite prime numbers" that the product won't be divisible by any of those numbers? I don't really understand how that works.

[@thecolossus_5917]

Do it by contradiction : if any p in {p1, p2, ..., pn} divides p1p2...pn+1, then since it also divides p1p2...pn (which is the product of all the prime numbers), then it must also divide (p1p2...pn+1)-p1p2...pn = 1, which is a contradiction. Thus, p1p2...pn+1 is either prime, or divisible by another prime number p that is not in the list {p1,...,pn} of all prime numbers, which contradict the fact that we had listed all the prime numbers.

[@star_ms]

Divide it by any of those numbers (long division). You'll always get a 1 in the reminder, so they're not divisible.

[@Gelsyviolet]

Che ammirazione!! Super!