The Last Digit of Prime Numbers - Numberphile

[@justinlasker6269]

If you try this in base 2, you'll notice a stunning bias toward primes ending in 1.

[@psych707]

underrated comment

[@AdamDavis444]

+Justin Lasker Wow. Just WOW. But get this, in ANY base, there's a bias AGAINST primes ending in 0. I know, right?!

[@chrismason1839]

+Justin Lasker You'd be correct, infact the only prime number [in base 2] that doesn't end in '1' is two (which is 10)

[@Jat_Bodyguards]

+Chris Mason in base 3 too 10 is a prime also in base 5 , base 7 , base 11 , base 13, base 17, base 23, base 29 , base 31 , base 37 ... ( I Think you got the PATTERN)

[@Maverician]

+Manhattan Project ... I seriously spent about 2-3 minutes trying to work out why that was. I need coffee.

[@wolken_bruch]

Dude, James is such a gift

[@wolken_bruch]

+Abdou Abdou thanks, they really are. i always get compliments

[@latinoheat300]

*tips fedora

[@wolken_bruch]

you made my day, sir. thank you very much.

[@wolken_bruch]

i know mate. i knew when he mentioned thread. was nice to read nevertheless. i know im beautiful

[@TechyBen]

I'm so like "0Oh, it's a base ten thing"... and James is "it happens in any base." I just got shot down. Your matter of fact answer is great. I love it!

[@prae197]

+TechyBen Well if it only happened in base 10 then I don't think anyone would care tbh

[@bvbinsane1vanity]

aww, you was hoping you'd be able to act like you're smarter then this guy :(((((

[@Leyrann]

First thing I thought "what about base 12?" Really love how, in base 12, you also have only 4 possible ending digits even though it's a larger number than 10.

[@simonmultiverse6349]

NOOOOOOOOOOO... if it happened ONLY IN BASE TEN, that would mean that there was something special about TEN.

[@s-nonymous0273]

@@Leyrann Well in base 12, 3 is not an option as 3 can divide 12. You only got one odd digit, 11 (it's B when written in base 12). So you gain one possible ending digit and you lost another, so the net change is 0.

[@ZimoNitrome]

I like solving equations since I can't really solve my life.

[@leedaniel2002]

Oh I feel like this is true for a lot of us

[@ruinenlust_]

So true.

[@BobStein]

+ZimoNitrome Keep at it, you'll get there. Seriously.

[@GegoXaren]

Skapa fler "How to" filmer. Det gör mig glad i allafall.

[@ericsbuds]

+ZimoNitrome life's variables are ever changing. there are few or no constants in our equation! be adaptive and youll go far!

[@robbowman8770]

It's amazing that this is new work. Given how numberphiles love primes, and given how many highly skilled mathematicians there are in the world, it just seems that this is the sort of thing that would have come to light 30 years ago.

[@janmdsdks]

3,7,9,1... Rings from LotR. Problem solved.

[@leo17921]

@@dielegende9141 r/woosh

[@diffway531]

jesteś geniuszem

[@EugeneKhutoryansky]

How can you focus on doing a video on prime numbers when there is an unsolved Rubik's Cube on the desk behind you? Whenever there is an unsolved Rubik's Cube sitting around, fixing it should always be the number one priority.

[@morris1818]

+Physics Videos by Eugene Khutoryansky So then it would be the... PRIME priority

[@tsunderesama9967]

It distracts me so much when I see an unsolved cube.

[@tsunderesama9967]

lol, you can learn how to solve a Rubik's cube on the internet. It took me about 3 days.

[@cobblegames]

+Maurice Merry dang u got me

[@brandonhall6084]

+Physics Videos by Eugene Khutoryansky You said the same thing on the Crash Course Physics video. Someone's passionate about Rubik's cubes. Lol.

[@akaRicoSanchez]

2:27 wait... why is the table symmetrical along the "wrong" diagonal?? If you tell me (7:1) is as likely as (1:7) I am like, sure... but that's not what is happening here: 7:1 is as likely as 9:3... Why? WHY?

[@leadnitrate2194]

If I tell you that 7,1 is as likely as 1,7 , you shouldn't be like, "sure". In the first instance, the difference between the two primes is 4, 14, 24 or generally 10n+4. In the second, it's 10n+6. In the case of 9,3 and 7,1 their differences are 10n+4 in both cases. So that might have something to do with it.

[@clam379]

I'm surprised we discovered this recently. It's interesting that we're still discovering more math properties to this day. Can you do a video on Chebyshev bias? I've never heard of it until today.

[@delve_]

I vote for this too.

[@simonmultiverse6349]

I still haven't heard of Chebyshev bias.

[@georgeivanov2322]

Dr James Prime

[@audeforcione-lambert7200]

It's symmetrical! Why is it symmetrical? Why is no one freaking out over this? This is crazy! It even works in base 3. Does anyone have an explanation?

[@ludvercz]

+Aude Forcione-Lambert I am totally freaking out about this since 2:20 and he doesn't say a word of it. Aaaaaaargh...

[@jasonneu81]

+Aude Forcione-Lambert Symmetrical in what way?

[@ludvercz]

+Jason Neu To the other diagonal. Also seems to work in any base I check.

[@jasonneu81]

ludvercz Oh yeah, I noticed that too! I was thinking more along the lines that they placed it like that on purpose or rather the way they placed the possible pairs resulted in the symmetry. Didn't really give it a second thought tho.

[@cecasiahaan6801]

Aude Forcione-Lambert it's also a parker square

[@digglebick]

i dunno what it is, but watching this guy get excited about numbers is really enjoyable. i have no idea about what hes talking about, im terrible with numbers and maths, but i watch a numberphile vid from time to time just to see this guy tell me things id be excited about if i understood

[@jamilhneini1002]

Omg this is really important it reduced the time usage of my prime number generator function a lot

[@Vulcapyro]

+Jamil Hneini What, were you still accounting for numbers ending in even digits and 5? lol

[@jamilhneini1002]

Vulcapyro well....kinda XD i was googling about it and found this XD

[@DiapaYY]

+Jamil Hneini And only check up to the square root of the number you're checking, it saves a lot on bigger numbers.

[@jamilhneini1002]

DiapaYY yeah i did that XD thank you very much^^^

[@thomasscrase7406]

+Jamil Hneini Also check only numbers you've already found to be prime since a number is either prime or the product of primes

[@DrRChandra]

Oh, boy...there's that "2 pi " again. That'll certainly encourage the tau fans to take notice.

[@hakkbak]

it's singingbanana's twin brother: singingPlantain !!

[@MrDasbinich123]

Just noticed that you look like the guy from singing banana. What a coincidence :O

[@ikasu00]

It's as if they are twins or something

[@rafabulsing]

I don't really see it

[@italego6]

+Ikasu0 almost the same person

[@cheongziyong8871]

It's almost as if they're the same guy. Oh look at what lunacy I'm spouting

[@ikasu00]

+Italego6 non-sense

[@Hedning1390]

3:00 you don't have to say prime gaps

[@WombatSlug]

I've enjoyed seeing Dr Grime's prime number sculpture in the background of the last few videos that you and he have done. I'm glad it's still around.

[@diegopalafoxalbacete8731]

Great video! And another amazing prime number's property. Loves when he says: "Who cares about base 10, right?(..).it would happen in any base...&...&... it does" hahahaha xD

[@ulob]

very clear and high level of enthusiasm. great vid

[@CaballusKnight]

I think the last phrase should've been one at the start, that the bias doesn't stand when things go to infinity. Because I've been in shock the whole video, I was like: This is monumental, Riemann Hypothesis itself might be disproved if the randomness aspect can be attacked. Like we just found neutrinos that move faster than the speed of light.

[@Airblader]

Agreed. But to be fair, the pattern not holding up is based on a conjecture.

[@muffinspuffinsEE]

+Awaclus that was his point :p

[@janeturner4196]

+Awaclus that's what he means. it's a big discovery that turned out to be fals

[@jbkralc13]

+CaballusKnight Agreed. This seems to be like saying, after three coin tosses of H-T-H, that getting heads is more likely when it would seem to be equally likely. A million or trillion primes seems like a large number, and indeed it is. But that's still an infinitesimally small sample of all primes.

[@IamGrimalkin]

+Jonathan Bennett But what if your coins follow Bose-Einstein statistics?

[@skakdosmer]

Why would anyone expect the numbers to be random? And since there are infinitely many primes, how can you be sure that the first few hundred millions are representative?

[@Airblader]

Watch the video until the end. The pattern doesn't actually hold (at least so it seems).

[@goldjoinery]

+Lau Bjerno Because primes resemble randomness. Many accurate heuristic arguments that eventually lead to conjectures rely on the primes' apparent randomness.

[@IllumTheMessage]

+Lau Bjerno One wouldn't expect Primes to be random. Which is exactly why their perceived randomness was interesting. In this case a bias is equally interesting. It raises an unanswered question. That being, what property of Primes could be said to cause this? A proven answer to that question would be a BIG deal.

[@volbla]

+Lau Bjerno Obviously the primes aren't random per se. Certain numbers are primes and others are not. But no one has figured out a Super Prime Formula which tells us exactly which numbers are indeed prime. There is no simple pattern to the primes, so any pattern we find is worth studying.

[@prae197]

+Lau Bjerno People want primes to be random because if we found an equation to come up with or 100% determine if a number is prime or not then they'd lose their glamour a bit. People love primes because it's so hard to prove anything about them because they DO seem to be random. Things like this question that randomness so it drives mathematicians up the wall. Primes are one of (if not THE) oldest and greatest mysteries of mathematics. Even Fermat's Last Theorem was proven, yet we're not sure if primes are random.

[@BigDBrian]

3:00 except that after 10 you still have a lower chance of the next number having the same last digit. for example, if you look at any of the cases of a prime difference of 18 or lower, then the same number(for instance, {9,9}) has almost half the chance of being the case of the others, since there's 2 more "chances" of the others to be prime. Although with larger gaps in primes this diminishes, it doesn't ever disappear(unless there's an infinite prime gap!)

[@MarcelloSevero]

+mrBorkD Yes, but what Grime said is that there is a large discrepancy between what is expected if this is the reason for the uneven distribution of primes and what is actually measured.

[@BigDBrian]

Lord Marcellus I agree, but he made it seem like there wasn't an expected gap there at all. He didn't address it well enough, in my opinion(only discussing gaps smaller than 10, then dismissing it as if it had 0 effect at all) It does not account for the entire thing, but probably for at least a bit.

[@n8style]

+mrBorkD I thought the same thing, sounded like he hadn't thought it through properly

[@jumpman8282]

+mrBorkD Technically, there is an infinite prime gap, between the "second-to-last" prime and the "last". :)

[@BigDBrian]

Jerry Nilsson False: there is no last prime, that's been proven.

[@xintakxvi]

Wow. It's amazing all the research that has gone into just prime numbers.

[@jjason18795]

4:44 "and it does" haha, don't know why I found that so funny

[@mladenstific2459]

+Numberphile could you do an episode on Chebyshev's bias? Sounds like a fascinating topic.

[@superj1e2z6]

This comment section is prime real estate this early

[@knexator_]

+superj1e2z6 That was a Parker Square of a joke :D

[@knexator_]

+superj1e2z6 That was a Parker Square of a joke :D

[@fakefirstnamefakelastname8305]

gu4t4f4c r/woooosh

[@stevetaylor4232]

i love discovery of therom like this biggest reason why i watch this channel great work more plz.

[@Ponsvoid]

I find your videos incredible... I find myself watching your videos for hours... and I have a history in the past of not liking math.

[@TiaKatt]

Yay! I was waiting for a Numberphile video on this pretty much since the announcement :)

[@DodderingOldMan]

My mathematical skills can be measured by how long it took me to realise why a prime had to end with those 4 digits. A while. Relatively.

[@AniketKumar-lw6su]

"We found a new pattern in primes!" "ONE THAT WE DIDN'T KNOW WAS THERE"

[@CaptainWolff]

Interesting that the percentages appear to be symmetric along the lower left to upper right diagonal of the tables.

[@LawatheMEid]

this channel made me humble, thanks for the statistical data... PRIME numbers were my minor hobby .. now they are the main hobby!

[@FooBarr1]

I was working on this very problem the other day. Someone complained and I was dragged off the plane and questioned by authorities for over an hour.

[@KnakuanaRka]

Idiot. What did he and the questioners think you were doing?

[@spootot]

kek

[@U014B]

If anyone ever touches you in a way that makes you feel uncomfortable, 3:19.

[@patrickhodson8715]

+Noel Goetowski whuuuu...?

[@Gameboygenius]

+Noel Goetowski I understood that reference!

[@hecko-yes]

Hah, classic Sonic.

[@darkseid3225]

+Noel Goetowski 3:44

[@whitherwhence]

Nice picture!

[@rogermouton2273]

This is the type of thing that fascinates me about mathematics: things that reveal fundamental properties of number, regardless of the base used. I find such things fascinating and mysterious, moreso than just about anything. It leads me to ask what is number? How does it exist? Why and how do mathematical properties such as this arise? And what is the mode of existence of these properties? I suspect this feeling of awe is what drives some people to make mathematics their life.

[@RinInABin]

Keep calm...! Count the primes!

[@AnnaMariaKorolczuksinger]

I just fell in love with your Brain Dr. Grime

[@tiuk23]

James Grime has his own channel singing banana. Head over to his channel for more.

[@ruinenlust_]

+tiuk23 Yeah!

[@nea89o]

i think he once said "there is a guy at numberphile which apparently lookls like me but... that's not me..." on singing bananas

[@PowerIsReal]

Fascinating. Thanks for this.

[@pegy6384]

No mention of the Octomus Primes--I had my fingers crossed, too! :(

[@simonmultiverse6349]

NOOOO!!!!!!! It's Octopus Plime!

[@midwestrailroadproductions8116]

You're such a funny math philosopher, Dr. Grime. I like your sense of humor.

[@lordgorko]

Okay, so I don´t know if someone else already pointed this out in the comments, but the prime ending pairs ((1,1), (1,3), (1,7), and so and so on), when arranged in that square manner shown in the video, form a magic square with their percentage of appearence, at least in the shown base 10 and base 3. It's not a perfect magic square (it doesn't add up in the diagonals), but still, a very neat thing.

[@IamGrimalkin]

+Lordgorko So it's a _______ square then? ^Insert word above.

[@ipassedtheturingtest1396]

+IamGrimalkin Parker \^-^/

[@Cityslickerr_]

+IamGrimalkin is your name based off the spooks series?

[@IamGrimalkin]

+AEON Shadow Yes.

[@lawrencebermudez]

So it's a parker square?

[@conradgarcia4850]

Hi James! Avid fan here. What a coincidence! I was up all night working on prime numbers. I stumbled upon the same discovery too but it's just the beginning. Yes, it's true that prime numbers and with 1, 3, 7 and 9 and i want to call 1, 3, 7, and 9 root primes because they are obviously the main numbers to identify primes with. meaning no prime ends with 0, 2 (except 2 itself) ,4,6, and 8. anyways, with this discovery, anyone, if informed by the idea can now identify any number if it is prime. but, right now, i could only give 40/60 chance. anyways, i've discovered more than the root primes but my research is just beginning so maybe i shouldn't be too sure yet, that they also have their partners. there are less 40 of these numbers that make up all the primes. one more thing, I'd like to express my opinion that one should be promoted as a prime number because of more than 2 digit prime numbers ending with 11, 21, 31, 41, 51, 61, 71, 81, 91 and 101. but that's just me. anyways, im glad that the mystery of primes are slowly getting unraveled.

[@bencrossley647]

James: "Primes don't like to repeat their last digits" Primes love to repeat their last digits! Just ask Binary!

[@CaptainBlue777]

I love what you did, there, in the description! More Prime and More Grime! Clever!

[@ExplosiveBrohoof]

Chebyshev's bias doesn't state that primes ending in 3 or 7 are slightly more likely. It states that in general, there are slightly more primes ending in 3 or 7. This is an important distinction. The proportion of primes ending in a 3 or 7 converges to 0.5, but most of the time, it sits above 0.5 (though sometimes it dips below).

[@liquidminds]

I like to explain it as "all primes larger than 10 end with ... " makes it a lot easier to explain... I was looking into what the longest prime, constructed of another prime, with another digit added to the end through concatenation was . Which in itself is a fun task, since it offers quite a surprise at the end. Recommend it to anyone who has a couple hours to waste.

[@rosiefay7283]

Heuristically, there should be infinitely may primes of the form 10p+3 where p is prime. Likewise 10p+9 (not necessarily the same p!). And likewise with 1 or 7 added to the end, but then p needs to have the form 6k+1, and then you get 60k+11 or 60k+17. If there are infinitely many, there is no longest one. But unfortunately, so I read, proving or disproving such a statement seems to be impossible. The task is about as difficult as proving the conjecture that there are infinitely many Sophie Germain primes (prime p where 2p+1 is also prime).

[@IIARROWS]

What about primes in base 2? I just found out that, except one 0, everyone else ends with 1!

[@happmacdonald]

+IIARROWS That's right. And surprisingly, pairs that both end in 1 (written (1,1) using the above notation) appear to be the most popular configuration by far. :D

[@TheWTFDubstep]

That's because odd numbers end with 1 when converted to any even base

[@IIARROWS]

@@TheWTFDubstep You don't say?

[@F1R3S74R73R]

2:51 It's further away: It does absolutely could have an effect if it is more than 10 away. If there is a prime X, and the prime following X is more than 10 away, lets say its 11-20 away. Lets say in this 11-20 away region, there are 2 primes, 12 away and 14 away, well, the prime following X will be the one that is 12 away

[@yqisq6966]

Nice to see video about contemporary research :3 How do these probabilities change if you make it hexadecimal? Or binary?

[@IchBinKeinBaum]

+Yang Qisq Binary is easy. All primes end in a 1 (except 2).

[@josephkahan7677]

+IchBinKeinBaum Great point ☺

[@therealzilch]

Lovely and convincing. Thanks.

[@Kricen]

I have a feeling that primes have a pattern, but it's probably too complex for mathematicians to figure it out.

[@reshpeck]

I agree. I don't think that they are random at all. Neither do I think pi or the square root of two are random.

[@Gauteamus]

2:21 - I wonder: Why would that table be symmetric across the diagonal from bottom left to top right?? There must be a pretty obvious reason, and I have not given this much thought, but at first glance, that is pretty creepy.

[@GoodTimeswithMinecraft69]

This is indeed an interesting find! I just read the arxiv paper, and must say that just another intriguing property of the prime numbers was added to an already endless list. Concerning the core observation, however, I'd carefully venture out and assert that it is less surprising if one views the occurrence of prime numbers as not being random. I know, statistical analysis shows that to be the case, but a look at the Ulam spiral alone clearly leads to the contrary impression. Statistics can be misleading. Now, in each finite interval, prime numbers are nothing but the result of a superposition of periodic functions (the simplest version is the sieve of Eratosthenes). With that, it is not hard to imagine that some residual structure stemming from the fully deterministic construction remains. Of course, this by itself does not at all explain quantitatively the specific observation, but makes it appear less mysterious. It is like the patterns appearing in the Ulam spiral. Apart from that, prime numbers remain possibly the most mysterious objects in mathematics, but in my humble opinion this is due to the disparity between a very simple rule of generation and the resulting properties of the sequence itself which continue to evade our explanation and understanding. The stuff out of which madness is made ... gosh, I love prime numbers :DDD

[@coopergates9680]

+Good Times with Minecraft Why does x(x+1) + 41 do so well... lol

[@aepokkvulpex]

I gotta say, you really got me with the base 10 argument. I wasn't expecting that theory to be turned down. So weird that it works in other bases!!!

[@Nalkahn]

What about base 6 ? Beside 2 and 3, all primes ending in 1 or 5, right ?

[@NonDelusional74611]

+Nalkahn I always thought base 6 was interesting for primes.

[@FekalniVuz]

i just love this guy

[@TGC40401]

I have a swinging banana... Oh, yeah.

[@333_BDON]

اشكرك على الترجمة 🌷

[@hydroxychloride9077]

I can't agree with the intuition of primes being random. Just by way we can find all primes by using the Sieve of Eratosthenes, it means to me that they have an intrinsic pattern.

[@Airblader]

The sieve works on the definition of primes, not on a pattern. It doesn't say anything about their distribution.

[@hydroxychloride9077]

+Airblader indeed it doesn't create a pattern but primes are anything but random. So even if they seem to be randomly spaced, the video and this idea show to my pov that it is not :)

[@fiona9891]

+HydroxyChloride The only reason the primes are primes is because they aren't multiples.

[@hydroxychloride9077]

so there is a reason, this is not a random choice among all the numbers, that's what I mean. Even if it's their definition, it's also the way they are chosen ^^

[@KrazyisSloth]

+HydroxyChloride It's an arbitrary definition. I can define loads of series of numbers that won't necessarily have any patters in them. The easiest way it obviously just to state a bunch of numbers at random. An infinite set is more tricky because you'd need to prove ideally that they are independant, but e.g. digits of pi and e are suspected to be.

[@Prasen1729]

This is crazy, math is most mysterious thing in and outside observable universe.

[@Enedrapvp]

Awesome, been bothered by prime problems in my brain for about a year. Even have had restless dreams... I'm not a mathematician, majored in comp sci. But logically, primes are important.

[@liesdamnlies3372]

"Primes are important." Goodness, in the context of comp sci, that's the understatement of the century. :p

[@Enedrapvp]

***** Guess so, then again the extent of my work with primes is maybe a sieve...

[@chadisbad6]

+lies damnlies I'm starting my comp sci course this fall, mind explaining to me, even in short, why that is?

[@liesdamnlies3372]

Chad M I'll give you two examples. For hash maps (data structures that use key-value pairs and a hash of each object being stored to determine "where" it will be stored; I'm summarizing a lot), primes are used as part of the algorithm to determine a piece of data's hash code, which helps to prevent collisions (two items having the same hash code, which then incurs a performance hit as the hash map figures-out how to resolve the conflict). Prime numbers are also the basis of RSA encryption. In general, primes are actually pretty handy for cryptography.

[@liesdamnlies3372]

Local Bakery "Guess so, then again the extent of my work with primes is maybe a sieve..." For the record: I lol'd. Very punny. :D

[@SpitefulAZ]

you make me love prime numbers.

[@waynelin592]

Interesting. even I can't understand whatever they said I can still feel this is interesting.

[@sk8rdman]

I expect the bias will become less noticeable more quickly in lower bases. Check the biases for the first 1,000,000,000 primes in base 10, and then check for the first 5,159,780,352 (1000000000 in base 12) primes in base 12. I Hypothesize that you will get similar results.

[@AGCDoctor]

No views? What is this madness?

[@adelin081]

+AGC_Fenrir same here xD

[@AstolfoGayming]

+AGC_Fenrir The same madness that made you get first comment. *Bows before your greatness* Also it's probably just a really new video. I'm here at 149 views.

[@elenap15227]

149 now

[@formsacello7988]

I would love you guys to do a video about primes and factorials. Could talk about Legendre's formula and that stuff! I'm curious what the numberphiles have to say about the relationship between base-p digits sums and p-adic valuations of factorials.

[@beliasphyre3497]

So what about bases larger than 10?

[@perf2.078]

+Belias Phyre Yes, it always puzzles me why so many people and even the mathematicians among them try to find some patterns in different sequences based on the decimal system, when it is absolutely arbitrary, and even not the best one for counting numbers. But he addresses it in his speech.

[@Le_Tchouck]

+Belias Phyre Hello, it's probably the same and you'll encounter extra digits (like the letters A to F used in hexadecimal/base 16).

[@panthar1]

+Belias Phyre I would say try base 30, base 210, and base 2310 to start. Then follow with larger primorial bases. I am a bit skeptical of this trend continuing, but could be wrong. Would love to see a follow up video where someone does just that.

[@virshirevirshul3083]

I will code this for base 30 over the weekend. Stay tune for my results :P

[@GeneralJarrett1997]

+Belias Phyre I was wondering the same thing.

[@enolastraight577]

Except for 2 and 3, all primes are of the form 6x+1 or 6x-1. If you add or subtract the 1 and end up with 6x, would you discover a more calculate-able pattern in the last digit distribution?

[@aliedperez]

but prime numbers are not random...

[@iZainAli]

true it just could be a complicated equation(just my opinion).

[@baekalfen]

Granted I'm not a mathematician, but "just" a CS student. When I look at an Ulam spiral, I can't help but feel that there must exist an exact equation for all prime numbers.

[@WR3ND]

I'm not a mathemagician either, but the idea makes me think of fractals none the less.

[@npip99]

+baekalfen Not sure what that rly means. You can simply define an equation to be the nth prime number. Equations are just mappings. There are any number of ways to otherwise define such a function, here's two methods primes.utm.edu/notes/faq/p_n.html

[@coopergates9680]

There are equations from that spiral that are highly rich in primes, but none can always give primes.

[@remixener22]

Found something groundbreaking about primes in base 2 and it is that none of them end in 3 4 5 6 7 8 or 9 and I think we should make a video about this mind blowing pattern

[@hpekristiansen]

Ok - so nothing strange going on at infinity!? /Who cares about the first few primes!?/

[@muffinspuffinsEE]

^^ haha indd!

[@RylanEdlin]

We don't know for certain what happens when we go to infinity, we can only guess

[@tarcal87]

+hpekristiansen No, think of it this way: If the apparent random nature of primes were true, we wouldn't need a million or trillion of them for them to even out. The fact they still don't even out means there is an underlying pattern (not random), which would the ultimate goal anyway - the key to understand/predict primes. Think of it the other way around: if those endings were statistically random - it would be all the harder to even imagine any pattern behind it. At least my wee opinion.

[@z-beeblebrox]

+hpekristiansen The thing about randomness is it's supposed to be homogeneous. You can't trend *toward* randomness, because that, inherently, isn't random :P

[@joshignatowski8207]

+hpekristiansen If it was just a random pattern that happened in the first few primes, it wouldn't remain constant when the base is changed.

[@mostafamahmoud2658]

a set of consecutive primes create apatterns of composite numbers

[@MrMakae90]

"Do not try bending the spoon. That's impossible. Instead... only try to realize the truth. There is no spoon."

[@nerdalert226]

+Lucas Balaminut Relevance?

[@MrMakae90]

nerdalert226 seeing a pattern, or the shadows of a pattern, in the sequence of Primes would be like seeing the code behind the Matrix. Primes (the spoon) would not be independent and individual entities, but a collection of consequences of a much deeper logic that generates those values (like a specific function). Assuming, of course, that there is a pattern in the first place ;)

[@MrMakae90]

***** no. That is not what I said. There is the possibility that the primes are NOT the result of a formula. But if they are, and you can see this formula, then it is like seeing the code behind the logic that governs our world - like in the Matrix. Read again.

[@emmanuelramirez3482]

Hey Brady you should do a video exploring the uses of the trigonometry unit circle, it's very interesting and many could make good use of it!!

[@xiaoruli5996]

That's awesome. But I'm just meanwhile wondering, what if this is just a joke that the nature plays with us (or we played with ourselves)? Even if we checked the first 1,000,000 or the first googolplex primes, that's still just a small amount of numbers which only we humans conceive as huge. I mean, we are checking against infinity, not that we are calculating whether it's safer to travel by train or by plane. Maybe this is only a coincidence that just kind of occurs in the human checkable primes. We are only seeing numbers that are "practical", but if some bulk beings talk and think in googolplex^googolplex or other non-human-practical numbers as a daily basis, they may find other fun stuff about the nature. There may also be some inferior beings finding it surprising that primes simply make 40% the population of positive integers, when they are able to understand infinity, but in their reality they just checked until 20, deciding that it's already an astronomical number. I'm not saying this discovery is nonsense at all, on the contrary I'm fascinated and excited. But without logical proof, we shall always be ready to get a disappointing answer someday. :)

[@coopergates9680]

+Xiaoru Li For any base larger than 8, there are no prime gaps big enough for consecutive same digit endings to even be possible until the gap of 14 between 113 and 127. If you write a program to test this, go ahead and rule out any primes smaller than the first two that produce a gap of >=(base number). The *average* gap is probably still somewhat smaller than the base size when that 1st gap is reached. You could eliminate even more small primes. Of course, you argue for how small of a fraction of all primes are actually being tested, but this gap size issue explains one possible reason for the bias that isn't what the conjecture is going for. You guessed it, for odd bases, the prime gap needs to be at least 2*(base number) since n+odd = m would make either n or m even.

[@shack8110]

Yes, that was my thought also, and I think several people have wondered about that in the comments section as well. It's an interesting property, but there is an infinite set of prime numbers. Looking at the last digit of the 1st 5 million primes, for example, is almost a null sample size compared to the entire universe of all primes. Since humans only know of, say 5 million primes, and we know the number of primes is infinite, then isn't the sample just too small to conclude there is some bias? I'm sure the Professor has thought of this also and I wonder if he had given some response? Maybe the primes humans know of is large enough to conclude that there is a bias?

[@rainjar]

There is a certain symmetry - proportion of (1,3) + (3,1) = 13.4%, p of (1,7) + (7,1) = 13.9% p of (3,7) + (7,3) = 13.8%, p of (3,9) + (9,3) = 13.9% and p of (7,9) + (9,7) = 13.4% and p of (1,9) + (9,1) = 13.4%. And p of (1,1) = p of (9,9) (4.6%) while p of (3,3) = p of (7,7) (4.4%)

[@MrGoatflakes]

It's almost as if the primes have structure xD

[@noslohcinkin]

+MrGoatflakes ...they do, it's the non-random structure of non-divisible numbers. I'm not clever enough, but a savant might be able to see it...

[@TheSweBoo]

The unsolved Rubiks cube in the background buggered me for the whole video xD

[@Neueregel]

I analysed the first 1 Quadrillion primes. It's (a,a) = 5.3% and (a,b) = 7.3%. Apparently they both converge to 6.25%, so this theorem is BS

[@Vulcapyro]

+Neueregel can confirm, analyzed the first quintillion myself just today 10/10

[@liesdamnlies3372]

+Vulcapyro Pffffftt. I just analysed the first quintillion-quintillion over dinner with a pencil and some paper.

[@Vulcapyro]

+lies damnlies appropriate username for this thread

[@Jat_Bodyguards]

+Neueregel Where did you got those quardillion primes . I have first 4 billion primes an that is a 10 GB File. 4 * 10^9 = 4 billion primes = 10 GB . And 1 quardillion primes = 1 * 10^15 ~ 5*10^6 GB 5000000 GB File? hack on earth?

[@Neueregel]

Manhattan Project *I got them in my digital REM dreams. Each prime calculation required 1 picosecond, so 1 quadrillion primes required just 10^15 x 10^(-12) seconds = 1000 seconds. That's not even even half an hour of dreaming. Then, I woke up and had a epiphany that this theorem is total BS, because my Empirical math common sense says that Statistical Law of Large Numbers converges the limit of all specific endings towards 100/16= 6.25% each. Any Casino hacker will tell you the same. Also, do you see that Feymman guy in your avatar? I have same math intuition as him, and also the same taste for strip girls like he had. How do you like them apples?*

[@josephyoung6749]

Interesting to see a log function and the pi constant in that same Hardy Littlewood equation.

[@alexabaxter3355]

That scared me. I thought they meant they had somehow found the last digit of some last prime number!

[@villanelo1987]

+Alexa Baxter Ah, yes, the legendary last prime number, in a system with infinite prime numbers! Of course you would be scared, you were breaking the laws of physics! :p

[@anvaybate3001]

Haha same!

[@godsadog]

WHY AM I SO EXCITED FOR REAL?

[@NickZack]

Hey its that singing banana

[@ratlinggull2223]

they're twins

[@Minecraftster148790]

It is singing banana

[@harrybarodawala3588]

rubilk's cube, dice, old calculator, abacus in background there to fit the stereotypes of smart people

[@brandonhall6084]

Why are Mathematicians so fascinated with Prime Numbers?

[@ADrunkPanda]

+Brandon Hall In part because primes are the 'building blocks' for all integers. The largest application of prime numbers today is in cryptography: one can use large prime numbers to encrypt data. Plenty of Numberphile videos on that very topic :)

[@LittlePeng9]

+Brandon Hall Another reason is that the prime numbers are extremely simple objects. Then you can ask seemingly very simple questions like "how many prime pairs are there which differ by 2?". Only then you realize that this question you've asked is an extremely complex one and reaches into many different parts of mathematics. This is what mathematicians love - simple ideas which force you to look at them from many different angles.

[@goldjoinery]

+Brandon Hall We just are.

[@gavin9088]

+Brandon Hall I think it's because prime numbers are so widely known. Everyone who's gone through elementary school knows the definition of a prime number and a few of them, so to have such a beloved concept like prime numbers still have a plethora of information that hasn't yet been understood about it means that there is always an interest. Same thing with Pi and other logically irrational concepts. One of Brady's other channels has a video of all the quack math papers people sent to the royal institute since its inception and even back then there was a huge interest in prime numbers.

[@firen777]

+Brandon Hall the same reason you have secure Internet communication.

[@rosiefay7283]

2:35 "If that's the case, then we're looking at prime gaps less than 10". The explanation you tried to refute can't be dismissed that way. Where p=10j+9 is prime, *it doesn't matter which decade* contains the next prime >p. Let's say it's the decade of 10k. Then we're given that every integer between p and 10k is composite. The next prime is *still* most likely to be 10k+1 and least likely to be 10k+9. Even if it's unlikely that 10k is just 10j+10.

[@captinobvious4705]

does the bias trend down the closer you get to infinity

[@Diotialate]

yes

[@JasonVoorheesFri13]

+PigSnowball I think he meant "tends" towards infinity

[@captinobvious4705]

Jason i meant does the bias go away the higher you go (approaching infinity)

[@error.418]

+Tiger King you mean is there a regression to the mean? Yes.

[@DiapaYY]

+Tiger King yes it does

[@luisa.machado6595]

Dr. James Grime, how about using base 6? After all, a lot of primes are either one more or one less than a multiple of six. E.g.: 5 and 7 are both next door neighbors of 6, 11& 13 are both "around" 12, 17 & 19 of 18, 23 (but not 25!) of 24, 29 & 31 of 30, and so on. Thus, the distribution in base 6 might be far more interesting!

[@JimFortune]

What about prime endings in bases other than 10? I guess I should have watched the whole video before commenting.

[@coopergates9680]

Can you code and test it yourself in more bases?

[@custersword7746]

I see your Primer in the back round This was a great video about the last digit of primes

[@Rippertear]

hmm. that reminds me, I solved pi. the last digit is 4.

[@jumpman8282]

+R!ppertear AKA SDL Benj!! TGPASOTS!SAPROA(THR)STTLOYTYVMDFTBYTBCTSAASP!YH Makes sense.

[@souravzzz]

The last digit of pi is 0. (in base pi).

[@Rippertear]

⚛ U Wot M8 ⚛ I believe it would be 10 in base pi, wouldn't it?

[@souravzzz]

That's correct! Everything is 10 in base itself.

[@TheMursk]

+R!ppertear AKA SDL Benj!! TGPASOTS!SAPROA(THR)STTLOYTYVMDFTBYTBCTSAASP!YH Last digit in 10 is 0, isn't it?

[@YamiBeast]

Can we please get a video similar to those on Sixty Symbols, but talking about Mathematics professors and how theories/conjectures are proven etc.? Thanks.

[@wfcyellow]

What's a prime?

[@ruinenlust_]

+whyaremeninmyshoes A prime is a number that's ONLY divisble by itself and 1.

[@James_20]

+whyaremeninmyshoes any number that is only divisible (leaving a whole number) by itself and one is prime. The first few are: 2,3,5,7,11,13...

[@HoTrEtArDeDcHiXx]

+whyaremeninmyshoes An indivisible number (divisible only by 1 and itself)

[@Standard37]

Why are there men in your shoes?

[@JNCressey]

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

[@Xeroxias]

This is a great topic!

[@cfgcfh6350]

a small sample of 1 000 000 000 000 primes

[@oz_jones]

I know that this is said in jest, but in reality, a sample of 1 000 000 000 000 primes is merely a fraction of all of the possible primes.

[@joshgadget]

I need to hear more about this discovery

[@Melomathics]

So basically this is a mere numerical illusion because we didn't analyse enough primes?

[@abohlson]

The issue is that we could never analyze enough primes. Because they are infinite, no matter how many we include it is an infinitesimally small sample size.

[@ZardoDhieldor]

+sKebess Maybe yes, maybe no. There is no proof yet.

[@drewfro666]

+sKebess Not exactly. All we know at the moment is that the bias approaches zero as the number of primes measured increases. It could never completely go away. If it covers the first few trillion prime numbers, there's definitely something fishy going on.

[@Melomathics]

A trillion compared to infinity is still pretty much nothing though.

[@drewfro666]

sKebess You're right, but "numerical illusion because we didn't analyze enough primes" is an exaggeration. It's just that the bias is inversely proportional to the number or size of primes measured. It's still a measurable mathematical phenomena. There's still no evidence that the bias ever does actually reach zero, just approach it to the point of being negligible.