The Last Digit of Prime Numbers - Numberphile

Dr James Grime on a new discovery in the Prime Numbers.
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[@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.

[@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.

[@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

[@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.

[@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.

[@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.

[@kajetanczerwinski3962]

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

[@leo17921]

@@dielegende9141 r/woosh

[@diffway531]

jesteś geniuszem

[@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

[@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

[@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.

[@hakkbak]

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

[@georgeivanov2322]

Dr James Prime

[@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

[@mladenstific2459]

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

[@DrRChandra]

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

[@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.

[@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.

[@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%)

[@Hedning1390]

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

[@xintakxvi]

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

[@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).

[@pegy6384]

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

[@simonmultiverse6349]

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

[@abedfadila9266]

i like listening to this guy, the way he explains things always gets me excited for math

[@TiaKatt]

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

[@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

[@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.

[@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!

[@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!

[@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?

[@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.

[@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

[@WorldCollections]

It's giving you mathematical properties of the base system you end up using AND properties of primes, both fascinating things.

[@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

[@AnnaMariaKorolczuksinger]

I just fell in love with your Brain Dr. Grime

[@RinInABin]

Keep calm...! Count the primes!

[@LawatheMEid]

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

[@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 ☺

[@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?

[@stephenbeck7222]

The first really cool thing I saw in the linked paper is that they credit Tadashi (yes, THAT Tadashi, with the cool toys and his foot) for the inspiration to do the research.

[@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

[@keeperofthegood]

I would want to know, if a prime ended in a 1, and a prime following ended in a 9, what would the next prime likely end in. The assumption would be that the bias works the same in both counting directions, but that is only an assumption.

[@PowerIsReal]

Fascinating. Thanks for this.

[@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.

[@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.

[@TGC40401]

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

[@harrybarodawala3588]

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

[@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

[@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

[@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.

[@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.

[@19750bob]

So every prime can be factored and/or displayed in standard form with a monad of either 1,2,3,5,7,9 which in turn can be made in just 1, 2, 3 [5 is 2+3, 7 is 2sq+3, 9 is 3sq- we dont do that to simplify 1 for obvious reasons]. So every prime can be factored with a monad of up to 2 of either 1,2,3 using basic functions. Using basic functions (+,-,x,/,exp.) and the first 3 numbers you can loosely factor primes

[@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.

[@Prasen1729]

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

[@waynelin592]

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

[@believe2000]

I believe if the range was shifted to 3 digits up, I think the change between 9,1 and others would reduce a bit. I think the base 10 9,0,1 pattern are higher earlier, due to this.

[@believe2000]

I realize 100/1000000000 is small.

[@jeremywhite6901]

Siudzinki and Manigaulte call those prime number endings - 1, 3, 7, and 9 - "cynosure numbers" and they argue that all bases (in number theory) are subject to, i.e., are going to reveal, "a prime-bias" because of something they call "nominative structuralism."

[@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.

[@wyboo2019]

primes are always one more or one less than a multiple of six. this fact is easy to realize when you notice that, in base six, a prime must end in a one or a five (ending in two or four means it's even, and ending in a three means the number is a multiple of three). i don't know if this is mentioned in the video but its a nice fact

[@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?

[@FekalniVuz]

i just love this guy

[@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

[@333_BDON]

اشكرك على الترجمة 🌷

[@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!!

[@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.

[@AniketKumar-lw6su]

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

[@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...

[@therealzilch]

Lovely and convincing. Thanks.

[@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

[@bad_manbot]

the 'prime gaps' explanation doesn't just hold for distances less than 10. It holds for *ALL* distances. To get from 9 to another 9, sure, requires to go through 1 3 7, - and there's some small bias there, but to get through the 9 after that one, you have to go through yet another 1, 3, 7-- so for every single time you look for the next 9, there's an additional set of 1,3, and 7 that you must get through before seeing that 9.

[@CaptainBlue777]

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

[@PhilBagels]

I'd like to know what other bases they investigated. I'd be particularly interested in highly-composite bases. like base 6, or base 12. Or perhaps factorial bases, like 6 and 24, and 120. In base 10, 4/10 digits are available for the final digit. In base 6, only 2/6 of the digits are available.

[@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?

[@godsadog]

WHY AM I SO EXCITED FOR REAL?

[@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!

[@sursurrus]

Since it deviates from equilibrium as plus or minus a log of a log divided by a log --> suggests that something about the construction of the natural numbers is sort of a flexible constraint. Since the effect is much stronger closer to Zero I suggest that the very concept of zero (itself undefined over the natural numbers) is sending a kind of perturbation down the number line. I don't remember very much physics but I wonder if the equation given could be modeled as some sort of vibrational effect with damping.

[@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?*

[@joshgadget]

I need to hear more about this discovery

[@NickZack]

Hey its that singing banana

[@ratlinggull2223]

they're twins

[@Minecraftster148790]

It is singing banana

[@marchenprinz9533]

Well, as there are infinite prime numbers, there will be sections, where one or another end digit is more often.

[@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.

[@billbillby5482]

The simplest, easily understood or done; presenting no difficulty. Answer, Feynman.Everything is random.

[@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.

[@rosiefay7283]

I see two forces at play here: + boosting (1,3), (3,7), (7,9) and (9,1): whenever the next "allowable" number (i.e. not a multiple of 2 or 5) is prime, it's the next prime. However, to produce any other digit sequence, at least 1, 2 or (in the case of two equal digits) 3 "allowable" numbers in between would have to be composite + boosting (1,7) and (3,9): with a 50% chance, the next "allowable" number after p is a multiple of 3 and is thus composite. But the second-next is p+6 which is certainly not (provided p>3). There is no comparable boost to (7,3) because if p ends in 7 then either p+2 or p+4 is not a multiple of 3.

[@lonestarr1490]

I find it intriguing that the matrices you draw for the pairs of endings appear to be symmetric w.r.t. the antidiagonal. Does this come from the rounding or do these proportions actually match? If the latter should be the case, I guess "the thing" accounts for that.

[@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?

[@messitup]

Randomness and pattern are essentially the same thing. for example, if you toss a coin 100 times, the coin will land "Randomly" 100 times. but its also so "Unrandom" that the likely hood of getting 50 heads and 50 tails are close to 100% It is the very reason why primes numbers are primes, because they have very "random" properties that guides them, in this case the 44% across the diagonal( the (a,a) (b,b) endings) turns out to be one of those randomly rigid patterns.

[@messitup]

also, without randomness pattern would never exist, vice versa.

[@PeterAuto1]

but there are bias through some patterns. If some thing is perfectly random each pattern should be equally likely

[@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.

[@philm5380]

Just based off the first minute and a half: They're considering a finite sample so it's probability and completely makes sense that some patterns would be more apparent than others. The randomness will flatten out as you expand the sample size, right?

[@ralpix6240]

If you continue the sequence of the last digits (1 3 7 9 1 3 7 ...) and consider the (wrap-around) difference between two subsequent digits (2 4 2 2 2 4 ...) the pair 9-1 is in the middle of a "streak" of 3 "twos". Could that middle position be significant, like the center of a "pile"?

[@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.

[@aliphim]

It is incredibly fascinating that pi (which deals with circles, seemingly nothing to do with anything related to primes), is involved in a formula that is used to find proportions of primes. Does this have to do with the fact that this formula is looking for proportions, and pi itself is a proportion?

[@daicon2k6]

Stop saying "random" when you mean "evenly distributed."

[@JNCressey]

+David Conrad, Yeah, it's annoying when people call definite numbers and sequences "random".

[@DanDart]

it is still both because without checking you'd not intuitively know the next number, there is no major pattern to speak of

[@JNCressey]

Dan Dart, It's not random though, the third prime will always be 5 no matter how many times you run through the primes. The next undiscovered prime has always been and will always be what it is. There's no randomness to it. There are no random variables anywhere near this.

[@uuu12343]

JNCressey Are you suddenly the mathematician professor with a PhD?

[@alephnull4044]

I mean, randomness doesn't exist full stop.

[@brianuuuSonicReborn]

How come the "distance" doesn't explain the bias? I mean you don't have to necessarily only look at gaps of ten, it's the same for any size of gaps. For gaps of ten, you have to pass through the 1,3,7 before it, same goes for gaps of 20, you STILL need to step through the 1,3,7 before the 9. You might say shouldn't it be 1,3,7,9,1,3,7? No, because it already covered in the gaps of 10 case.

[@SpitefulAZ]

you make me love prime numbers.

[@fpan5090]

I believe that the 'patterns' that we think we are seeing are really the build-in patterns imposed by the number systems we are useing...base 10, base 3, etc. If we use a unary number system (first few primes: **, ***, *****, *******), the digits used are no longer a mystery. Primes: The Pattern-less Pattern.

[@stevenr4]

has anyone studied this in different number bases? I'm sure base 8 or base 12 could hold some interesting results

[@Algoritmarte]

What about the last binary digit (LSB) of (P-1)/2 with P prime ... :-)

[@MrPipat]

this makes the twin prime conjecture more relevant

[@DustinRodriguez1_0]

I have several questions... this is very interesting. First, I had considered seeing what would happen if I trained a machine learning system to do primality testing, but I figured surely someone had already done such a thing already. Apparently they must not've, or they would have seen that it recognized this pattern pretty easily. I actually wanted to do it to see how the system behaved in an environment where they was actually nothing TO learn... Second, is it still the case that the endings themselves are evenly distributed, and the bias is only seen when you consider pairs? Could the fact that the ones which are not repeated as often as expected are also the ones which are identical regardless of their ordering affect anything? I don't think it should, but before this video I would have said I don't think there's any bias in this distribution either...

[@13thxenos]

+Dustin Rodriguez This is my friends proposal! Although he wouldn't finish it until next year. So you might beat him to it.

[@DavidVaughan00]

+Dustin Rodriguez That's tricky though, because in general you can't expect to get human-interpretable relationships out of a machine learning system. Maybe somewhere a system has recognized this pattern and used it to find primes, but perhaps the relationship is just stored as like a big matrix in the computer, so a human would have had to have been specifically looking for this sort of pattern anyway to get it out.

[@DustinRodriguez1_0]

David Vaughan That is true. Identifying what pattern is being recognized can be challenging, and it's different for different sorts of machine learning systems. It might be easy to see in a neural network, due to the weightings on nodes examining the last digit being different from the others... but even that might be difficult if you're not feeding the network with the digits as inputs individually. That SOME pattern exists should have been glaringly obvious, though. One would expect to get a very predictable and very flat performance as the system was trained - no improvement over an initial random weighting. But clearly, that wouldn't be what would happen... And people say there's no experimentation in computer science!