Why 82,000 is an extraordinary number - Numberphile
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- čas přidán 11. 06. 2015
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2,3,4,82000
Well that escalated quickly.
2,3,4,82000 and then after that (if there is another number in the sequence) it is over 2,000 digits long. Another sequence you can look at is 1,1,1,5,? where you look at the digits (in base 10) of the first sequence, so if there is an answer for ? in this sequence than it is over 2,000
@Exyll well he said quickly not outrageously :P
@ktbDash how about 0, 1, BH(3), BH(4)
More quickly than TREE? Maybe but I doubt it, it feels like there are more answers out there.
What is the next number in this sequence?
2,3,4,?
A)6
B)5
C)42
D)82,000
Profile pic = perfection
George Sorkin 5
D
B
George Sorkin C
how do pure mathematicians sleep with these many mysteries floating around :D
they count sheep in base thirteen
Chris Birch HA!
I get 20,230,686 = SHEEP in base 29. 28*29^4 + 17*29^3 + 14*29^2 + 14*29^1 + 25. Don't trust my math.
+Qermag that math is right.
they don´t sleep, THEY THINK TO MORE MATHEMATIC MIND BREAKERS
If we go on like this, all numbers will be special by the end of the year :P
Slartibartfass now you're talking!
Slartibartfass Ha
***** so because all people are special, none are? everyone (and every number) can be special in a different way, and still all be special.
Huntracony If all people are special, it's not special to be special xD
TheYackick the conversation is an open problem just like 82,000 xD
I like this guy. He is so relentlessly enthusiastic about everything. :D
He is amazing.
it's because of the meth... ehm, math
i'm using him for my therapy sessions haha you can't get tired for watching this guy talking and talking it is just soothing
I wish I loved my job as much. Congrats to him for doing something he loves in life -- and making math fun for the rest of us.
Have you seen Cliff Stoll?
Numberphile videos need a NSFW tag. It's hard to get any work done after watching them.
Nicolino Will NSFW means not suitable for work
***** ( ͡° ͜ʖ ( ͡ᵔ ͜ʖ ͡ᵔ )
Wild Academy NSFW means cursing or nudity is in the video.
You are describing another problem - NSFP.
Not Safe For Productivity.
Wild Academy lol
Wild Academy Lol.
Number 45 is also special as it is the only number of its kind, no other number is 45. Same goes for 62892, 80085, and 3 among others.
But what about 45?
Yes but 145 comes close so...
"Or is it?"
-Vsauce (michael here)
+Endymion777777 haha yes xD
but what's the best number in the universe?
69
+AndroxineVortex 52!
(52 factorial)
AndroxineVortex I discovered the best number in the universe, unfortunately the internet is too small to contain it.
why do programmers often confuse halloween and christmas? oct 31 = dec 25
true story, except that we don't actually confuse those dates, heh ;)
I know we don't :)
The Classic: There are 10 types of people. Those who understand binary numbers, and those who don't.
This is the most beautiful thing I've seen all day. Thank you.
There are 10 types of people in the world: those who understand binary, those who don't and those who didn't realize the joke was in base three :D
As a computer scientist, it pained me to see you put binary digits in groups of 3
I use groups of 4 because it just makes it easier to read and 4 is a nice round number.
hey, at least its easier to convert to octal that way ;)
More importantly, as both a computer scientist and a mathematician it was painful to me to watch them grouped by threes from the most significant digit. They should be grouped from the least significant.
@@kauhanen44 4 is a round number? But it is all edges, no round on it at all. In fact, a round 4 is actually a 9.
Not a fan of octal?
This video reminds me of a mathematical joke: "Why do mathematicians celebrate Christmas on Halloween?"
.
.
.
.
"Because Oct. 31 = Dec. 25"
The thing I like most about James, and also everyone else associated with Numberphile, is that he doesn't seem to talk down to you as a viewer. Mathematicians are known for being condescending to people who aren't equally as or more intelligent than them. And that's why I like Numberphile so much. They actually apply their skills from the mathematical world and make it actually somewhat interesting and perhaps even fun. Keep it up guys! :)
Since they searched up to 2k digits, let's say they find another one that's 2000 digits. You'd expect its representation to be random digits in base 6 (or following Benford's law), so each digit has a 2/6 chance of being 0 or 1. Therefore, you'd expect to have to find 3^2000 numbers that work in base 5 before finding one that works in base 6 (about 10^1000). But the number of digits increases very quickly as well! If there are any real mathematicians here, it comes out to be E = p+p*sum(n=1 to inf, p^n*(n+1)*prod(m=1 to n, 1-p^m)) where p = 1/3^2000; find E to get expected # of digits for the next number.
Yeah I had my suspicions about the conjecture that there were no more numbers for the sequence. I mean there are infinite numbers, checking a relatively infinitesimal fraction of all the numbers doesn't lead to a very convincing reason to assume no number can fulfill the requirements. Of course it is possible there aren't anymore and that their are simply finite numbers for the sequence ending at it's 4th number, but this certainly seems to be one of those cases where you will need true mathematics not computers to prove this one way or the other. Mostly since if you find a number for base 2-6 then what about 7? We need a proper proof for a real answer to this mystery.
Really liked this video . Brought us back to the fundamentals of what makes numberphile videos fascinating . Also , can't ever go wrong with Dr. James Grime ; always contagious with his enthusiasm .
I am currently writing a C# program that will hopefully solve this mystery (of what comes after 82,000). I will return.
Alyosha Romanov Thanks!
IAmAPerson are you done yet ?
3bjdarras Yea, program's complete. Debugging it now. Then I'll put it to work.
***** It's worth a try ;-; (Probably not)
IAmAPerson sauce pls
I love this guy, he is allways smilling
Why is 73658 an extraordinary number? Because no other numbers are like it.
Useless but fascinating - almost like art. I'm beginning to understand the attraction of pure maths.
Watching your videos is soooo relaxing to me !! All problems out there seem gone for 5 minutes !!!
Love it. Please post more videos about open problems. It's a part of math people don't generally get experience with in high school, but it make math seem so much more approachable I think.
no one:
no one ever:
nobody ever in the universe:
numberphile: why is 82,000 special?
Numberphile videos. I don´t get a thing, but they make me feel more intelligent. ;)
that didn't make sense xD
I feel ya, bro. Same here.
Well, to answer your question, you said stupider which isn't a word. I hope this answered your question!
InFact NoOne it is
stu·pid
ˈst(y)o͞opəd/
adjective
comparative adjective: stupider
having or showing a great lack of intelligence or common sense.
Haha
Such an amazing channel keep up the amazing work
As per usual, another great video by Numberphile. Always learning something interesting on this channel :-)
There is another number, it's 0
Glad i solved this for you and i didn't even need fancy computers and stuff!
milo He said that numbers on this sequence must be greater 1, so no 0 or 1. It would be too easy and boring otherwise.
KAND0RIS Oh can we select numbers that count and ones that don't? Then i choose to exclude 82000.
:D
milo Well, mathematics are all about definitions so yes you can decide what numbers can belong to the sequence or not, it is the definition of the sequence. Here it is "the Nth number in the sequence is the smaller number which can be written with only 1 and 0 in base 2, ..., 1+N. Excluding 1 and 0". But you have to define this sequence so it has an interest. Without 82000 there won't be much to say, the sequence would be 2,3,4 and that's it. But 2, 3, 4, 82000 is much more interesting because why does it go from 4 to 82000 all of a sudden ?
I know m8 :) it was just a bit of friendly Friday internet trolling ;)
milo No problem ;)
Someone on reddit came up with a really clever algorithm for this, and using it, they actually proved that the next number has to be more than 11 million digits long :O
qqwref2 linkkk please!!!!!!!
qqwref link
I kinda need a weekly Numberphile podcast in my life.
***** Whaaaaaaaaaaaaaaaaaaaaaat
Why have I not been informed of this sooner! I must investigate!
On Wisconsin! Glad to see UW-Madison getting some numberphile love.
I've discovered a truly marvelous proof that there is no such number above 82,000 , but this comments section is too small to contain the proof
Put a link!
@@Anonymous-df8it bro my comment was a joke too and you didn't even get it
It seems to me that the number of candidates can be significantly reduced by looking only at those numbers which can be written using only 1's and 0's in base 6. Create this list (which shouldn't be at all difficult), then start examining.
But that list is endless
I LOVE every time Dr James Grime is a guest
Thank you for your amazing explanation.
Yaaay, James is back! :D
please do a video on fractional or irrational bases... base pi anyone?
great book, learned a lot about how to use probabilities and statistics to look at everyday things differently.
I'm not ashamed that I had to watch this twice to understand xD
This stuff is fun ^^ Keep 'em coming :D
Could you make a similar sequence except only using the *highest* two digits from every base?
How is it so round? 82,000...
Is there a reason?
Yoni B Because the last digit is 0 in Base 5 you only get multiples of 5 with only 1 and 0, and the same is with 4, so its a multiple of 4 and 5 --> a multiple of 20
Of course the last digit doesn't have to be 0, but it's not that unlikely if you only have 1 and 0 ;)
DorFuchs But why 3 zeroes?
Yoni B
It ends with 000 because that means its a multiple of 1000 or 2x2x2x5x5x5 and on top of that it is even divisible by 2 one more time so it is divisible by 2x2x2x2x5x5x5 which means it will end with 0000 in base 2 and it will end with 000 in base 5 and it will end 00 in base 4 cuz 2x2=4 so 2x2x2x2=4x4. All these 0s help cuz u can only have 0s and 1s. For the rest there need to be a lot of coincidences so the rest will also be 0s and 1s and 82000 seems to fit. :) This is all said by someone that just watched the video and did some logical thinking so I might be wrong. I was wondering the same thing.
SmileyMPV So why is it not divisible by 3? :)
Anyways, consider that we're only using base 10 to show the number in a way we're familiar with. Base 10 isn't really important in this problem. It looks like a bunch of 1s and 0s in the bases we care about for the problem.
zeidrichthorene In base 3, it has a 1 at the end - if you take that 1 away, you get 81,999, which is divisible by 3. That last number in what determines whether a number is divisible by the base number: if it's zero, it's divisible, if it's something else - it's not. Works for any base you could think of.
Thank you numberphile for keeping my mathematics sharp during this summer break.
Thanks! I had forgotten how to convert exponential with different bases between each other. A piece of knowledge long forgotten since forgotten grade school.
Try Graham's Number, I bet that will fit the sequence.
Actually graham's number ends in a 2 in base 5 - nice try though
How do people with OCD sleep with such unsolved mysteries?
Numbah'
I'm in love with this channel already.
I love this guy
He always seems so happy 😁
Actually there is an actual definition of an extraordinary number. A number is an extraordinary number if the biggest prime factor that the number has is bigger than the square root of the number. so the first extraordinary numbers are 2, 3, 5, 6, 7, 10, 11, 13, 14... (yes all prime numbers are extraordinary numbers because their biggest prime factor are always themselves). However the prime factors that 82000 has is 82000 = 2*2*2*2*5*5*5*41 so the biggest prime factor that 82000 has is 41 and the square root of 82000 is about 286,3564 so the biggest prime factor that 82000 has is NOT larger than the square root that 82000 is I´m sorry numberphile but 82000 is NOT an extraordinary number :)
+Bengt Bengt i should have seen it coming! but i didn't.
@@Joffrerap I don't get it.
Why does he sometimes pronounce it noomber and other times nuhmber? For example, at 0:04 and 0:06 .
Dan Sanger It's always "nuhmber" but sometimes his mouth just feels a bit number.
Penny Lane Pure genius.
Dan Sanger I think it's because at 0:06 he says it quicker.
Dan Sanger He answers this himself at 0:05: "kind-of arbitrary-sounding _number_"
***** Are you saying British people can't decide which dialect to talk in? ;)
this actually helped me understand bases now!
I love how enthusiastic Grime is when he talks about maths. =)
Why can't you use 5 for base 5 wouldn't it just be 1 0 as well
stevj15 5 in base 3 is 12
stevj15 its 12 in base 3
stevj15 well, 5 in base three is 12
stevj15 in base 5 , 12 is 22
stevj15 Because it's 12 in base three.
From a computing background, it looks so weird to me to see binary in three digit groupings, rather than four digit ones. *shrug*
+Paul Bottomley Not to weird for you I hope though, remember, octal is grouped in three. ;)
Don't treat like hexadecimal who treats you like octal.
You are young. You work in hexadecimal. If you had worked earlier you would have worked more in octal and been used to three "bit" groupings. (not decimal)
I love that book! Everyone who likes this channel should consider purchasing it!
I enjoy your enthusiasm.
What if we take "e" as the base?
1123aka What do you want to do with "e" as a base? You can't describe integers with e^1,2,3... Bases are used to represent integer quantities mostly :)
i was just wondering,as its also an important number from natural point of view. May be its the nature's base :)
David Seelmann if you took an irrational number as the base, then all natural numbers would be irrational in that counting system. which means that "irrationality" is very much bound to integer number systems. Maybe there is an irrational base system in which pi and e are both rational....
en.wikipedia.org/wiki/Non-integer_representation
1123aka basically you can use it and other irrational numbera. But it is not convenient, since we have to use infinite numbers just to count.
Phúc Minh Nguyễn Lê They aren't infinite, they are just irrational.
Noombah :)
As a Finn, I *_REALLY_* love the chapter name: ”Less like Sweden” 🇫🇮⚡️🇸🇪.
It is both comforting and disturbing, that someone is looking into problems like that!
Hmm ... this problem is understandable ... and open ... and not too well-known at this point ... I really want to prove it now. XD I doubt I'll get anywhere though, it looks like some bright minds have already looked into it.
Some super computers sure must have looked into it.
Why 82,000 is an extraordinary number? Because it is larger than all of the numbers before itself... duh!
You are dutch
Amazing... thanks for the interesting video:))
May I ask, how you're converting the numbers in your head? Ive always had problems converting large numbers to base 2.
as impressive as the lesson is he does math with sharpies.
Just write the number zero in all the bases!
+RedsBoneStuff Number one also works.
GudGod, that sound of the marker being grinded across the surface of that paper gives me the chills.
Wow! This is amazing!
82,000; 8+2+0+0+0 = 10
10 in base 3 is 3
triangle has 3 sides
ILLUMINATI CONFIRMED
1 = 01
2 = 02
3= 10
Can't you just crank up some supercomputer and check for numbers larger than 2000 digits?
fifthgear93 Pretty sure that's how they were able to check numbers with 2000 digits. Doing that by hand would be ridiculous. For perspective, here's what a 2000 digit number looks like.
14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837297804995105973173281609631859502445945534690830264252230825334468503526193118817101000313783875288658753320838142061717766914730359825349042875546873115956286388235378759375195778185778053217122680661300192787661119590921642019893809525720106548586327886593615338182796823030195203530185296899577362259941389124972177528347913151557485724245415069595082953311686172785588907509838175463746493931925506040092770167113900984882401285836160356370766010471018194295559619894676783744944825537977472684710404753464620804668425906949129331367702898915210475216205696602405803815019351125338243003558764024749647326391419927260426992279678235478163600934172164121992458631503028618297455570674983850549458858692699569092721079750930295532116534498720275596023648066549911988183479775356636980742654252786255181841757467289097777279380008164706001614524919217321721477235014144197356854816136115735255213347574184946843852332390739414333454776241686251898356948556209921922218427255025425688767179049460165346680498862723279178608578438382796797668145410095388378636095068006422512520511739298489608412848862694560424196528502221066118630674427862203919494504712371378696095636437191728746776465757396241389086583264599581339047802759010
^ I'm not sure where you copied and pasted that from, but you should probably tell whoever wrote it that the 24th and 25th digits are the same, so it can't be Pi 😂 unless you only stuck with it up until the 23rd digit, and then just typed random numbers. I only know Pi to 20 digits anyway 😂
*****
(function(){var digits = ""; for(var i = 0; i < 2000; i++){ digits+= Math.floor(Math.random() * 10).toString(); } return digits;})()
26031708629072021616365640016223779005855639626572804032044862925723521511661282923760545413683312327176445496787115176611722012319861334578054052457744217209681278844934067886653571037096108282224148046981968821974690912899121861109203291306842984615430177588438315972824186616939398045499348320905777595823327654164007953674906753734327998173932649589059336663466742241740201393042118450409193120885754917838239321380802379017654546043640598431741475932914390810232701149779311868086163163503553854587154373948139592829964981980258600850116927574662341070779749275287527293377635591854970651247686688156888592461917129161437095894693233470721103444761126654603361986507237418189107610699449553113027567815520660705803579057449046541132146338494360438000613282988155602009179873041925114702353938753385458755984399237818566602843354913569471296160240716774235161420424510882852386245664930823342956810545273792022861797843758701977343075163701460851064337308676923318285520665858642542529964087903965525239043182946632544851672918555070863284315421797049285921704436589659966755680671413896820887367816533573517053972088106088772488879841037514580864258883444076274686836013105991064124250446808448076706374646947776275985115702205378075270864052368301947702404534626052225082486302947512856253437848115910193947811896889273808065135555395818665771958426156082452541116950009511148077096505256981908397146142160957219584560362514911243476794330293424918223659095121299896405986977495035202857107597696099325435080067691781779979791518350516862365880614105123385463474408525490869442681804122728810148976236474351120458423163339318982814438585584101120234776419030664604236251596991035831850444677851152783467564329437784879023538771349311841738670356533377181788719571630842360928630413511660804464394434832344497274962388685560894888193857293873763631210631447527283136739863849052091827371802145130625730981721000831146711370498603274985328721680274214249769768827982354708720583478926303936694162
***** It is digits of pi. There are repeated consecutive numbers in the digits of pi.
I remember the first time I learned about Pi in grade 8, which is also when I memorized it to about 130 digits (which is why the first 20 are seared into my brain 😂), the teacher told us it doesn't repeat, but I looked it up and kept seeing digits repeating. It was fun listening to him explain it and confuse everyone even more lol, but no I was just incredibly sleepy, about to pass out and unfortunately my brain doesn't like being smart under those circumstances 😭 haha.
this was so informative
I just one-upped you: 1 s only written with a 1 in ANY base!!! WOAHHHH AMAZINGGG i did all the calculations (it was a mess; there was calculus involved :D)
Aren't all "bases" technically base 10 if you write their "base number" in their own numbering system? :P
TinyRoboticPiggies Well, "ten" means "the number after nine", not "the number written 10" :P
You, sir, have outwitted me.
TinyRoboticPiggies i guess what you are saying is that the base-n representation of the number n is 10 for all n
IceMetalPunk The word "hundred" actually used to mean "the number written as 100" back when the English speakers didn't use the base-10 system. The value was actually equal to today's 120. It wasn't a complete base-n system as understood today, it had quirks.
zwz • zdenek
Ugh, people were confusing before they learned mathematical analysis >_<
(For those interested in more info about hundred=120: en.wikipedia.org/wiki/Long_hundred )
What about 1? It's written the same in all bases.
QuantumBraced He excluded 1 since it would be too easy.
QuantumBraced 12 is 3 in base 5
QuantumBraced Well, they excluded 1 because the problem is to find a number that contains both 0 AND 1 in a meaningful way. So you can't just take 1. You don't need a 0 to write it.
jk991234 12 in base 5 would be written 22
He's leaking the joke from another comment chain into this one. xD
I've come up with my own counting system. I call it Base 0. Everything is in zeroes, and the only way to figure out what anything is is to use crude hand gestures.
Brady is so good with the composition, I feel like I'm sitting next to James talking with him.
69 is an extraordinary number and i really love it
is there a number that isnt extraordinary??
6274
1^1823066305350523
9 ajsjsjsjdjdhdhdhdhdhdhdhdjdjdjdjdjdjdjy ajsjsjsjdjdhdhdhdhdhdhdhdjdjdjdjdjdjdj indistinguishability stsisrysyizkfsfyskyfksktuskfskrysryksorussfysgususufstusfisgsgjsgjgsgjgagjsgjsgisgisigsigsgisgisgisgissghskfakkkkhfsafhkkfaradhkadyadyakdyakfyakyayjayarayrhrhrfssftgdtuxfdigstusydajzksfajariayddjsjsrahryshrys
The longest commentary ever and Nunberphile knows binary
xd
AND:
=D-
NOT:
->o-
Logic Gates R Fun :D
xxgusfustusfsutsgusahaykjafsussusfshsusugsusig
THE END OF THIS COMMENTARY!
nine hundred ninety nine vigintillion nine hundred ninety nine novemdecillion nine hundred ninety nine octdecillion nine hundred ninety nine septdecillion nine hundred ninety nine sexdecillion nine hundred ninety nine quindecillion nine hundred ninety nine quattuordecillion nine hundred ninety nine tredecillion nine hundred ninety nine duodecillion nine hundred ninety nine undecillion nine hundred ninety nine decillion nine hundred ninety nine nonillion nine hundred ninety nine octillion nine hundred ninety nine septillion nine hundred ninety nine sextillion nine hundred ninety nine quintillion nine hundred ninety nine quadrillion nine hundred ninety nine trillion nine hundred ninety nine billion nine hundred ninety nine million nine hundred ninety nine thousand nine hundred ninety nine
+Golden Freddy (F.F.) I won!
Brady, which lavalier mic are you using for your outro, and how are you connecting it to your computer, please?
Question. Using this same idea is what the higher number that can be represented in a Base 3 answer? ie the answer can have the digit 0,1 and 2 and how high can the answer base go? Can it go higher than base 5?
Since there are infinite numbers, doesn't that mean that there HAS to be a number that satisfies this?
Bane? That's not how that works
Sure it is. There are infinite numbers, so there should be infinite numbers that can be represented with 1s and 0s in any given base. Since we're talking about infinity, of those infinite numbers that are represented with only 1s and 0s in any base, there should be an infinite amount of numbers which have this property in multiple bases.
Skyhunter That's not what I'm saying at all. Obviously there aren't infinite solutions to every equation just because numbers are infinite, but in this situation it should work.
Example, there must by definition be an infinite amount of numbers that can be written with only 1's in base 10, yes? The same can be said for any digit in any base, i.e an infinite amount of numbers that can be written with only 4's in base six and so on. If there are infinite numbers that can be written only with 1 in any whole number base, that should mean there is a number to satisfy this requirement.
Listen, big guy.
There aren't infinite even primes, are there?
Big Guy For You Again, completely different situation.
I always love videos from James, because it is always just pure maths fun. Like, he see's math differently than I ever did, and I love that :D
Came back to this because I parked my car and saw it was at 82,000 miles
is there an update to this problem? has the base-6 number been found? this is very interesting!
Sequence of numbers n>2
with no 2 in any of base b in 2..10 :
3,4,31,58130496,58130499,58130500,58130703,58183951
HAPPY NEW YEAR
I remember a while ago you made a video about a prime generating machine. When you guys reach 492,113 The video of the longest gap the machine has would be made!
This will be solved next year with some whacky circle math involving the Riemann Zeta function along with David Eisenbud's compass. I will check back.
Hi, 2021 here, is it solved yet?
O my gosh I already have that book!
Cool video mate
I have to say that is fascinating! I don't really know what to make of it. It's just this oddly arbitrary number that has this unique property.
Good on yer, Brady. You're a good bloke, mate.
Hey, man who says stuff at the end of the video, can you tell us what other books you have on that shelf? Thank you.
JAMES!! Where have you been?!?!
1:37 "shall we do that in a different base? Yeah!! Yeah!! "
Hé ladies and men of Numberphile ! Any news on this one after 8 years ?
how that was discovered dude? amazing!!!
Coming from a computer science/engineering background, this concept was easy to pick up since we have to fiddle around with base 2 and 16 constantly. :)
Sergio M Medina The moment he said let's write 82,000 in base 2 I instinctively knew it's 17 bits long and starts with 101. I dream in binary.
Another sequence: 2,3,4,5,82000
The pattern is, all of the bases in which 82000 can be written in using only 1’s and zeros, since 82000 is 10 in base 82000.
Numberphile or Computerphile needs to do a video about doing calculations with more digits then the hardware supports.
error079 I believe (though I'm not sure) that the way arbitrary precision numbers are created and operated on is that they're stored as strings and operated on digit-wise, the same way people do math. I think. At least, that's how I coded up my own arbitrary precision library XD
82,000 in base 6:
“I prefer 3s and 4s over 0s”
they should look for it in the reversed way day should count in bas 6 upwards only with 1 and 0 that way they are much faster because they are not calculating each other possibility because the first rule is that it fits in base 6 and than you can look if it fits to the other bases
Fantastic
I like the base!