Liar Numbers - Numberphile
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- Опубліковано 2 жов 2024
- Fermat's "Little" Theorem is great - but beware of Fermat Liars and tricky Carmichael Numbers.
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I love how cheeky James got about the $620. "Think of what you could do with all that money!"
He's my favorite face on this channel.
shotguntornado uhuhuhhhh you could buy yourself a chocolaqte iceream every day
Pol SP Twice on sundays!
+shotguntornado I think he's the favourite of most subsribers
That completely depends on if it is £ or $.
+MelodyFluff it's $ check it out
“…is called a fermat liar. Ooooh, it's not really prime, it's lying. It tells me it's prime and it's not. Naughty!”
James talks are my favorites :)
I wonder what CGPGrey would say about the "Naughty!" part :'D
He said that right as I read that
LOL same
2:50
This is a lesson. If you are a number reading this, don’t lie.
We need to pay more salary to mathematicians. He almost lost his mind over 620$ :D
there's probably just something mathematically amazing about the number 620.
schadenfreudebuddha yes, it's the number that allows you to buy ice cream X days a week, twice on Sundays...
Any Nobel man knows that the opposite is true ;)
Every nobel man thinks the opposite ;)
Depends on where you live.
I'm a doctor in India and I need to work 2.5 months to earn 620$
620 dollars, I'm regretting not becoming a mathematician already. It seems to be a life of pure decadence and unequaled wealth.
Quickly becoming my favorite youtube channel.
wow, thanks
Numberphile Don't be so surprised. For someone like me who doesn't have a lot of experience in mathematics, these are absolutely amazing videos. Very inspiring and skillfully made to reveal the awesomeness of mathematics. I think just about any discipline can be presented an a boring manner if you're not particularly clever. Needless to say, you guys are indeed very clever. Thanks so much for doing what you do!
+Numberphile i love math. im in 4th grade and everyone asks me about math but when i dont know,i come to you
+Colin Huggins yep same learnt so much new stuff from this one over the last few weeks
It's everybody's favorite channel!
This is so Parker Square test.
Matt "Carmichael" Parker primes
Didnt street
Woah u have 341 likes 👀
$630 wouldn't even cover the cost of electricity to compute a counter example I bet haha
James is great. I love all the people you have on this channel, Brady, keep up the good work.
thanks. will do.
James is really great.
+thihal123 Yeah I love his enthusiasm and passion. Many professors in college lack that.
Moaiz Shahzad A lot of professors in college are not too interested in the pedagogical side of being a professor. The tendency is to prefer the research and writing parts of professorship.
Maybe this test is actually for Carmaecal composites, and every prime number is a liar.
That's just a conspiracy theorem.
No because the primes are what define the Carmichael numbers. A number that passes this test, but isn't prime, is a Carmichael number. You can't define that without primes.
CrazyOrc Someone missed the joke
@@Zwijger Actually, you can. A Carmichael number is a number _n_ that is divisible by some _m < n_ and for all integers _b_ with _(n,b)=1_ satisfies _b^(n-1) = 1 (mod n) ._
See, no primes here.
@@lonestarr1490 saying that n is divisible by some m
How do you get a mathematician to solve a difficult problem? Offer a years supply of Klondike bars! That will get it solved fast.
Clearly superior than offering $1M. We would have proven the RH by now, damn it! :)
What would you do for a Klondike bar?
It's a little known fact that the guy who proved Poincare's Conjecture didn't actually refuse the million dollar prize, he just asked to be paid in Klondike bars.
How many Klondikes bars in a years supply?
I've spent the majority of my weekend watching Numberphile videos. Call me insane, you guys make numbers way more interesting than school ever did. I feel excited to learn again.
so true!
+Logan Fehr That's because the brain loves learning. It addicted it learning and it's amazing how school can manage to make it boring for people
***** why would it be?
+Leen B No I don't blame the school for not being able to teach. I would blame the parents that can't punish there children. And I understand that it's difficult to make a lesson interesting when you have people like that.
+isaac heaton It has been my experience the problem is parents who live to punish their children. and other people's also.
$30 at the beginning and now $620? Why they´ve picked such random values? Why not $561? Or $1105? Or $1729?
Why they pick composite numbers?
They should offer the $ value of the counterexample.
@@josebobadilla-ortiz7405 A 7 hour-old reply to a 7 year-old comment with two replies, interesting...
@@Korpionix with 2 replies?
@@josebobadilla-ortiz7405 that’ll probably be the whole earth
Sometimes i wonder after watching a numberphile video
i ask my self. Am i a prime?
"Prime" in portuguese is translated as "cousin", so if you aren't, just start a family in Brazil.
Lázaro Carvalhaes not only in portuguese but also in spanish.
I was born in 1999, which is a prime number. I guess I'm a prime.
Luis Espinoza Cousin in spanish is primo. Close enoguh I guess
*ItsAMb * if you have to ask your not.
Twice on sundays!? *sets to work*
I want Mr. Grimes to call me naughty, over and over again.
Change your name to 2
@@hanel6662 lol😁
It’s 2:43 am, I have class in the morning, I'm a freaking *geography* major…and here I am bingeing Numberphile.
i live in california and it is 5:22 right now and i just woke up. my mother walks in my room and yells YOU ARE STILL UP? HAVE YOU BEEN WATCHING PORN ALL NIGHT? and i reply with no mother im actually watching a video on fermat's 'little' theorem. now im punished for 2 weeks. thanks math
yep that totally happened...
George G what does it mean?
wait why did you get punished
what does your mom have against maths videos
The thumbnail is brilliant...
Running time: 7:09
709 is a prime
I see what you did there.
Waxwing Slain Its 7:08 after the update...
it's a Parker prime then
(sorry, I could not resist)
Insert Channel Name o
Thanks for another wonderful video! It raises a couple of questions for me that I'm hoping you or a commenter can answer.
1) Why does the test stop at p? For example, if you're testing whether 5 is prime, why isn't it of interest whether 6^5-6 is divisible by 5? I'm sure there's something simple I'm missing.
2) The Wikipedia article on Carmichael numbers uses a different formulation: a Carmichael number is a composite n such that b^n = b (mod n) for 1
And here I am the 1st person who liked your comment 7 years later.
@@andreasl3974 youre not alone😉
Oh wow nobody gave an answer in 8 years. Well I'm bored so:
The triple equal sign is known as a "congruence". It is used to emphasize when things aren't quite equal but practically equal with respect to some condition (in this case having the same remainder). It is pretty redundant to use that symbol when it's clear we are working mod n but it is a matter of taste.
As for why we can stop before checking 6^5 = 6 (mod 5) this is a beautiful thing about modular arithmetic, all that matters are the remainders when dividing by 5.
Notice that 6^5 = (5+1)^5 = 5^5 + 5^4*1 + ... + 1. It might seem intimidating expanding a fifth power but the exact value is unimportant. Basically when expanding you get a lonely 1 from each 1 in the five brackets and everything else has to be a multiple of 5. Modulo 5 all multiples of 5 zero out and so we get 6^5 = (5+1)^5 = 1 = 1^5 (mod 5). In general it can be shown a=b (mod n) means a^k = b^k (mod n). So checking just 0,1,2,3,4 accounts for all numbers mod 5.
Here's something fun if you've read this far: In practice mathematicians just know modular arithmetic is convenient and considering just remainders is sufficient. A more general notion is quotient spaces where we create mathematical structures that are determined by representatives (in this context remainders). All this proving that a=b implies a^k=b^k is abstracted away by just knowing "relationships between elements of a quotient space are independent of choice of representative".
One day I hope to find someone that makes as happy as primes make Dr. Grimes
He should change his name to Dr. James Prime
@@PoweDiePie Grime is fine. He can make a show called "Grime Prime Time".
Damn, I want a choc-ice everyday for the next few years!
James is hilariousxD
I have the number, just holding out for more icecream treats :3
First bagels now ice cream. This channel's making me hungry.
$620. That obviously was raised from $30 due to inflation.
Thank you captain obvious.
Thank you brady for adding subtitles to your videos, I have shown your channel to several hearing-impaired friends and they absolutely love it =)
1 actually follows Fermat's Little Theorem perfectly (a^1 - a = 0) but also isn't a prime number. That means the theorem works perfectly for every number with 2 or less factors.
Also, 5^4-5=620 and 2^5-2=30. Is that the significance of those amounts of money?
I vaguely remember Fermat's little theorem being used to re-arrange some equations for RSA cryptography. A video on that would be nice as it's quite interesting, but might be quite difficult to follow for some people.
i am in love with this guy
mia kablan lol i read your name as mia khalifa
Thumbs up for the choc ice
For 561 this extensive procedure isn't necessary, since its dividers are obvious:
5 + 6 + 1 = 12, it's divisible by 3.
5 - 6 + 1 = 0, it's divisible by 11.
So it's unfair to call it a liar! ;-)
Also work with 11 (but not 3):
341: 3 - 4 + 1 = 0
41041: 4 - 1 + 0 - 4 + 1 = 0
75361: 7 - 5 + 3 - 6 + 1 = 0
101101: 1 + 1 - 1 - 1 = 0
1729 works with 7:
29 (the last 2 digits) + 2 * 17 (2 * the rest) = 63, which is divisible by 7.
And with 19:
2 * 9 (2 * last digit) + 172 (the rest) = 190
Also for computers, wouldn't it be the easiest way to check divisibility using the perfect reliability of the cross sums?
Amazing videos by the way, you have a new subscriber. :-)
561 is not a liar
2 is
That darn three. Noone likes a snitch.
"620$, think about what you can do with that money, ooo. You can buy yourself a choc ice every day. Every day for a year. Twice on Sundays "
You're telling me Optimus Prime had time to pass all these tests?
I guess the Prime Minister gave him a honorary prime state!
I love that 101101 is a Carmichael number and a palindrome.
Bradah you make Math hella fun and interesting. LEGIT!
How wonderful of a person is Dr Grime! What a fantastic likable human. I wish I could always be around people like him.
620$? ... why not 561, 1105 or 1729$?
maybe after taxes it's 561$ :P
that's a lot of choc ices, mmmmm. here's my pen and paper?
james's sense of Irony/Satire enraptures me, substantially.
*Ssss, **_oooooooo_* - indeed, Mr. Grime; *Ssss, **_oooooooo_* - indeed.
No disrespect for the other people that show in this youtube channel, but james is the best (and I just found out he have another channel now I have hours of videos to see =D)
This is one reason that you shouldn't necessarily trust what numbers tell you. The other is that some of them, while not actually liars are completely irrational.
561^561 - 561 is surely divisible by 561 too right?
if you want to chick if a number is prime , it would be much easier to use the definition:
(n is prime) equivalent to ((n/m) is not a natural number 1
You have to do this again with or about Daniel Larson's new proof on carmichael numbers.
This is probably one of the hardest ways to earn $600
never been a math nerd but i´m starting to like this.
2^341 - 2 = 2 * (2^340 - 1) = 2 * (2^10 - 1) * (2^330 + 2^320 + 2^310 + ... + 2^10 + 1). So, 2^10 - 1 = 1023 is a factor of 2^341 - 2. Since 1023 = 3 * 341, we see that 341 is a factor of 2^341 - 2 without having to calculate 2^341.
One way to check if a number is prime is to try dividing it by everything up to its square root.
1^n - 1 = 0 :D always
The handsome $620 prize is ample reason to pursue a doctorate in mathematics, if you ask me.
You'll be spending a lot more money than 620 dollars to get through collage, my friend.
he and standupmaths are my favorites
Why do you have to impose the restriction 1p as each a is congruent to some a' mod p where a' is between 1 and p inclusive...
If it works for all a < p (and obviously for p), then it works for all numbers, as any number 'c' can then be represented as c = (p + b), where b < p (or if c is big enough you just take some integer amount of p instead: c = (kp +b) - it doesn't really matter). What you then have to check is whether (p + b)^p - (p + b) is divisible by p, but all the summands in (p + b)^p when you multiply it by itself p times will contain p to some power exept for one, which is b^p. So you will have a bunch of numbers that are (p to some power)*(b to some power), but these are naturally divisible by p, and the only questionable thing left is b^p, but b is less than p and by assumption any number less than p satisfies the desired property, that is: b^p - b can be divided by p.
Yes that's exactly that. Doing the test with a is the same thing as doing the test with a' its class mod p. If a number passes the test for all a between 1 and p then it passes it for all a.
You don't obtain more information.
Let's say a^p - a ≡ 0 (mod p), where 0
I didn't go to school, and I watch this, oh god.
Great video guys! I always love hearing your new math facts. Keep up the great work!
3:36 does anyone else here watch the show Chuck?
Cuz in chuck, there’s a guy named Chuck (duh) and he’s a spy, and his cover name is Charles Carmichael. I thought that was interesting because “Charles Carmichael” was lying about his name and Carmichael numbers were lying about being primes.
Fermat said all the primes will fit in this theorem but never said all the numbers fit in it are all primes, i take it?
Yep
Interestingly, most of the Fermat numbers, which he did claim to be all primes, are liars. It is easy to prove that every Fermat number F divides 2^F - 2. So, other than the Fermat primes, all other Fermat numbers are liars.
aww, whose the cutest little theowum in the world; you are little theowum, you are!
If I had a friend, I would want him to be James Grime.
From the definition, wouldn't 1 also be a Carmichael number? Every number minus itself equals 0, which is divisble by 1 :P
If you like this guy James he also has his own channel (singingbanana) that's just as interesting.
... Son, I dont want you hanging around that fermat liar !
green hair??
I also have green hair!!!! GREEN HAIR FTW!!!!!! (not colored, its a mixture of blond, orange and green, no kidding..)
Color correction magic! XD
tggt00 Pic or it didn't happened!
tggt00 green isn't a creative colour!
30 bucks?? i'll become a mcdonalds clerk then...
wait, 600$??
lemme think....
I used a numberphile on the coins in my pocket and got short changed
Perfect
"the first carmichael number 561"
1 is carmichael, right?
i mean, its not prime (which im pretty sure is the meaning of composite), and it passes 1^1-1=0 (which is divisible by 1)
and it passed all tests up to its value there look
numbers
I was like "wait 341 isnt a prime, its divisible by 11 because the hundreds plus the ones equals the tens"(i like to check if numbers are divisible by 11 or 3)
‘Imagine what you can do with 620 dollars.... You can get a choc ice everyday and 2 on some days’ like James choc ices are the boy right but come on
I'm afraid James has made an error. Carmichael numbers pass the Fermat Little Theorem (FLT) for all integers less than the Carmichael number that are RELATIVELY PRIME to the Carmichael number, NOT ALL INTEGERS as indicated at 4:00. For example, 406 ^ 560 = 1 mod 561, but 407 ^ 560 = 154 mod 561. Notice that 561 = 3 x 11 x 17, 406 = 2 x 7 x 29, and 407 = 11 x 37. Notice that 406 has no factor in common with 561, that is, 407 is relatively prime to 561. There are 240 integers (none have 3, 11, or 17 as a factor) less than 561 that correctly "witness" 561 as a composite, and 319 "liars" that falsely suggest that 561 is prime. If EVERY integer a < p satisfies a ^(p-1) = 1 mod p, then p is truly a prime. But of course is is highly compute intensive to check that, so it isn't a practical test.
Really this makes me so happy how joyful he is about math. Sometimes I get stuck on something, and then I get it. and then the best part comes I get to explain it in the same joyful manner. :D Great!
6:58 do they both say fast at the same time? It confused me so hard bc I was looking at the wrong guy.
whoa whoa whoa ! whats going on . am i a prime?
I put both of these numbers in my prime tester, which assumes that x^{p-1}%p == 1 and it identified them as composite
Just in case someone is wondering:
2^341 - 2 = 4479489484355608421114884561136888556243290994469299069799978201927583742360321890761754986543214231550
and thats equal to
13136332798696798888899954724741608669335164206654835981818117894215788100763407304286671514789484550 * 341
Yea i hate math, but watching people get geeked out over the science of numbers is very entertaining, and makes me excited about math too😂
1|2¹-2=0,3¹-3,4¹-4,5¹-5,6¹-6,7¹-7,...,999¹-999,...,∞1-∞,... So, Is 1 a prime, a composite (1×1×1×...×1=1), or, is 1 a Carmichael number... And does 1 pass the 100% Test...?
I love your videos and cool prime tests, keep making great videos! Also do a video on the number 73 the best number
Is this Fermat's test actually useful? Aside from chance of hiting Carmicheal numbers.
Computing 2^341 - 2 and checking if it is divisible by 341 seems more computationally complex than just dividing by primes lesser than 20. Or is it just my puny human brain?
How do you get a mathematician to solve a hard problem?
Offer them a new prime number!
numberphile, do you think you can make a video on cryptographic hash functions / cryptocurrency (bitcoin)? :)
So, do all these Carmichael numbers break down into the product of three primes?
1:29 Fermat did not prove this theorem, Euler did. Fermat merely stated it.
Euler proved pretty much everything though. So just give some credit to Fermat, Euler gets more than enough as is.
Sebasfort yeah I get your point. Didn't really think while I was posting. 👍
Whenever I hear Fermat I always picture Andrew Wiles, It threw me off when the painting of Fermat came up :P
101101 is such a cool Carmichael number.
The prize shouldn't be $620, it should be $561.
Tell'em that when you win. (rudeness uninteded)
No increase it to $1105
He's the friendly version of Dr. Sheldon Cooper
My favourite Carmichael number is 101101.
Also the prize should be $561
Does having a composite base 'a' add any new information that its prime factors dont? Does 4^p - 4 ever provide meaningful information that 2^p - 2 didnt? I have this thought that only prime bases for 'a' need to be checked but I dont really have any mathematical argument for why.
Well if you used a = 1 then evry number will pass the test because it will always be 1-1 which is 0. Right?
Why was the intro like James was held at gunpoint?
Watching numberphile videos and scrolling its comment box make my day.
This theorem works like the scientific method: question (is this integer a prime number?) --> hypothesis (if it is, then it passes this test) --> experiment (plug it in) --> observation (did it work?) --> conclusion (it did(n't), so it is(n't) a prime). And like science, it isn't perfect. It's just the best we have.
Well Fermat's Little Theorem just says that if p is prime, then the test works. All this video shows really is that the converse isn't true, not the theorem itself. I get what you're saying though
When you've watched so many math videos on youtube you already know what Liar numbers are lol
Fermat's Last Theorem was solved in 1996 by a teenager with a pentium based computer. Was it a prime number = no
do they actually have a proof for the theorem? (which layman can understand)
41041 and 101101.... nice
TWICE ON SUNDAYS!?
brb, disproving maths