Finite Fields & Return of The Parker Square - Numberphile
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- เผยแพร่เมื่อ 14 มิ.ย. 2024
- Matt Parker introducing Finite Fields and re-visits the infamous Parker Square...
Extra footage & become a millionaire by winning The Parker Prize: • Become a MILLIONAIRE b...
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The Original Parker Square video: • The Parker Square - Nu...
The paper featured in this video: arxiv.org/abs/1908.03236
MATT PARKER STUFF
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Extra footage & become a millionaire by winning The Parker Prize: th-cam.com/video/hn8SwBhhDvU/w-d-xo.html
The Original Parker Square video: th-cam.com/video/aOT_bG-vWyg/w-d-xo.html
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Matt's Books (Amazon): amzn.to/3absFfV
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A millionaire, or a Parker Millionaire?
Btw, Matt Parker got something wrong! Z mod powers of primes are *not* fields! For instance, in Z_4, 2x2 = 4 = 0, so that Z_4 has zero-divisors. Hence, since it has zero-divisors, it cannot be a field
@@felixlaroche8039 Exactly - given a power of a prime, there is a finite field of that size, but it's NOT just modular arithmetic (it's a bit more complicated than that)
Man I feel attacked...
How do we get the article you referenced?
I don't mind Numberphile's filler episodes, but I love it when they seriously advance the main plot like this.
The Parker Square 2: The Parkering
Underrated comment
Numberphile is my favorite anime
100pi likes!
also, LOOOOORE
That P vs NP killed me
That was golden :D
Had to pause for my lols to come to a side-stiched stop
Best part 😁
surely, NP should be rebranded IP - Inverse Parker.... ;-)
+
When I saw that "Parker" was a property of something in an actual, published research paper I legitimately doubled over laughing. The parker square is officially a real mathematical term!! I never thought I'd see the day.
See, at first I was surprised, but after a certain point it's like... Damn Matt and Numberphile's fans have gotta include a significant fractions of budding mathematicians
Same
@@tsawy6 I feel like there are 10 types of mathematicians watching Numberphile; those that came here because they know maths, those that were brought into maths by Numberphile and those who forgot this comment isn't supposed to be a spin on the classic binary joke.
@@DomenBremecXCVI That's a real Parker list, if I do say so myself.
It's not published though
In the Parker Square video, Matt said something like "In mathematics, fame is different. It's when someone looks you up once a century.". This must mean Matt is REALLY famous now.
he is some hybrid of maths-famous and regular famous which is both more famous than maths-famous and less famous than celebrity-status
Parker Famous.
@@custodeon TL;DR a superposition of different famousnesses
@@custodeon he’s a Parker square of a celebrity
He's on a coffee mug fer gosh sake. Millennia from now, archaeologists (probably alien) will dig them up and he'll still be famous.
8:33
Kid: “Mom can I have P vs NP“
Mom: “No, we have P vs NP at home“
P vs NP at home: Parker vs Non-Parker
The Parker P vs NP
@@elementalsheep2672 that's the one! 😂
brilliant
The better version
let me just quickly validate this joke... done
It's so rare, and incredibly delightful, to see a grown man beaming with joy at what is literally a consequence of being mocked in front of an audience of millions.
He is a meme.
I would be happy too...
Honestly I think it’s quite sweet that they named it after him.
But affectionally mocked.
Well, self-mocked, but yep. Matt is awesome.
All this mockery just earned him a place in mathematics for posterity.
Also:
"They're all non-Parker - because they work." *dies inside*
??
Everybody seem excited that "parker" has been mentioned in real, published research paper - but I think most of you underestimate how exited the authors of the paper are, that their paper has been featured in real, published Numberphile video.
It's basically numberphile bait
And we got tricked.
I love how mathematicians use Parker as an adjective meaning "almost works"...
If the large mathematical community finally caught it... Parker will be a legend.
idk if I would be happy or sad if my name was given that definition. On one hand, my name has become an ACTUAL property in math. Like in a published paper - it will live on forever. but on the other hand, the property my name describes is "doesnt work" LOLOL
Matt Parker is a comedian. Some of the best jokes in life are where things almost work.
I’m sure he’s elated.
@@pvic6959 At the very least, they have a sense of humor...
I would be tremendously honored to have my name used in math in any capacity. Matt seems pretty jazzed about it.
Parker square video was one of the most fun video I've ever watched. I never thought how a simple mathematical puzzle can be so enchanting.
1/7 is a cool number with 6 recurring digits and the 0 is the FP function
How many are there for 1/49 ? 😎
@@goldnutter412 but not in integer fields
14:48 there's 13 ? really ? 🤣
This seems right to me for personal reasons hahaha also distribution wise you wouldn't expect, but possibly suddenly another group appears wayyyy up there in the giant numbers.. hm
I look forward to seeing Matt being awarded the Inverse Fields Medal
Would that involve paying $15,000 for damages done to the field of mathematics?
I would much more like to see a Parker Medal for mathematical innovations that almost work.
The Parker Finite Fields Medal
The medal for math that doesn't work but you gave it a go.
@@MattMcIrvin Damn! You got there before me!
Everytime he said "Non-Parker.. because.. it's working" you can see in his eyes, a part of him dies. :D
It's fame... don't knock it!
@@simonmultiverse6349 Anti-Science is on the Rise. Uneducation causes Muffled Logic to be be more and more accepted, so casual B.S. is getting more and more popular.
People embarass themselves all the time now by claming NASA is faking the Sun,
the moon is a hologram,
the Earth is flat,
Aura and Chakra are kinda Science, so trust me bro, i know we are all immortal - oh, and one last thing: Koalas are Fake; they are ALL CGI. All.
A part of him becomes Parker.
@Irony What a silly comment, Irony.
false :D.
The 6x6 table says 3*2 = 1 mod 6, but I guess that is a parker-one.
The Parker Times Table
Lol, I caught that too. Seems closeups of that table were edited out due to the mistakes in it.
Plus he also circled that one when circling all the ones in the table ':D
Can we have a Parker Timetable, (not "Times Table") where the trains almost but not quite arrive at the times they're supposed to?
@@simonmultiverse6349 I think we already have that :D
It's been 5 years, but Matt Parker is still Matt Parker.
Matt Parker + 5 years = Matt Parker ? :D
And that's great!
Matt Parker is officially invariant wrt time
88th like!
And I'm still non-Parker.
Finally someone explaining P vs. NP in a way everyone can easily understand.
false.
I love the fact that "Parker" defined as "not working" is an actual term in a math research. I just started laughing so much, this was awesome.
6:50 ( left square 3;2 )(seen the meme, just was about to comment on the Parker square, and was informed that it's actually a Parker Parker square. The circle later really helped )
lets gooo you watch numberphile too
@@RaiinWing Hi rainwing 😀
??
I think I'm going to start saying "don't go trivial" randomly to people.
Just answer any complex question with relativity
Meaning of life ? relativity (or 369)
For String Theorists, every sequence of "Why" questions leads ultimately to the answer "String theory".
@@MrAlRats but they have to be "strings" of physical matter, with 2 dimensions 😅
For finite fields of prime power orders there was some confusion in this video. The integers mod 49 or 4 or 8 etc don't produce finite fields of those orders. It's just that there do exist other finite fields of those orders with different structure to them.
Eg. In Z mod 4 the multiples of 2 are 2x1 = 2, 2x2=0, 2x3=2 and 2x0=0 and so there is no inverse for 2.
The true Parker Finite Fields
Thanks for pointing this out! :)
And I think the way you can get finite fields of prime power order p^k is by adding zeros of particular polynomials to the finite field Z/pZ, much like you can add i (one of the zeros of x^2 + 1) to the real numbers to get a new, bigger field: The complex numbers
I was thinking that finite fields of order of "powers of primes" could be things other than Z mod (p^r), but note that the paper says: Finite Fields and Rings - which implies to me that they're claiming that magic squares of squares don't just work in (most) finite fields of the form Z mod (p), but also some rings of the form Z mod (p^r) where p is prime...
Yes, that is a very important point, hope they correct that.
For those who might've missed a pun at 8:38:
P v/s NP (Which in video is used as a short form for Parker v/s Non-Parker) is actually one of the seven millenium problems by the Clay University. Each problem worth a million dollars. That means if you solve it you'll get a million dollars.
I'm feeling the need to hear the word "Parkericity"
"Hey how about the Parkericity of that field ?"
Parkerness?
This. This needs to become a thing.
Why not have degrees of Parker for how far off from working it is
Margin of error is now called "Parker approximation".
@@Games_and_Music approximation is already Parker property (Parker action?)
Parkerximation?
"Every real number has a buddy real number, where if they multiply together, you get 1."
1: "Am I a joke to you?"
0: "Yes."
1 is such a lonely number. so powerful they wont even let it have its proper title of prime of primes.
@@pulsefel9210 You could even say that one is the loneliest number
-1:
1? One’s buddy number is 1.
0? Zero is the same as n (limit as n goes to zero). So its buddy number in that case is 1/n (limit as n goes to 0).
It is after all an ancient mathematical proposition that one is not a number but the unit first enunciated by Aristotle. If 1 is not a number then 0 is right out.
He gave it a go, he tried, and finally he's achieved infamy in actual mathematical research! Kudos to you Matt
5:13 The finite field with 49 elements is not actually the integers mod 49 (Z/49Z), because 7 has no inverse. The construction of this field is more complicated.
For the curious - to actually construct that finite field, consider the set of polynomials with coefficients modulo 7. You can get a field with 49 elements by taking all polynomials of the form ax + b, and then doing arithmetic on them modulo x^2 - 3 (again all the coefficients are modulo 7). 7 choices for a and 7 choices for b make 49 elements, and you can never multiply two polynomials to get zero because x^2 - 3 doesn't factor modulo 7.
You can get finite fields whose sizes are higher prime powers (i.e. 7^n) by doing arithmetic modulo some irreducible polynomial of degree n.
Same with any number that isn't a prime.
@@jaredbitz why modulo x^2-3 and not x^2?
@@jaredbitz no wait, it's because you can imagine x=sqrt(3)
@@jaredbitz , or for people who don't know how to do that with finite fields, but do know how complex numbers work, imagine that i is the square root of 3 mod 7, and consider things of the form a + bi where a and b are in Z7.
"Technology has moved on since", showing a 3D-printed version of what he once wrote on brown paper.
The fact that you still have the mug at 7:52 makes me super happy to follow math community
3 blue 1 brown
Matt Parker: You cannot find a whole number inverse of an integer.
1: I will pretend I didn't see that.
1*1=1
Unless it’s the identity. Just like the only non-negative number with a non-negative additive inverse is 0.
I tip my hat to the author of this paper. Well done.
This is the most clear explanation of N vs NP I've ever seen.
"Return of the Parker square"
This is probably the most clickbaity title possible, for numberfile fans ;)
Is it clickbait if it's true?
It's the only reason I clicked instantly
Return of The Pink Parker?
...featuring Peter Parker? (different superhero, I know)
it did reappear, not as main focus though
more like a cameo old character in the new series
Matt Parker is a great teacher and quite funny too. I love seeing him here.
PARKER SQUARE LES GOOOO
Z mod 49 and Z mod 25 are NOT fields. There exist fields with 49 or 25 elements but they aren't simply integers modulo some number.
THANK YOU for pointing this out. An uncharacteristic error from Matt :(
Correction: only "integers mod a prime" is a field, not "integers mod a power of a prime". There are finite fields of size "power of a prime" but they are not a quotient of the integers.
Thank you! I didn't think they would miss such an obvious mistake..
It was a Parker-explanation
These are the Parker finite "fields"
The P vs NP reference killed me
This is one of the greatest character arcs I've ever seen!
“If you’ve got a number, I dunno… a.”
Can’t wait to see that one out of context
I wish there was a compilation of every time Matt says "big fan..."
This is one of the best gags of this channel lol
The Parker prize needs to become a reality, surely!
For all maths research whose results do not accomplish what they aimed for,...
...but which do make some headway towards it, which gives an insight into the subject, which explores useful perspective on the subject, or which studies the hardship of proving what you are trying to prove,...
... so that maybe one day we can make more informed maths research that DOES achieve what it was trying to do.
In other words, for all the disappointing, unglamorous near-misses which might eventually lead to actual results.
Not a bad thing to have a prize for, actually.
If this approach of near misses does at some point answer the question whether the integers are parker or not, then it actually becomes a serious proposal: the approach worked.
@@nielskorpel8860 Basically, "Give it a go"
14:46 'Parker' being in Comic Sans is the cherry on the top of this video.
;)
5:09 - Is he implying that the integers mod 49 are equivalent to the finite field of order 49? Because as far as I know this only work for primes, for prime *powers* the multiplicative structure is actually different.
Yea, just noticed that, in Z_49 you have 7*7=0, and a field doesn't have zero divisors, so its not a field. I guess he kinda confused it with the fields of order equal to that prime power.
I am a bit confused, does a Z7 field for example, have 0 in it? Cause 0 doesn't have a multiplicative inverse does it?
@@pianissimo7121 Yes. All fields must have a 0. The rule for multiplicative inverse doesn't include 0, as with usual real numbers, rationals, etc.
@@pianissimo7121 In fields, zero is a special number that follows different rules. In every field, 0*a=0, for any a in the field, and 0 is the only number that doesn't have a multiplicative inverse, because a field needs 0 to work. That's true in the real numbers, complex numbers, and any other field. Hope that clears it all up.
@@pianissimo7121 The statement of being a field is that: "Every nonzero element has a multiplicative inverse", where zero is defined as the element satisfying 0+x=x+0=x for all x in the field (in other words, 0 is the additive identity). So yes Z_7 has the elements {0,1,2,3,4,5,6}, where all but 0 have multiplicative inverses
I’m waiting for the OEIS entry for Parker Numbers
A308838, the Orders of Parker finite fields of odd characteristic, aka the list shown ignoring 2. The "state of the art" has improved and it was shown 243 is a Parker finite field.
5:04 Just to clarify the the integers mod a power of a prime do NOT form a field in general (for example, 7 does not have an inverse mod 49). It is only the case when the power is 1 (that is, the integers mod a prime). There exist finite fields of size p^k for p prime and k > 1, but they are constructed differently.
"What about _infinite_ rings?" Well, if a magic square of squares "works" in ℤ, then it must also work modulo n ∀ n∈ℕ. However, in some of those ℤₙ, the square may have repeated entries that weren't there in ℤ; in particular we know that this must be the case for all n for which ℤₙ is Parker. (As the paper points out, and as you mention in the extra footage, a solution in ℤ would imply there are only finitely many Parker rings.) Thus those rings give us constraints on any possible solution in ℤ; for instance, ℤ₆₇ being Parker implies that a magic square of squares in ℤ cannot have all nine numbers distinct modulo 67, because otherwise it would imply a solution in ℤ₆₇. It's the Parker rings, and _only_ those rings, which help us by cutting down the search space for ℤ; Parker rings are _useful_ because they help us identify what _won't_ work, and that can be valuable in itself :)
Hope that helps Matt feel a little better about his eponymy.
It's not true that integers mod 49 (or any non-trivial prime power) form a field. For example, 7 doesn't have an inverse mod 49.
I think Matt got confused by the notation F_{49} for a finite field with 49 elements.
8:09 I am made up, and enormously proud of you, Matt! Edited: doubly proud of your joke at 8:33 🤣🤣
Important note (for anyone who, like me, is going to spend a few hours looking into this): The finite field F_(p^k) is NOT the integers mod p^k. For example, F_9 = {0, 1, 2, i, 1+i, 2+i, 2i, 1+2i, 2+2i} where i = sqrt(-1).
well you know for pedantry that it's actually that i^2 = -1, thats the technical definition
Darn I just went and typed all that out less clearly and then I saw your comment!
Please also not that the field F_(p^k) has character p ie. np = 0 for all n in the field
@@redapplefour6223 that's not the technical definition either. Or, well, it is part of the definition, but of the technical definition of the _multiplication operation_ in ℂ, not of i. The imaginary unit can't be defined like this. (Note that e.g. in the quaternions there are three distinct values that all fulfill this equation!) To make it a technical definition, you need to first define ℂ as a 2-dimensional vector space with unit vectors 1 and i, and only then equip it with the multiplication that has this property, in order to form a field.
@@leftaroundabout right, thanks! makes sense that that's how that works. so are field extensions are just unit vectors in disguise?
This guy has such comical facial expressions, he would probably do well in comedy movies if he did acting.
He does math related stand-up
Well he does do stand up about math lol
He's really the Jim Carrey of math(s) educators.
Matt Parker is a comedian after all
Mallu spotted 😂
I am in love with this whole saga
I love it when they bring back season 1 characters!
14:10 i don't know why but seeing those parkers pop up on the screen cracks me up
Parker and non-Parker being used in an actual paper was an hilarious twist ahahahah
We tease because we love you, Matt. Your enthusiasm is infectious. I consider myself, to be a Parker Person.
I just love the fact that "Parker" is a term accepted by most if not all mathematicians.
I burst out laughing when you put "P vs. NP" as an overlay on the screen for Parker vs. Non Parker fields
This is about right. We had the "Mould effect" so now Matt is just catching up to Steve with the "Parker property". I assume this is the omen that Matt will catch up with a million subs soon. 😘
This was great, having in-depth math on a higher level than usual! Please do more of this!
Ok. When you get named in a paper that actually delivers, and sets a new standard for maths... This is amazing.
It would be hilarious if Parker vs Non-Parker becomes an elemental part in solving the P vs NP issue.
Truly the most troll-y way to get something professionally named after you. I love it.
I actually find the the fact that Parker is rare a really cool thing. Sure they “don’t work” but they got people talking first, and there aren’t that many
This amazing 1 in the column of 2.
Made my day Mr Parker.
Thank you.
Many years from now when you're pushing up the daisies, at least you will be forever remembered having had a mathematical property (even a duff one) named after you. Quite an honour.
A Numberphile video with Matt Parker AND a Stand-Up Maths video on the same day? Nice!
Maybe a Mathematician gets to be upset when their name is associated with a kind of failure, but a Standup Mathematician is just happy to setup a punchline.
This video made me happy! Not that any other Numberphile video makes me otherwise, but this one's special. Congratulations Matt!
10:58 nice surprise seeing myself in a Numberphile video!!
Polynomial rings typically aren't fields, but you can make fractions of them (rational functions) and those will be a field.
True, but those aren’t _finite_ fields. You have to mod an irreducible polynomial to get a field.
I've always enjoyed how the multiplication symbol is the addition symbol nudged over 45° and the division symbol is the minus symbol with some dots or recently also just pushed over at an angle /
This made me so happy. I can't believe this is actually in the paper - what a wonderful thing the community has created here.
15:26 I love that the previous video is in the citations for this paper!
Being diagnosed with Parker finite-fieldness is a truly heartbreaking event, my condolences.
I've been following these channels forever and I'm like, look at you Matt! Congrats!
8:00 I read the title of the video and knew what I was getting into, but I didn't realize they did poor Matt THAT dirty with the section title
awesome stuff as usual, thanks so much for making these sorta vids i love them
That's absolutely insane! Parker is not only a scientist, but also a living meme - we know that for quite some time. But the fact, that he's not just an ordinary walking meme (albeit this in itself is something to be proud of), but a meme which is included in scientific papers. Incredibly amazing!
Are mathematicians scientists? And if so (or not so), what exactly are the criteria we're using to define what a scientist is?
@@jd9119they do research in universities in a scientific field. Difficult to be more of a scientist....
@@fregattenkapitan Except scientists usualy apply the mathematics to another discipline.
a small mistake at 5:04. It only works for prime numbers. If you take a power of prime numbers, it is not modular arithmetic anymore. So basically if you are working in the finite field with four elements, 1+1 is still 0 just like the field with two elements, but you have an extra element x which satisfies x^3=1.
What a coincidence I was wearing my Parker Square shirt today!
The sequel we always knew we needed
5:17 that's not exactly true, the integers mod 49 do not work as a finite field. However there is indeed a finite field of 49 elements, which can be constructed as 1st-degree polynomials over the integers mod 7.
In fact [Theorem 1] the integers mod n are a field if and only if n is prime, and [Theorem 2] there exists a finite field with n elements if and only if n is the power of a prime p (constructed as polynomials over integers mod p)
Ahhh - useful comment. Since 2 is a prime and powers of 2 crop up in computers this creates lots of possibilities once you realise the fields are more complex than just mod n. Now I need to look up polynomials over integers as fields - well that's this afternoon gone.
It's a parker field.
Correct, the powers of primes correspond to extension fields, i.e. ordered n-tuples of elements of the base prime field. It's analogous to how the complex numbers may be viewed as ordered pairs of real numbers.
@@twohoos Yeah exactly. Now that I think of it complex numbers are essentially polynomials modulo x²+1, which is really similar to the way we construct finite fields of order p^n.
I think convergence almost everywhere should be called parker convergence.
Excellent comme d'habitude ! Un plaisir de regarder cette chaine.
Translation for non french people : " Hi, it's sunny today but it depends where you live actually"
I haven’t started the video yet and this is the best explication of fields I’ve heard
P vs NP has just been solved 😂😂
15:25 hey...brady's name is on a paper now (if this has already happened before i didn't notice)
(also...it would technically be better if the link was archived since stuff on youtube can disappear)
It brings me joy that the Parker Square has left the numberphile bubble and ventured into general mathematics and is being used in published research papers
11:51 Maybe I'm too accustomed to seeing numbers in different forms, but adding those numbers mod 29, I just mentally read 28 as -1.
This is the equivalent of how Gary Larson is now credited as naming "the spiny bits on the end of a Stegosaurus" the Thagomizer because before him nobody had a name for it. It was done as a joke and then someone saw value outside of it being funny.
This was soooo interesting, thank you Parker for being very knowledgeble and funny. I wish i was able to sit with you with a glass of beer and just ask basic questions about math, which i'm terrible at, and the answers would be probably unexpected. Yeah, thanks again!
parker is finally a true mathematician, he has a thing named after himself
Love seeing the TH-cam url put in as a reference in the paper.
As the Trojans became famous for their horse, so too shall Parker be known for his square
Well. I can only guess that Matt Parker’s ego went from finite to non-finite after being established as an (in)famous legend of mathematics! I love this guy!
Dude, you’ve made it! This is like having a unit named after you in physics (my field)!
Actually, finite fields with order p^n are not Z mod p^n (check for yourself 2^2). They are instead constructed using polynomials, as outlined on wikipedia
Finite field F₄ isn’t technically integers mod 4, it’s a bit more complicated than that. Example: 2² = 3, it’s not mod 4 because 2² = 0 mod 4. This is true for all non-prime order fields.
I also showed that integers mod 4 has zero divisors and therefore NOT a field
Imagine in 100 years and students are asking why Parker means “something that almost works but isn’t quite there”
They probably ask:
Grandad, what does shoddy mean?
- That's an ancient word for Parker.
This is the greatest video I've watched this year by far 👏🏻👏🏻👏🏻.
Wow this is so cool! Awesome sequel to parker square. Can’t wait for part 3 in a few years