Squaring Primes - Numberphile

[@numberphile]

NEW: Belphegor Prime T-Shirt and Poster --- www.bradyharanblog.com/blog/belphegors-prime-t-shirt

[@kyakarogenaamjankar898]

Tysm

[@murk1e]

17²=139.... the Parker Prime Square

[@marthak1618]

The appropriate phrase is "more elegant".

[@joshuaadamstithakayoutubel2490]

7*7=49 7*1+1*7=14 14+4=18 1*1=1 1+1=2 17*17=289

[@Pacotille_6529]

Hi, can we hear more about sub-primes numbers, is there really a way to define them?

[@clearlyc0nfus3d19]

Watching a PhD mathematician struggle to get 17^2 was reassuring.

[@TheVillan1980]

It was a Parker square...

[@sylviaxx3574]

does he have a PhD?

[@lawrencedoliveiro9104]

Ah, binomial theorem, my old nemesis, we meet again.

[@joshuabradford8372]

@@williambiggs3699 unfortunately that’s not how multiplication works. For example, 7 x 7 is 49, but 6 x 8 is 48, and 5 x 9 is 45, and 4 x 10 is 40, and 3 x 11 is 33, and 2 x 12 is 24, and 1 x 13 is 13, and 0 x 14 is 0, and -1 x 15 is -15, and -2 x 16 is -32, and it just keeps decreasing by more every time. If you don’t believe that, try 8 x 8 is 64, 7 x 9 is 63. This means that 17 x 17 DOES NOT EQUAL 14 x 20. Sorry about that.

[@Jordan-zk2wd]

@@joshuabradford8372 They didn't say that 17×17=14×20, they said 17×17=14×20+3×3. Their argument was a little unclear to me at first too, maybe give it a reread. I'll summarize my understanding below though: 17-3=14 17+3=20 14×20=(17+3)×(17-3) 14×20=17^2-3^2 (because (a+b)×(a-b)=a^2+b^2, you can foil it n check) 14×20+3^2=17^2 280+9=289=17^2, which is true. Basically their method is taking a number x, adding or subtracting some number y to find the closest number of the form n×10^m for n is an integerbetween 1-9 and m is an integer greater than -1, then multiplying x+y and x-y and adding y^2 to that to get x^2. Since y is smaller than x you don't have to memorize as much, and if you don't know y^2 you could do the same trick there. Feasibly you could reduce every problem of large squares to an arithmetical calculation using smaller squares and numbers of the form n×10^m and not need to know any squares above a certain value even.

[@jerry3790]

*Sees first two primes don’t follow his rule *Calls them sub primes.

[@nimmin8094]

To be fair, 2 is the only even prime. It breaks a lot of rules. You can modify it and use a different equation, and see it works for inverted positive integers: 2^2=(1/8)(24)+1 3^2=(1/3)(24)+1

[@mrmimeisfunny]

If you read the book, you know he calls them sub primes because they are prime by default and don't even have an opportunity to be divided by anything.

[@qwertyman1511]

@@nimmin8094 evenness is a poor property to use.

[@rosiefay7283]

@@mrmimeisfunny *No* prime can be divided by anything -- if it could, it wouldn't be prime. [By any positive integer except itself and 1, of course.]

[@macronencer]

@@rosiefay7283 I think the point is that 5 is the first prime greater than 2x2 (the first compound number), though I don't remember reading that part of the book so I might be wrong.

[@N.I.R.A.T.I.A.S.]

Video starts with Matt trying to Parker square 17.

[@Israel220500]

17 Parker squared

[@WhattheHectogon]

You're my hero.

[@grandpaobvious]

Algorithms are for computer nerds.

[@besserwisser4055]

and more

[@kuromurasaki5273]

@@grandpaobvious algorithms are a key part to how the (human) world sustains itself. They are part of every facet of our technological life; from your mcdonalds order and grocery stores to space x and mars rovers.

[@Jacob-yg7lz]

I like how hard it was for him to do the math in his head. It reminds me of the saying "the more math you know, the harder it is to do math"

[@KemonoFren]

Who said that?

[@kryptoknight992]

@@KemonoFren Joe

[@GroovingPict]

mental arithmetic is not "doing math" and you completely missed what that saying is, well, saying

[@Rishnai]

@@GroovingPict Aye that’s the fun part about sayings, they gain power as both their original meaning and its inverse over time

[@messagedeleted1922]

knowing how to do math, and actually doing math are two different things.

[@munjee2]

It amazing how matt did all the mental maths perfectly and then said 170 +70 AND 49 is somehow less than the original

[@jacksonr260]

🤣yup

[@Richard_is_cool]

0:58 I love how he rewrote the 139 to make it read 289 after he scored out that calculation so he could say "Dammit I was right". Parker convincing.

[@ojaskumar521]

No of likes 468 . Divide it by 2 you get 234. Well now you have increased his like count

[@faizanmohsin3685]

The sum was actually 289. Check it again

[@sameldacamel3889]

He wrote the 170 so it looked like a 110 that is why his maths is wrong. And he carried the 1 wrong.

[@JakubS]

Hello Richard.

[@matkomajstorovic6935]

That was a real parker square of a calculation.

[@seanfraser3125]

“I like to argue that 2 and 3 are not real primes” Goodbye, fundamental theorem of arithmetic

[@grandpaobvious]

is it iconoclasm or nihilism? We report, you decide!

[@quaternaryyy]

CogitoErgoCogitoSum Its called a joke lol

[@blackflash9935]

@@quaternaryyy I am pretty sure he was joking too so... double r/wooosh for you, I guess.

[@ThomasNimmesgern]

welniok There is one reason to prefer Fahrenheit: Compared to Celsius, you usually get much more degrees in Fahrenheit.

[@israelRaizer]

​@@e11eohe11e 2 and 3 ARE real primes: a number is prime when it is divisible by only 2 numbers, 1 and itself. 2 is divisible only by 1 and itself, thus fulfilling the criteria, there's no such things as real or non-real primes. The only other category related to primes that I can remember now is semiprimes, which you get when multiplying 2 primes together.

[@BrooksMoses]

I like how the four categories from your proof also show up in the "easier" proof. Either the multiple of 4 is above or below the prime, and either the multiple of 3 is above or below the prime, giving four possibilities that directly correspond to your categories.

[@KingdaToro]

4:31 Matt received weapons of math instruction

[@PsyChoLogicZ]

Said Mike Tyson

[@52flyingbicycles]

One of the few cases when the British “maths” (🤮) would be better because “maths” sounds more like “mass”

[@simonmultiverse6349]

@@52flyingbicycles so you could have maths confusion and maths debate...?

[@sietsejohannes]

Matt: There is a pattern with prime squares, they are all multiples of 24 plus 1. All? Matt: Almost all. So it's a parker squares pattern then...

[@alexvandenbroek5587]

It is truly a parker square because if they're all a multiple of 24+1, surely they are all also a multiple of 2+1. It's more graceful cause you don't have to call 3 subprime. Whatever even number above 1 you pick, it'll always have some cut off point where a prime is too small for it to work. Unless you pick 1+1, which includes all the primes because it is literally the definition of primes. Matt is just defining prime numbers here in a very weird and unnecessary way..

[@alexvandenbroek5587]

@Urrcreavesh I never claimed that it has anything to do with squares because it doesn't. I'm referencing a meme about the parker square because it's used whenever Matt tries to do something clever which is unimpressive and doesn't work very well. I think this is such an occasion. Look it up, it's on KZbin somewhere

[@jovianarsenic6893]

@@alexvandenbroek5587 multiples of 2 + 1 does not sound impressive at all since that is the definition of an odd number.

[@dannygjk]

Many primes do not follow his rule not just 2 and 3.

[@debarshidas8072]

@@dannygjk example?

[@j_sum1]

17^2=139 I think I just witnessed Parker Squaring.

[@rosepinkskyblue]

Underrated comment

[@yuvalne]

17^2=139. C'mon. Now you're just begging us to make a Parker Square joke.

[@masansr]

And the way he did that, wth. (a+b)^2=a^2+2ab+b^2 is much easier to square numbers!

[@karolakkolo123]

@@masansr yep. 20 * 14 = 280 and then + 3^2 = 289 Because basically you have (n+3)(n-3) = n^2 - 9. Then you just add 9 to get n^2

[@moadot720]

139 dislikes on the video...

[@mateuszm7882]

17 x 17 is the easiest way, lol

[@simonvanprooijen]

@@mateuszm7882 yeah haha

[@dan_tr4pd00r]

Only three things are certain: Death, Taxes, and Parker Square jokes.

[@macronencer]

Don't forget rice pudding.

[@Pacvalham]

Parker Squares from the Nile; does anybody else get the second reference?

[@priyanshusudhakar5206]

Man you stole it from a scientist whose name i cant remember .The real statement goes like this “there is nothing certain in this universe except death ,taxes and the second law of thermodynamics”

[@alfredo.zauce1892]

Priyanshu Sudhakar No, the real statement is from Benjamin Franklin and it’s only the first two.

[@renatoherren4217]

The fourth one are Uranus jokes. 😜😜😜😜

[@HunterJE]

That second proof gave me chills down my spine when I saw where it was going, you might even say it was an Amazingly Satisfying Mathematical Result

[@yoni5919]

I am currently doing my degree in maths and one of the things we need to prove is that all primes squared (above 3) are one more than a multiple of 6, and I know how to prove it because of your video. love you matt!!!!

[@nimmin8094]

3*3 is one more than 8, 1/3 of 24. 2*2 is one more than 3, 1/8 of 24. Pretty neat!

[@slartbarg]

exactly, he didn't say that it had to be a whole number integer multiple of 24

[@TheDGomezzi]

Hahahah Slartbarg, in that case, all numbers are multiples of 24

[@jamespfp]

*LULZ* So Yeah -- I thought I caught a mistake in your maths there Nimmin, but I double-checked what you wrote -- 3*3 isn't the same as 3^3, my bad. *BUT LOOK.* 2^3 = 8 ; 3^3 = 27; 5^3 = 125; 7^3 = 343 Or, 1/3 of 24; 24 + 3; (5*24) + 5; (14*24) + 7.... :D

[@nimmin8094]

@@TheDGomezzi I was just going by inversions of integers

[@nimmin8094]

@@jamespfp My brains a bit slow this morning. I'm interested! I'll have a proper look this afternoon :)

[@christianp7200]

17²=139, nice to start the video with a parker equation!

[@NOTNOTJON]

can someone make this T-shirt please?

[@MithunGaming]

Can you explain what a Parker equation is? Please

[@zucc4764]

@@MithunGaming it's a running joke (meme if you will) when a calculation/classification is a miss, recategorizing them as a "Parker square or equation" etc. instead of identifying it as a miss.

[@hingedelephant]

Mithun Gaming - Parker Square or Parker Equation: A joke that has outlived it’s humor and should die.

[@zahidshabir4038]

the easiest way to work it out for me is just work out (17*20)-(17*3) and 17*10 is 170 which is above the 139 he worked out

[@B3Band]

Nice sneaky edit of the 139 on the paper

[@ractheworld]

I just love your guests, every one of them. Listening to them is such a treat. Thanks

[@MrYairosh]

my theorem: every prime cubed is one more than a multiple of 2.

[@ronindebeatrice]

Well yes. A prime will be an odd number. The product of 3 odd numbers will be odd. Was this a joke? I'm dim.

[@MrYairosh]

of course it's a joke @@ronindebeatrice

[@patricksalhany8787]

yair koskas wrong. 2 is a prime. 2^3=8. 8 is not 1 more than a multiple of 2. Your theorem is wrong.

[@MrYairosh]

@@patricksalhany8787 so this is the only prime that doesn't follow my theorem

[@patricksalhany8787]

@@MrYairosh yeah, but you said EVERY prime, so including 2.

[@Thomas-vn6cr]

I sure hope the maths related items are for review and unboxing purposes.

[@CompactStar]

You have more likes than one of Numberphile's pinned comments.

[@skeletonrowdie1768]

no hate, but i don't get unbox excitement and i'm jealous

[@beardedemperor]

@@skeletonrowdie1768 I generally agree, but calculator unboxing is a whole different beast.

[@andyb6177]

MathSSSS

[@munjee2]

They could just for his store though

[@SD-el9wj]

The fact that you don't count 2 and 3 as proper prime numbers is the REAL subprime crisis.

[@haidynwendlandt2479]

S D to be fair, both 2 and 3 are the only prime numbers divisible by 2 and 3 respectively

[@akshataggarwal4002]

@@haidynwendlandt2479 Dude,do u even know the definition of prime numbers?

[@haidynwendlandt2479]

Akshat Aggarwal The proof forces the numbers not be divisible by 2 or 3, so every prime number greater than 3 works

[@akshataggarwal4002]

@@haidynwendlandt2479 that doesn't explain ur 1st comment,it doesn't make any sense.

[@haidynwendlandt2479]

Akshat Aggarwal I literally said in my first comment that 2 and 3 were prime numbers. I was explaining that one of the reasons why he didn’t include them was because the proof doesn’t allow for the numbers to be divisible by 2 or 3.

[@takonyka]

lol i love how he failed 17^2

[@patricksalhany8787]

What's weird is that he said that 17^2 is 170 plus something, but he got at the end 139 which is less than 170. Aliens.

[@edskev7696]

Parker square!

[@patricksalhany8787]

@diego maradonna I thought you were only a footballer, but you also do maths. Wow! Keep up the great work dude !

[@patricksalhany8787]

@diego maradonna ohhhhh. That is sad.

[@VWftw82]

And the dude has a PhD in mathematics!

[@sb-hf7tw]

STEP 01: Make a RULE STEP 02: When u find any element not following RULE, simply call them exceptions. STEP 03: When u find infinite such exceptions, say it's a COROLLARY of the main RULE! Now, u R done!!!

[@trombonemunroe]

There is a point to be made, though, that 2 and 3 are the only primes which are smaller than the lowest compound number (which is 4). So they are kind of special in that way.

[@Green24152]

Any number plus half of itself is odd.

[@Green24152]

@ABHINAV JAIN That's an exeption.

[@Green24152]

@ABHINAV JAIN That's just a corollary of the main thing.

[@maxwellsequation4887]

@ABHINAV JAIN that's the joke

[@kat-oh3hx]

> supposed to work for all primes > works for almost all (not 2 and 3) > the parker square of prime patterns

[@oz_jones]

They aren't subprimes, they are Parker Primes :)

[@demonking86420]

upvote that man

[@JamesSonOfBaboonzo]

This comment is under appreciated.

[@HBMmaster]

every prime being adjacent to a multiple of six is yet another reason why seximal is the best numbering system (all primes end with 1 or 5!)

[@effuah]

Fails at 2 and 3

[@dermathze700]

@@galoomba5559 I prefer the unary number system, since every prime including 2 ends in 1.

[@The_Guy_]

@@galoomba5559 correct

[@fatsquirrel75]

@@effuah Every prime adjacent to a multiple of six does not include 2 and 3.

[@stuartofblyth]

Just to spell it out for fatsquirrel75 5 is seximal 5 (0 x 6 + 5) 7 is seximal 11 (1 x 6 + 1) 11 is seximal 15 (1 x 6 + 5) 13 is seximal 21 (2 x 6 + 1) 17 is seximal 25 (2 x 6 + 5) 19 is seximal 31 (3 x 6 + 1) and so on. I love it! Thank you, @@HBMmaster.

[@ancbi]

8:15 "I did this way. This is mine. I love it." That's the sipirit! of a classic Parker Squarer. Keep calm and square on.

[@NOTNOTJON]

I watch a lot of mathy channels, this one, loads of sci show, 3 blue 1 brown etc.. Somehow it has taken until today for me to realize that though I love these videos, deep down, I come here for the comments section.

[@sattat3705]

P^2 - 1 way of proving is so very elegant. It really melts my heart. Simple & Brilliant

[@abhinavs2484]

91 = 24*345 + 1, but 91 is not a prime :)

[@karthikeyank132010]

@@abhinavs2484 91 is not a square either. 91 = 7 x 13

[@Wontervandoorn]

A prettier (or at least quicker) version of the first proof: (6m ± 1)^2 = 36m^2 ± 12m + 1 = 12m(3m ± 1) + 1 and 12m(3m ± 1) is divisible by 24 as either m is even, or 3m ± 1 is even

[@anon6514]

Just posted similar comment - yours is better.

[@Myrus_MBG]

Also just posted a similar comment, yours is better since I don’t know how to do +/- without copy and pasting it online which I was too lazy to do.

[@krowa1010]

yeah this is much better, cause at least you dont need to assume that we have 2m or 2m+1 which is not necessarily true, just that even or odd which is 100% true

[@kourii]

@@krowa1010 Um, even numbers can all be written as 2m, and all odd numbers can be written 2m+1. What are you trying to say?

[@patricksalhany8787]

2 and 3 are not primes but subprimes? Mmmmm I too like to live dangerously.

[@innactive1407]

@CogitoErgoCogitoSum because you can-t divide it by 1 and itself since it's the same. Also we can do it from truth by contradiction. Let's say we have a prime p p is divisable by p and 1. if 1 is a prime then it is the only prime thus since having 1 singular prime is useless 1 is not a prime

[@015Fede]

@@patricksalhany8787 this is circular reasoning. The fundamental theorem of arithmetics assumes 1 is not prime. Then, you can't prove it with the fundamental theorem of arithmetics. 1 is not prime, because we have defined prime numbers to be such that they have exactly 2 divisors. 1 has only one divisor, so it is not a prime number.

[@unfetteredparacosmian]

@CogitoErgoCogitoSum Because we define them to have exactly 2 divisors: 1 and themselves

[@An_Amazing_Login5036]

Ok, say i think i like the idea of 1 being prime. I put on my magic hat and make everyone use the definition of prime as A prime is any positive integer factorisable only with itself and 1. What, except the trivial loss of the fundamental theorem of algebra, (which i would like to restate as every number can be written as a unique, simplest possible prime factorisation. Why would it not work?) what breaks? Please enlighten me in how our naturalistic understanding of math (i don’t have any clues about the ground-floor of peano-arithmetic, only that it is how i usually count and use numbers).

[@lunafoxfire]

@@An_Amazing_Login5036 I mean, mathematicians used to consider 1 as a prime number, but as number theory evolved it was generally agreed upon that it's easier to just say that it isn't one. That way you avoid constantly saying "every prime except for 1". Also primes are interesting solely because of the Fundamental Theorem of Arithmetic. You could say they were "invented" as part of the theorem. So it would be kinda counterproductive to say 1 is prime but then also make an exception for it in the theorem. In a sense, 1 is "too special" to be "just" a prime number... it's sort of a foundational concept that's _even more_ fundamental than primes.

[@thomasi.4981]

I paused at 9 seconds to work it out with algebra. It makes tons of sense! I knew right away that it was reasonable since prime numbers themselves have a similar multiple+offset pattern, where they are 6n+-1

[@thetntsheep4075]

So Matt's method, in its inferiority, could be called "The Parker Proof".

[@Piaseczno1]

Right, but someone earlier called it the Parker goof.

[@dannygjk]

Many primes do not follow that rule not just 2 and 3.

[@Solarsooo]

@@dannygjk like which one?

[@richardnanis]

I love numbers theory, esp. with primes! So amazing and easy to follow! Keep it on!

[@DrSnap23]

Aaaaand Matt Parker failed a square again. Typical.

[@DanielVCOliveira]

Lots of Parker Square jokes, but your wording was the best lol

[@DrSnap23]

Thanks xD

[@Jemtus]

-Typical- Classic.

[@stapler942]

As he would say, at least he gave it a try.

[@Yoshiyosh]

Horrendous!

[@staffehn]

I'm a simple man. I see Parker and squares, I click like!

[@gyroninjamodder]

staffehn I remember when you still made videos

[@YellowBunny]

It's always cool to find other KZbinrs you (used to) watch in the comments.

[@DrSnap23]

So 139 is the Parker square of 17, huh.

[@suvanshsharma7878]

DrSnap23 underated comment

[@hkayakh]

I recently found a marvelous pattern in the prime numbers! Every prime number is a prime number!

[@stultar]

did you know all primes are indivisible by all numbers except itself and one?

[@pierredenis2482]

... and conversely!

[@JedidiahWB]

I think the word for the second proof is "elegant", it's compact, gets the job done. But elegance in design often comes after the working out and pruning of things that are unnecessary, and are often not the route that is taken by a pathfinder; instead, it's the shortest route that you can really only clearly see after you've made it to the destination. I always think of when I would be off-trail in the mountains and come across something interesting. The path I would take people on to come and view the interesting thing was usually much shorter than the route I took to discover it, because now I have the destination and you can find the "shortest route" to it. I think the mental path of discovery is very analogous, and I'm happy that Matt has made a point of showing the more circuitous paths, I think it really makes the journey seem more accessible to people and de-mystifies math and knowledge, which is all too often held up as unattainable and some sort of magic. Yea, once you point something out to other agents and experts in your space, people will start optimizing immediately, and the result of that peer-engagement usually has that sort of elegant and beautiful quality. But, often the most innovative ideas come from a mind that is just bent on finding "A" better way or "A" solution, and it's great to showcase that grit and brute-force and inelegance are not enemies of furthering understanding and knowledge, while at the same time, showing how engagement with other experts takes a "cool" idea, and turns it into something beautiful. --- Thanks Matt (If you're still reading comments on here 4 years later)

[@KappaClaus]

Makes me feel human even mathematicians trouble with head calculations!

[@EGarrett01]

Einstein used to carry a cheat sheet around with various fundamental constants written down and Ramanujan lost a mental-calculation contest to a random guy at Cambridge.

[@charlesmartin1972]

The professor who got me to understand calculus couldn't tie his shoes

[@EricPetersen2922]

Charles M - I’m a successful biz man and can’t tie a necktie. We all have our strengths & weaknesses

[@greenoftreeblackofblue6625]

Nah it's just a Parker Square he meant to do that.

[@Peter_1986]

Some mathematicians love to make themselves appear all mighty and invincible, but they ALSO struggle with math every now and then. Like Matt Parker himself has said a few times - math nerds don't necessarily love math just because it's "easy", they love it because they enjoy its difficulty.

[@sacredbolero]

I was so proud that my proof is the “simpler” proof. Although being in secondary school... maybe I had a headstart with the p^2 - 1 part.

[@pickleballer1729]

Great video. I've always been fascinated with primes. The first thing I did when I got my forst computer(a Commodore 64 (khz processor speed) was to write a prime number generator and then tweak it until it would run really fast. Gees, what a geek.

[@SkippiiKai]

You might like Dave's Garage channel. He talks a lot about programming prime number finders as a kid on very early computers and optimizing the code and now he uses the same code to test the speeds between 100 different programming languages.

[@pickleballer1729]

@@SkippiiKai Thanks, I'll check that out.

[@Myrus_MBG]

You can do it with (6n+1)^2: 36n^2+12n+1 12(3n^2+n) If you remember that n^2 is odd if n is odd and even if n is even, then you can see that 3*odd+odd will be even and 3*even+even is also even. So, it’s 12(even) which is a multiple of 24. You could also just factor it as 12n(3n+1), and either n or 3n+1 has to be even since if n is odd, 3*odd+1 is even

[@ThomasGodart]

Wow, the second demonstration is very clever. I wouldn't have found it

[@iateyourgranny]

You can do it all at the same time: (6k +- 1)^2 = 36 k^2 +- 12k + 1 Then factor out the common stuff in the first two terms: = 12k(3k +- 1) + 1 Either k is even, or, if k is odd, then (3k +- 1) is even. In either case, 12k(3k +- 1) is a multiple of 24.

[@genewirchenko347]

I did about the same. A lot simpler than his four cases.

[@rogerwang21]

Just say “For primes 5 and greater”

[@DarthTaiter79]

I was thinking what could be added "if p^2 > 24, then...."

[@anand.suralkar]

U mean all the primes? 2 and 3 are subprimes. According to matt Parker

[@nathana2898]

Bruh mathematicians will pull some bogus like “this number has to either be equal to 1 or not equal to 1” and it somehow shows them the answer

[@52flyingbicycles]

Strange but true. Proof by cases can be very helpful. It’s also why most mathematicians do their best work while they are young and creative. The genius of many mathematicians comes from clever ways to rethink of problems in (relatively) simpler terms

[@wayneyadams]

I've been interested in and studies prime numbers since I was 14 years old, and next month I will be 74, so that's 60 years. I've found all sorts of interesting, quirky facts about them. They are some of the most fascinating numbers to study, because it seems like there should be no patterns and yet they are everywhere.

[@DaC10101]

Parker: Squaring Primes

[@mememem]

Also known as the Parker 24

[@tasin7777]

We should name everything he comes up with after him

[@Kolkritan]

I'd argue it's just another type of Parker square.

[@rewrose2838]

Actually the p^2= 24k-1 part works It's the 6k+1 and 6k-1 being equal to p part that's worthy of being called 'Le Parker 6'

[@vincentwilliamrodriguez3572]

parker 139

[@Fregmazors]

I had no idea the primes could be divided into categories like this! In my (admittedly limited) maths education I got the impression that the defining characteristic is being absolutely without patterns. This video, as well as another video where you actually directly state that primes do have patterns, have enlightened me! Thank you. :)

[@leong108]

Its not a generator, because not every (24k + 1 ) is prime. So its really not showing a pattern. Its created a pattern for possible primes, just the same as "not even" creates a pattern for possible primes. Now show a pattern to ALL the primes and ONLY the primes.

[@aashutoshmurthy]

I was writing a program to check if a number is prime or not and I used this mathematical concept over there. I just realized that though 2 and 3 don't fit into Matt's theory, but they can be applied to the concept in reverse manner, i.e, (2*2 -1) and (3*3-1) divide 24 perfectly. That helped in optimization of my solution.

[@dickballsour]

Does that mean 2 and 3 are Parker primes?

[@Archimedes115]

"2 and 3, I call them the subprimes" ~Matt Parker "Square"

[@kevinjones5001]

"elegant" ... as in ... "The friend's proof seems more elegant." ... might serve better, in context, than "easier".

[@profdaniel1787]

Searched the comments to find this one. Elegant was the word he was searching for.

[@One0ldGeek]

The first is brute force, the second is elegant

[@numbr6]

Elegant proofs when clearly explained are usually more understandable. The brute force approach is arguably a stronger demonstration of primes occur next to 6. The elegant version requires the explanation to follow.

[@viliml2763]

Want an even more elegant one? All primes are +-1 mod 3, which means all prime number squares are 1 mod 3. All primes are +-1 or +-3 mod 8 which means all prime number squares are 1 or 9 mod 8, and 9 is also 1 mod 8. Combine those two facts to get that all prime number squares are 1 mod 24.

[@pepesworld2995]

thing about this dude is that hes real genuine. hes really skilled in what he teaches - because he enjoys it. hes real. and i appreciate that

[@maninalift]

Ooh! This is two years old and I have no idea what's in it but I love square primes

[@222tarot3]

Hello everyone, for more codes number required send a message on my whatssap [phone]

[@TheDabol51]

Here's an algebraic simpler version: (6k +/- 1)^2=36k^2 +/- 12k + 1 Rearange to: 24k^2 + 12(k^2 +/- k) +1 = 24k^2 + 12(k(k +/- 1)) + 1 Now, either k or k+/-1 is even so we can write : 24k^2 + 24(k(k +/- 1)/2) + 1 = 24(k^2 + k(k +/- 1)/2) + 1 = 24N +1, where N must be an integer since both k^2 and k(k +/- 1)/2 are. QED

[@michalbreznicky7460]

I did something midway between yours and the one in the video: (6k +/- 1)^2=36k^2 +/- 12k + 1 = 12k(3k +- 1) +1. Since k(3k +-1) is divisible by 2 as either k or (3k +-1) must be, then 12k(3k +-1) must be divisible by 24.

[@louiswouters71]

There's a far easier method. The squares of 1 3 5 7 mod 8 are all 1. And the squares of 1 2 mod 3 are all 1. Combine the two and it must be one more than a multiple of 24.

[@Aaron-P]

2 & 3 aren't *real* primes?!! And I suppose hydrogen & helium aren't real elements? 😉

[@TheGeneralThings]

Only real elements are uranium and above.

[@MrMichiel1983]

Aaron P.. They are real primes, but different from all the others. There is no way a non prime number can be in between 1 and 2 or 1 and 3, so it's a bit obvious that 2 and 3 must be prime. 5 is the first prime that has a non prime between it and 1 (namely 4)

[@patrickgono6043]

No. See, hydrogen and helium are the only real elements. Everything heavier are just metals *astronomy intensifies*

[@Joe_Payne]

And gold isn't an element? As it's not a prime?

[@haniyasu8236]

They're Parker primes. They fail to square to one more than a multiple of 24, but at least they gave it a go.

[@gustavoexel5569]

Actually it is possible to prove that a multiple of 6 +- 1 has rest 1 in the division by 24. x = (6k+-1)^2 mod 24 x = 36k^2 +- 12k + 1 mod 24 x = 12k^2 +- 12k +1 mod 24 x = 12 * k*(k +- 1) + 1 mod 24 And since k*(k +- 1)=0 mod 2, because it is the product of two consecutive integers (and therefore must be even) x = 1 mod 24

[@deept3215]

Haha, yeah, that's basically what I did too and was wondering why he said it was too complicated... Started to think I did something wrong

[@rabbitpiet7182]

Gustavo Exel are you German?

[@user-tn2dk2pg2p]

@@deept3215 Lol, I proved it too and was confused how you could make a 13 minute video on the properties without realizing it was trivial.

[@Jooolse]

You missed a factor 3: x = 12*k*(3*k +/- 1) + 1 mod 24

[@EchosTackyTiki]

Whenever he was trying to compute 17^2 and was coming up with an easy way to do it, I immediately thought "that's gonna be 170 times 2, minus 3 lots of 17." I even paused the video and heard it in Matt's voice in my head. "170 times 2 is 340, 3 lots of 17, 51, 340 minus 51........ 289." You can hear it in his voice now, can't you?

[@TheFakeVIP]

I'm definitely a Matt Parker type of maths enthusiast. I love maths, and I really appreciate the beauty of that second proof, but I would've for sure gone down the route of the first proof if I was solving this. I wish I had the intuition to solve problems the way the second proof does, but I don't.

[@LetMeRetort]

2 and 3 work too. 2^2 is (24 * 1/8 + 1), and 3^2 is (24* 1/3 + 1). And since the multiplier is a fraction less than 1, I am with Matt on calling these two numbers as sub-prime.

[@rayscotchcoulton]

I'm sure someone somewhere said this (and I haven't finished watching the video, so maybe they'll cover it?) but 2^2 - 1 = 3, and 3^2 - 1 = 8 .... and 3 x 8 is 24 :)

[@Thomas-vn6cr]

Nice haircut.

[@fawadmirza.]

😂😂😂

[@eileenvilaca]

Almost balding, not quite... could call it a parker cut.

[@kgipe]

The ears could still use a trim

[@pleindespoir]

@@kgipe how would he look without ears ? ;)

[@kgipe]

Pleindespoir 🙉😂

[@laxrulz7]

I like the second proof better not because it's "easier" but because it also shows why 2 and 3 don't square to multiples of 24 which is nice

[@zackszekely6618]

Using the same method as the second (more creative) proof, it also turns out that if you take the square of a prime number and multiply it by that same square minus five, you'll always end up with four less than a multiple of 360. Example (using the prime number 7): 49 × 41 = 2156 = 2160 - 4, and 2160 = 360 × 6. The proof comes from multiplying the factors (p - 2) (p - 1) (p + 1) and (p + 2). You'd end up with a polynomial that looks like p^4 - 5p^2 + 4, which can be rewritten as p^2 (p^2 - 5) + 4. When you look at the four factors on a number line, in addition to having a multiple of 2, 3, and 4, the newly added (p - 2) and (p + 2) also guarantee a second multiple of 3 as well as a multiple of 5 (but only if you're using prime numbers higher than 5). Therefore, since 2 × 3 × 3 × 4 × 5 = 360, you can guarantee that multiplying all four factors will give you a multiple of 360.

[@user-tn2dk2pg2p]

This is just a really easy number theory problem. We just use the fact that all primes can be written as 6k+-1 excluding 2 and 3. This fact is simple: We could have 6k,6k+1,6k+2,6k+3,6k+4,6k+5. Unless the prime is 2 or 3, we must have P=6k+5 or 6k+1. 6k+5 is the same as a number of the form 6k-1. So we're just squaring 6k+-1. We just get P^2= 12(3k^2+-k)+1 3k^2-k is the same as (3k-1)k. Either k or 3k-1 will be even (If k is odd 3k-1 is even and if k is even k is even). Then 3k^2+-k is the same as 2n for some integer n. Plugging in gives us P^2=12(2n)+1=24n+1 with the exceptions of 2 and 3. This isn't special about primes- Any number of the form (6k+-1)^2 is one more than a multiple of 24.

[@bobingabout]

24 used to be my favorite number. Many of the reasons why it was my favorite number is basically the same reason why some people suggest Dozenal is a better number system than Decimal, it just divides nicely by a lot of single digit numbers.

[@willmungas8964]

What is your current favorite number?

[@bobingabout]

@@willmungas8964 Not sure I even have one any more. though I do like the powers of 2, like 16, 32 etc, and I do still like 24.

[@leonhardeuler9028]

Hey Matt, it's a way shorter to show that (6n+1)² or (6n-1)² are Multiples of 24 plus 1 For Example (6n+1)² = 36n²+12n+1 = 12( 3n²+n) +1 3n²+n is always a Even number because if n is uneven you have 3*uneven²+uneven which alswes ends up beeing even because uneven+uneven = even and if n is even you have 3*even²+even which is even, too Therefore there is always a k from the natural numbers such that 3n²+n = 2k With that you have 12( 3n²+n) +1 = 12*(2k)+1 = 24k+1 You can do the same with (6n-1)²

[@EnteiFire4]

I prefer factoring to 12n(3n±1) + 1. For 12n(3n±1) to be a multiple of 24, you need n or (3n±1) to be even. If n is even, we're done. If n is odd, then 3n is odd, and adding or subtracting 1 gives an even number, so (3n±1) is even.

[@richardfredlund3802]

@@EnteiFire4 you can also use the p=6 plus or minus 1 fact, and note that of p-1 and p+1 in the factorization p^2-1=(p-1)(p+1), one is going to be a multiple of 6 and the other a multiple of 6 plus or minus 2 and so is a multiple of 4.

[@Tim3.14]

@@richardfredlund3802 I like that! Although I think the pair is either a multiple of 6 and a multiple of 4, *or* a multiple of 12 and a multiple of 2. That still works, though.

[@Tim3.14]

To put it another way, the product of any two consecutive even numbers is a multiple of 8. So the square of any odd number is one more than a multiple of 8. And since all primes past 2 are odd, all you need is that one of those factors is a multiple of 3.

[@Tim3.14]

Or more concisely: If 2 doesn't divide p, 8 divides p^2-1. If 3 doesn't divide p, 3 divides p^2-1. So if neither 2 nor 3 divide p, then 24 divides p^2-1.

[@cameronbaydock5712]

Open question: I’m from Canada and when we talk about mathematics we shorten it to “math” not “maths” the way you do in UK, Aus, etc. Any reason why 4:28 said “Math-related items” vs “maths” despite Matt and Brady’s Aus backgrounds? Am I up too late again?

[@dannygjk]

IKR they are not being consistent.

[@Algebrodadio]

The most instructive thing about this video is Matt explaining the difference between doing a proof the "easy" way and doing it the "hard" way.

[@liviousgameplay1755]

Probably mentioned before, but I do like how 2^2-1=3 and 3^2-1=8, multiplying to form a familiar number.

[@JordanMetroidManiac]

My teacher actually had me and his other students prove this on a test. He expected us to use equivalence classes in mod 24. The proof follows these steps: 1) Partition the set of all integers by all of the equivalence classes in mod 24. 2) Consider the classes as the range of numbers from -11 to 12 (these numbers are actually equivalence classes, so they represent the set of all integers). 3) Cross out all of the multiples of two and all of the multiples of three. (We’re left with the equivalence classes -11, -7, -5, -1, 1, 5, 7, and 11, all still in mod 24). 4) Square each number and minus one. The new numbers are 0, 24, 48, and 120, which are all multiples of 24. Of course, this proof does not show that only primes have this property. It only shows that numbers which are not multiples of two or three have this property, and since all primes are not multiples of two or three, they have this property. So, there are definitely numbers that aren’t multiples of two or three but are not prime, just like Matt showed in the video (e.g. 25). Such numbers are those of which there are multiple prime factors and none of the prime factors are two or three. In the case of 25, its prime factorization is 5 and 5, so it is one of the numbers that is not a multiple of two or three and is not a prime number. But it is definitely true that prime numbers are not multiples of two or three, so they can be squared and end up being one more than a multiple of 24.

[@kristofferssondavid]

Why don't use mod30? Then you are left with 8 possible primes every 30 numbers. 30 +-(1,7,11,13) Just like in mod 24 bur you seive out more numbers.

[@tomaszjachimczak]

A simple proof can better be described as an elegant proof.

[@Thedeadbeatmatt]

I had a geometry professor in community college always say, "Matthew, make this proof more elegant." At the time I didn't know what he meant. It wasn't until my capstone math course that I finally got what he meant. No other professor ever said it. I have my bachelors in math now. I'm with you. When he said easier, I immediately thought, nah that's more elegant.

[@louisvictor3473]

@@Thedeadbeatmatt tbf, what is "easier" depends on where you're coming from at the moment. For me, the whole proposition seemed almost trivial and the p^2 - 1 approach sounded very similar to something I would try first. But that is because of something I have been working on that is actually very related to that, so of course I would try something more like it (that likely would quickly reduce to it itself).

[@firefist3684]

Every fourth power of a prime except for 2, 3, and 5 is one more than a multiple of 240.

[@unfetteredparacosmian]

Every sixth power of a prime except for 2, 3, or 7 is one more than a multiple of 504.

[@sergiokorochinsky49]

Let k be integer and p(n) be the n-th prime number, then: p(n>2)^2-1 = 1 x 24 x k p(n>3)^4-1 = 10 x 24 x k p(n>4)^6-1 = 21 x 24 x k p(n>3)^8-1 = 20 x 24 x k p(n>5)^10-1 = 11 x 24 x k p(n>3)^12-1 = 2730 x 24 x k p(n>2)^14-1 = 1 x 24 x k p(n>7)^16-1 = 680 x 24 x k p(n>8)^18-1 = 1197 x 24 x k p(n>5)^20-1 = 550 x 24 x k p(n>9)^22-1 = 23 x 24 x k p(n>6)^24-1 = 5460 x 24 x k ... As usual, the 24th power is a show off...

[@customarylover3857]

@@unfetteredparacosmian Mind=blown 5^6=15625=31*504+1 11^6=1771561=3515*504+1 13^6=4826809=9577*504+1

[@asheep7797]

Every zeroth power of a prime is one more than a multiple of 8,200,601.

[@cjhuffpuff]

One thing I found interesting about 3 is that even tho you don’t get one more than a multiple of 24, you still get one more than a factor of 24, kinda cool

[@liviousgameplay1755]

Yeah, I noticed that too! :D. 2 makes 3, 3 makes 8, 3*8= 24. It kinda looks more beautiful to me that they are all related to 2 and 3.

[@anon6514]

You can do it from the 6k+1 and 6k-1 cases. Squaring 6k+1 gives 36kk + 12k + 1, which is 24(1.5kk + 0.5k) + 1. Squaring 6k-1 gives 36kk - 12k + 1, which is 24(1.5kk - 0.5k) + 1. If k is odd then k squared is odd, if k is even, then k squared is even - therefore the bit in brackets is an integer. QED.

[@AkshayKumar-be7jl]

Hello

[@johnfmartin2576]

Hi Matt-- Thank you for this interesting episode. I really dig your presentation

[@blazingfire7517]

I did 17 squares in my head and got it right first try. I’m proud of myself.

[@trejkaz]

Sounds easy enough. Just do it as (16+1)².

[@anonnymouse3058]

I am the 17th like of this comment. I am proud of myself.

[@will4not]

This is some Grade-A prime content. I love prime facts.

[@GetMeThere1]

I was just amazed to find that the square of ANY number is 1 more than the product of the adjacent numbers.

[@mikeguitar9769]

(n+1)(n-1) = n^2 - 1 n^2 = (n+1)(n-1) + 1

[@GetMeThere1]

@@mikeguitar9769 Right. But when I think of "squares" in the context of variables my mind goes into "algebra mode," and away from straight "number" relationships, lol. My math education beyond high school was only a year of college calculus.

[@mikeguitar9769]

Consider graphing n on the x axis and n on the y axis. The area of this square-shape is n^2. Then graph n+1 on the x axis. We’ve added n units of area. Then graph n-1 on the y axis. We’ve subtracted n+1 units of area. So, The area of the rectangle is the same as the square minus the 1x1 corner piece. Algebraicly : n n = n^2 ( n + 1) n = n^2 + n ( n + 1) (n - 1) = ( n^2 + n ) - ( n + 1 ) (n+1)(n-1) = n^2 - 1

[@isaaczackary364]

it makes me so happy that he tried doing the equations by hand before using the calculator

[@soyitiel]

4:06 wow

[@camelopardalis84]

Guy's secretly a cat.

[@KpxUrz5745]

Very interesting. I already knew that 17^2 is 289 because, well, I like numbers, especially primes, and just happened to know that. Incidentally, genius savant Daniel Tammet called 289 an "ugly" number (in his incredible synesthetic mind), but I find the number 289 quite lovely.

[@billborrowed3939]

Still pretty sure, that delivery actually was a new role of wrapping paper to write on and a bunch of sharpies.

[@paulvanderveen4309]

If you square any prime (greater than 5), then square it again, you get 1 more than a multiple of 240!

[@haal0361]

My approach (which is very close to the (p+1)*(p-1) explanation: 1) Every prime number can be either expressed by 3a +1 or by 3a + 2. (a is an integer) (3a+1)² = 9a² + 6a + 1 -> (3a+1)² - 1 can be divided by 3 (3a+2)² = 9a² + 12a + 4 = 9a² + 12a + 3 +1 -> (3a+2)² - 1 can be divided by 3 2) Every prime number can be expressed as 2b+1 (2b+1)² = 4b² + 4b + 1 -> if b=2c (i.e. b is even), then (2b+1)² = 16c² + 8c + 1 -> (2b+1)²-1 can be divided by 8 if b is even -> if b=2c+1 (i.e. b is odd), then (2b+1)² = (4c+3)² = 16c² + 24c + 9 = 16c² + 24 + 8 + 1 -> (2b+1)²-1 can be divided by 8 if b is odd. -> p²-1 can be divided by 8 and by 3 and therfore by 24...

[@johnchessant3012]

I found a remarkable pattern in the primes: Every prime number is a number!

[@An_Amazing_Login5036]

John Chessant Incredible! How did you stumble across this remarkable fact?!

[@YellowBunny]

Every prime number minus one is still a number.

[@Frandahab]

Oh boy! Goldbach conjecture here I come

[@Grozdor]

Bold statement right there.

[@sacredbolero]

Fields medal incoming!

[@MrBoubource]

0:50 parkerSquare(17) = 149. Should we create a new OEIS sequence to collect all the parker squared Matt has discovered over the years?

[@isavenewspapers8890]

Parker wrote 139, not 149.

[@jimbig3997]

I think the "slight of hand" is in calling the subject primes when ANY number not a factor of 2 or 3 will fit that pattern.

[@TheKirbs01]

I have been doing something similar as a easy trick to multiply squared numbers in my head. The difference of squares thing can be generalized. So, a^2, can be modified to a^2 - s^2 and it be changed to (a-s)(a+s). To solve for a^2, just add the s^2 back on to the answer. So, 19^2, can be rewritten as (19-1)(19+1) + 1 (or 361). 22^2 can be rewritten as (25)(19) + 9 or 484.

[@WindowsXP_YT]

The powers of prime numbers and the numbers that can only be divided by the different prime numbers are very common. For example: 2^x, 6^x, 30^x, 210^x, etc.

[@bonecanoe86]

Since a multiple of two is called even, a multiple of 3 should be called threeven.

[@hackermub2598]

Then every(1) multiple of 0 should be called oven... 🤔🤔🤔

[@Usage_of_Sawdust_In_South_Caro]

@@hackermub2598 no! Even and threeven are ideals, oven is just .... nothing

[@pdxfrog]

Oven should be multiples of one, then oven is everything.

[@serglian8558]

Even Trine Quadrate Quine Sextate Septate Octate

[@NoActuallyGo-KCUF-Yourself]

Since I speak English, I call multiples of 2, "doubles" and multiples of three, "triples."

[@thegabrielchannel818]

6:16 that looked like e (euler’s number) at first.

[@NotJaydenix]

0:17 i was told in school a bit ago that the were no patterns to prime numbers

[@frabol02]

they lied to you

[@nathanisbored]

they probably meant that theres no explicit formula to generate the sequence of primes (except something something mill's constant)

[@NotJaydenix]

@@nathanisbored yeah i think thats what they meant

[@avananana]

TECHNICALLY there is not any pattern. But when we talk about patterns, we mean patterns like the Fibonacci sequence, 1, 1, 2, 3, 5, 8... and so on. Primes doesn't have an algebraic formula to follow, but there are ways to figure out the nth prime number in other ways. It really comes down to how you define a pattern though, which is a problem since everyone has unique definitions for just that.

[@dhay3982]

You can set square root (24*n-1), n positive integer and create a prime sequence, removing irational numbers. It's a pattern, isn't it?

[@Darwinion]

"so the primes are always above and below all the multiples of 6... except it doesn't always work" LOL

[@RodelIturalde]

It is a "parker always". Sometimes it is always, sometimes it isn't.

[@numcrun]

All primes ARE 1 above or below a multiple of 6. But not all numbers 1 above or below a multiple of 6 are prime.

[@iain9821]

This is unrelated to squaring primes. I think it can be shown that all the composites of form 6n±1 can be revealed to be of the form 6n±n. 6n±n can be factored as (6a±1)(6b±1). Thus, these composites can be eliminated from consideration as possible primes. For instance, 25, which is 6(4)+1, can be eliminated by considering that it is also equal to 6(5)-5. 35, or 6(6)-1, can be eliminated by considering that it is equal to 6(5)+5, or, similarly, that it is equal to 6(7)-7. Values of n (multiplied by 6) must also be considered which are multiples of 2, 3, 4, 5, etc. For instance, where n=2[6(2)-1]: 6×2[6(2)-1]-[6(2)-1]=121, which is 11², is eliminated. Similarly, 6×2[6(2)-1]+[6(2)-1]=143, another number that is not otherwise eliminated if we consider only multiple of 6 that fit the the simple form 6n±1. Another example is the composite 187. This number is equal to 6×3[6(2)-1]-[6(2)-1]. It is interesting that this formula can be factored as [6(2)-1](6×3-1), which reveals the two prime factors of 187 right of the bat! Similarly, the square 169 can be eliminated by observing that 6×2[6(2)+1]+[6(2)+1], or [6(2)+1](6×2+1), which is 13×13. To investigate this, it helps to make a chart to count by six, starting with 0-5 in the first column. Based on these considerations, it seems that an efficient algorithm could be generated to sift for primes by automatic elimination of composites of the form 6n±n, where n is itself of the form 6n±1. As shown, the algorithm would also automatically factor at least some composites of the form 6n±1.

[@rohansharma1250]

Parker squaring prime numbers

[@Magnus_Deus]

Random guy: DO 17 SQUARED Parker: mk p0rk3r Brain: SYNTACK EARRIR SOLF DISTRUKT EIN FIV FUR THRE Random guy: Um BRON: OEN *911 happens*