What is the biggest tangent of a prime?

[@GameCyborgCh]

this whole video is just going on a tangent

[@_mels_]

I literally screamed. Well played.

[@heyandy889]

gottem

[@nickcarter4006]

I’ll co-sign this

[@RubyPiec]

Oh I think the tan function threw your comment to the near top

[@HagenvonEitzen]

nobody primed me for such witty comments

[@AkimKumok]

These numbers are indeed quite rare! Here is another one :) 230835870782558831561617186504559084198719501221763995608082253627620752053749345488376393822837250198036536001853828659466202612019525543362322174085744303421231446484541625047630462908919109308644634605051209877750956648014568322183373423523622941806761765245932401727973436579786298208782013178059220103271409347616696556052706562092799953175234183483071403726145726928572372071037042523626350312132351311366806233135093893271182587352730075523143635168510803804031460442796778933680674070124730971307185688425634077096234482442639666385695677866015904370207368846631450100939158029908242779848800640038255592227473300237596577845602369215568916732445980431078426390412264603773550384039765410088966381694110344811198325354315338629604946794192217817288101344643511450133142277670683067655250506551517767422160650566385017503208608678491109517443585115317845289832567015746473548492179557935154400719019569904865219030736244089287736334048402066257337090606092966121806567484954460809024219605952851728610326005069 Here is how this one was found: consider continued fraction approximations of Pi (3, 22/7, 333,106, ...) - series of a_n/b_n and look for the approximation where 1. a_n is even and is equal to 2 * p (where p is prime number; this is the number we are looking for); 2. n is even, so that tg(p) is very large positive number (as opposed to very large negative number). Then p satisfies these properties, reason being is that since a_n/b_n approximates Pi really well*, Pi/2 + Pi*n ~= p where n=(b_n-1)/2. *something something math

[@dmitrykrachun179]

Yeah, arguing along this lines and assuming pi has a continued fraction typical for a random number (or at least not too atypical) we see that the sequence of numbers n such that tan(n)>n grows exponentially and then standard probabilistic heuristics suggests that infinitely many of them are prime

[@rebeccaguarino3666]

Mines 6064949221531200

[@arnerob123]

Really cool! I calculated with this number p = 2.30835... * 10^1016 that tan(p) = 6.97387... * 10^1016, more than 3 times as large, but at the same time only 3 times as large. round(tan(p)) = 697387082468190756312896165657475777401722593807004041244417834044684513689353455969238234883329761276613035399154477164822740843775528672871686016906222348302918485696098278741889053565204547955838146993087426491097157173961334426093384280949249681534198827042641520481222606152417323294985819270935244525832596163943328138490884919729052743501300053209067668233228630703020638067994990803394563901660053860962206989758669406572172370025308289983032062451873257306200583906818459413663276259366117272350390515954244080244209624618548133624761980677252925810129236569449670092960717222817797906696185625949502357217561339308257734934133471331689607278070413968688395155572704153048270522415656862906092677241038009222445126506478904006196064330923777299991050061823583312810636162694420057024709721553267613003185652590938730343222126271388322411857815350911672819345406458309559422095703341858197439344141618854543580925622466480328055550310788028217273823243659051635438015619858609951634766195403415287151052947183

[@andrej2375]

1 would've worked if it was prime 😂😬😂

[@joeymurphy2464]

Everyone should be very careful when taking the tangent of very large numbers. Usually computers do this by subtracting pi many times until in the range of -pi to pi, and then taking tan of the result (relying on periodicity). However, if you're just using pi as a 32 bit float, you may not have enough digits to accurately find tan, after shifting by pi (but not exactly pi) over and over. Your number may land in the wrong spot on the very very steep tan function.

[@audiomystic]

Wow this is an excellent point. Thanks you!

[@Djake3tooth]

Well, you could do it with the Taylor series expansion of tan but that’ll take a LOOOOOOOOOOOOT longer...

[@poissonsumac7922]

@@Djake3tooth But does the Taylor series converge everywhere?

[@Djake3tooth]

@@poissonsumac7922 googled it and, i didn't know that, no it only converges for x=-pi/2 to x=pi/2

[@poissonsumac7922]

@@Djake3tooth I figured as much. If it did though that'd be epic.

[@amyshaw893]

whoever wrote in like that is very wrong, the tan bit is hilarious

[@HonkeyKongLive]

That whole bit was absolutely gold.

[@deecewan]

I laughed so much. The paper flew away! 😂😂😂😂

[@glenecollins]

I don’t know if I would mess with someone who can heckle via paper note as a video is being made.

[@KittenRaee]

I wholeheartedly agree with the statement 4:57

[@matthewfuerst6456]

Matt always has the funniest, stupidest bits in his videos

[@lucasdomingue9319]

3:32 "technically interesting" is the new thing I'll use to annoy all my non-maths friends from now on. This is pure gold.

[@Anklejbiter]

Technically interesting, the best kind of interesting!

[@DirkDjently]

Lucas Domingue iirc, he has a video several years ago with numberphile where he uses “appears on the OEIS” as a baseline requirement for a number or set of numbers to be designated “interesting” , or interesting enough to warrant a video. I think this was a callback to that

[@sebastianjost]

@@DirkDjently he probably hasn't changed his opinion about that. Can't blame him much.

[@jerberus5563]

Yeah, he seemed quite unenthusiastic this time around.

[@bengtbengt3850]

Yes, that's perfect. Usually math people (I might more or less be speaking of myself) hate the question "okay but why should anyone care?". Now we have the perfect answer.

[@CGreyEminence]

My girlfriend: "that bit with the tangent line is cringy" Matt: *reads an email that is flung away by the tangent line* girlfriend: *can't stop laughing* :)

[@orvinal2883]

its at 69 likes sorry i cant

[@efulmer8675]

Me: When will Matt get speared by the tan function?

[@bryanchandler3486]

@@efulmer8675 kinky

[@alexandermcclure6185]

@@bryanchandler3486 i'll tangent YOU. HAHAHAHAHA!!!

[@hallfiry]

There's actually a way to generate those hits efficiently: 1. Get a rational approximation for pi = a/b, where a is even. 2. Your magic number x with tan(x) = large is calculated as: x = a*(k+1/2). 3. Cranking up k will make the approximation worse and worse, so at some point you'll have to find the next rational approximation for pi to generate more numbers. 4. Regarding primality: Since a has to be even, multiplying by (k+1/2) will always result in a composite number for k>0. So for each family of magic numbers, only the very first can be prime. Or in other words: only those apprimations of pi where the numerator is 2*prime will give us primes in this process.

[@patrickwienhoft7987]

Similarly, with x = a*k you can find really good approximations for tan(x) = 0, or alternatively create a sister series for cot(x) > x. What I find weird is that, because we no longer have the restriction for a to be even, there are roughly twice as many approximations of pi that yield a solution to cot(x) > x as there are approximations that yield tan(x) > x which is pretty counterintuitive given that cot and tan have equally spaced asymptotes.

[@Jigkuro]

@@patrickwienhoft7987 that IS interestingly counterintuitive.

[@frankharr9466]

I'd like to see a video on this.

[@HagenvonEitzen]

@@Jigkuro and thereby technically interesting

[@lowthg]

There’s not necessarily more solutions to cot x > x. Note, since we use a rather than a/2, which would roughly halve the number of solutions in a given bounded range, cancelling out the doubling effect you noticed

[@wbfaulk]

Okay, this should clearly be "favorite mega-number", not "mega-favorite number". It's the number that's mega, not the favorite-ness.

[@ericklucasmenezesdelima570]

Man, you must be fun at parties

[@SharienGaming]

hmmm yes quite right my old chap - this is maths after all, we cant just move modifiers about willy nilly

[@Ice_Karma]

Clearly, in Matt's head, adjectives are commutative. 😹

[@ivanjones6957]

mega-favourite mega-number?

[@SharienGaming]

@@ivanjones6957 that can just be simplified to mega(-favorite -number) XD

[@jameswkirk]

17 as a favorite small number sparked a memory… A math I knew once did a standard equation derivation and told the class “now plug in a value for X and see what you get…” One student asked “which value?” to which the teacher replied… “Well, zero times anything is zero so that’s out. One times anything doesn’t change so forget that. Two is the only even prime, three is the smallest odd prime, four is two squared. Five is the only prime that ends in 5, six is a perfect number, seven is too lucky. Eight is a perfect cube, nine is a perfect square, any number that ends in zero is out. Eleven is too close to ten, twelve is divisible by too many things, thirteen is unlucky, fourteen is twice as lucky as seven, fifteen, well, any number that ends ends in 5 or 0 is probably too special. Sixteen is a perfect square and its square root is a perfect square. Eighteen is like twelve, it’s just too divisible. By the time you get to nineteen and raise it to a power things just get too big, so seventeen. Plug in seventeen.”

[@nanigopalsaha2408]

Wow, you know a math! (Amazing story btw)

[@gavinswift4329]

Sounds like you had a great teacher.

[@bokocop5725]

couldn't come up with anything for 11 huh

[@antwerp7970]

"It's the square root of a perfect square" I'm definitely gonna use that to justify why any arbitrary number is cool lmao

[@Zaros262]

@@antwerp7970 They said 16's square root is a perfect square. That is, 4 is also a perfect square Although your version is also funny

[@tommysimpson6662]

Absolutely love the bit about graphing tan.

[@flymypg]

My first mega-number was a simple one: 10^100 Why? Because that was the largest number that could be displayed by my first scientific calculator, back in 1974 (well, actually 9.999999999e99, but let's not quibble). And I also enjoyed the simple fact that 10^100 had a name: A googol. A name that, two decades later, would be misspelled when naming what has become one of the largest tech companies in the world. The real question I asked of my calculator was this: What was the simplest non-trivial calculation that could cause the calculator to generate that value? The calculator had an exponentiation function (^), which lent itself to a subsequent question: What value raised to itself would max out my calculator and equal 10^100? That is, for which value x does x^x=10^100? I needed a place to start: Clearly, x had to be be greater than 10 and less than 100, so I started at 50, which turned out to be a surprisingly good place to start, as the first digit was correct! Finding all the other digits was a long and boring iterative process that required close focus to correctly append a digit to the prior closest value that didn't overflow. And I still remember each digit of that number to this day, 46 years later: 56.96124842 So 56.96124842 is my "favorite mega-generating-number".

[@Alex_Meadows]

Very nice! It's funny how these things stick in the mind - thirty-odd years ago I had to crank out square roots by hand, and I still remember the roots of 2 and 3 to nine decimal places. If I had a pound for every time that's come in useful during my lifetime, I'd have exactly as much money as I have today. My own favourite meganumber is also related to the googol. Specifically, it is one googolplex and one, or 10^(10^100) + 1. Unlike you, I don't have an interesting mathematical background to my choice - in my case it's pure whimsy. I think that a googolplex is such a stupidly large number that it amuses me to imagine a scenario in which one googolplex of something is just not quite sufficient for my purposes, and I need to add one to get the required amount of whatever it is.

[@NoNameAtAll2]

@Firstname Lastname what's productlog? are you talking about W?

[@deejayaech4519]

@@NoNameAtAll2 productlog is a special function that is the inverse of xe^x

[@benjaminshatokhin4725]

@@NoNameAtAll2 yes, w is product log

[@michaelh.mertens7915]

Using continued fractions as suggested by Moritz Ernst Jacob one can find a bigger prime p with p

[@genessab]

But tan(11) is -226, not 226. Signs matter

[@EvanBartholomew]

Genessa He’s talking about absolute value, so signs do not matter.

[@bur2000]

1:40 this number is relatively small for primality testing. They probably use the Adleman-Pomerance-Rumely primality test, which is a deterministic version of the Miller test. Which was the topic of a recent Numberphile video (re witness numbers). Primality testing of general integers only begins to take a significant amount of time once you get close to 1000 digits or more and have to use ECPP test.

[@oscarsmith3942]

Also worth noting that Miller-Rabin can be used to fairly quickly get very confident that the number you're about to test is actually prime.

[@fibbooo1123]

Alright but that bit where the paper flew out of your hand made me laugh so much. It was a good bit

[@bsheaves]

I’m a chemist, therefore my favorite mega number is Avogadro’s number

[@markrichards9646]

My favorite number is 5,318,008. Put it in a calculator and turn the display upside-down.

[@tobybartels8426]

You must have been excited last year when they finally decided exactly what number it is!

[@jamesjennings3312]

I'm not sure that Avogadro's Number is an integer. It's the number of carbon-12 atoms that weigh 12 grams, and that could very easily require a fractional atom. Or so I would think.

[@suokkos]

@@jamesjennings3312 , It has been well defined integer since 2019 when the definition changed.

[@tobybartels8426]

@@jamesjennings3312 : To add to what @Pauli said, they defined a mole to be precisely 6.02214076E23 things, with the consequence that a mole of carbon-12 atoms doesn't weigh precisely 12 grams anymore (although it does to the precision that we can measure for now). They also redefined the gram at the same time, and at one point they were thinking of defining the gram to be exactly 1/12 of the mass of mole of carbon-12 atoms, which would have made 12 grams of carbon-12 contain an exact integer number of atoms, by definition (of the gram). Sadly, they abandoned that approach and redefined the gram in terms of Planck's constant instead. However, an atomic mass unit is still defined to be 1/12 of the mass of a single carbon-12 atom, so you can now say that the mass of a mole of carbon-12 is exact integer number of atomic mass units, by definition.

[@haniyasu8236]

1:30 Idk precisely how Wolfram Alpha does it, but the Miller-Rabin primality test is a commonly used algorithm that can (probably) compute the primality of an integer in Õ(log(n)^4) time. For reference, I coded up a single-threaded implementation of this test in Rust at one point, and for actual bonafide 64-bit primes on my mediocre computer, the time taken to complete the test wasn't even noticeable, so even though this particular number is about twice the digits (and thus 16x the time), considering they probably have much much better computers *and* use parallelism, Wolfram-Alpha should have no problem figuring out its primality . Also, for those wondering, _technically_ Miller-Rabin is not yet _guaranteed_ to be polynomial in the number of digits as the proof of it depends on the generalized Riemann Hypothesis. However, for sufficiently "small" integers, this doesn't matter as all primes up to a certain point have been proven to work. Additionally, despite this, primality testing actually _has_ been proven to reside in P and can be solved for sure in Õ(log(n)^6) using the AKS primality test.

[@ben1996123]

using miller rabin to test primality only takes a fraction of a second for numbers with well over a thousand digits

[@uncirtyne]

Or it just....checks it against the list of known primes.

[@haniyasu8236]

@@uncirtyne but.... there's literally around 10^43 primes below the number in the vid... The total amount of storage on the entire planet isn't even remotely close to enough to store all of them..... Sure, there's probably caching, but like... there's probably caching for _any_ query it receives enough copies of.

[@JohnDlugosz]

@@uncirtyne Considering the time difference between accessing memory in Level 0 cache vs main RAM, certainly vs seeking and reading from a file on SSD, I think the computation to test would be *much* faster. Actually, given a _big enough_ list the look-up, even from a slow HDD, would win eventually, as it's O(log(n)). It's just that the CPU is so fast, complete with multiple cores and vector arithmetic registers to perform multiple iterations at once, that what we learned about algorithms in school is not correct anymore; e.g. linked lists and trees are _slow_ and a C++ std::vector still wins even with tens of thousands of elements.

[@MatthijsvanDuin]

A nice overview of primality/compositeness testing algorithms with various trade-offs: cr.yp.to/primetests.html

[@hiimemily]

Big fan of 2³¹-1 personally. Maxed out my score in a game once, can't remember which, and was mystified when it wouldn't go past 2,147,483,647. As an added bonus, it's a Mersenne prime!

[@SunroseStudios]

nice! also hi lol

[@hiimemily]

@@SunroseStudios oh hey!

[@okidclol3633]

It’s because the maximum 32 bit number (the default number of bits for an integer on x86 architectures) is 2^32-1! it’s pretty neat

[@YoshiAsk]

@@okidclol3633 To be clear, the maximum number you can store with 32-bits is 2³²-1 for unsigned (only positive) integers. 2³¹-1 is the maximum for 32-bit signed (both positive and negative) integers, because the first bit is used to encode the sign of the integer.

[@RichConnerGMN]

nice pfp. i think i've seen you before at some point. hi emily

[@franciskatende1566]

That guy really created a sequence for tan(p)>p. Must be a hobby

[@cadekachelmeier7251]

My hobby is searching for positive integers where sin(x)>x. I still haven't found any, but maybe someday.

[@XPimKossibleX]

@@cadekachelmeier7251 pro-tip: try the negatives

[@anaru3416]

@@cadekachelmeier7251 |sin(i)|>|i|, good enough?

[@arpitdas4263]

@@cadekachelmeier7251 😶

[@danielhua8964]

@@cadekachelmeier7251 Good luck !

[@davidhumphreys3028]

#MegaFavNumbers 1,000,017 The smallest technically uninteresting MegaFavNumber.

[@frederf3227]

That makes it interesting! But then the next highest will be... new theorm, all integers are intergesting.

[@SimonBuchanNz]

@@frederf3227 when all numbers are interesting, no numbers are interesting.

[@kostyapesterew1068]

@@SimonBuchanNz NO CAPES!

[@matteovasta2326]

prove it

[@davidhumphreys3028]

@@matteovasta2326 Brute force search of OEIS starting at 1,000,001.

[@Erin-ks4jp]

I have been well and truly nerd-sniped, and have yet to find another prime with p < tan(p) - Will update if I get one.

[@althaz]

I've checked a few hundred million numbers. Haven't found one yet

[@frederf3227]

Very good, impatiently awaiting a second.

[@5hape5hift3r]

What about tan(n) > n^2 Or tan(n) > exp(n)

[@spycnub5737]

@@5hape5hift3r To answer the question tan(n) > n^2: (n is in radians) I checked for all positive integers with 8 digits or less. The only solution (I believe) is n = 1, with tan(1) = 1.557 and 1^2 = 1. But if you take |tan(n)| < n^2, then n = 11 is another solution. I will run my program again for bigger numbers, but at night.

[@standupmaths]

No spoilers but I can confirm that another one exists and so far two people have found it. Good luck!

[@DukeBG]

1:34 Matt, it looks like you have no idea what size of numbers are quick to prove prime and what are not. Here's an infodump from me who does primality-proves A LOT and has it on my fingertips: Anything below 300 decimal digits is less than a second on modern hardware. Your prime in question is measly 46 digits. You can fully factor a number this size trivially.

[@ngc-fo5te]

I don't think you know what prove means.

[@DukeBG]

@@ngc-fo5te ?

[@SpartaSpartan117]

Matt I'm already busy with the land area video!

[@algc19]

The primeness would be quite interesting if we were studying tan(n-π/2) > n. That way, we would get n ≈ kπ and so π ≈ n/k, a new approximation of pi with an irreducible fraction and to a ridiculous accuracy.

[@landsgevaer]

That is cotan(n)>n actually (looks more elegant that way).

[@algc19]

@@landsgevaer Wouldn't it be cotan(n)< -n ?

[@landsgevaer]

@@algc19 ah yes, true, not identical, I stand corrected. But my version also finds great approximations for pi though, although bounded from the other side. So maybe |cotan(n)| > n is the more productive one, depending on how you value 'elegance'...

[@nujuat]

Engineers: "wait, doesnt tan(n) = n?"

[@lakejizzio7777]

As an engineer I have another question: If tangent function is periodical and tan(89)=57,29ish then how can tangent of super big INTEGERS bigger than that? Shouldn't you put in rationals to get bigger results? I am utterly confused.

[@undergroundmonorail]

@@lakejizzio7777 it's because the period of tan is isn't a whole number, so it's not in sync with the integers. you're trying to find an integer that's really close to 0.5 + a multiple of pi, and that extra .14 or so will cause it to drift in and out of sync with the integers, like two car's turn signals

[@jhgvvetyjj6589]

@@lakejizzio7777 tan in integer radians, not degrees

[@jacks5kids]

It's a very good approximation for small n. For example, tan(0.000000123456) = 0.0000001234560000000006272...

[@anti-loquax2758]

Matt: calls people who stayed until the end the hardcore end of the video gang Me who just stayed for the music: yeah

[@youtubersingingmoments4402]

6:19 I never thought I'd find a Shania Twain reference to any channel I'm subscribed to, let alone Matt Parker. What could have possibly driven you to do such a thing?

[@carbonwolf3865]

I took a double take for it

[@JayTheYggdrasil]

Ok that Tan joke was interesting. I'm not entirely sure how it made me laugh but it did. Well done.

[@Joe_Payne]

I saw it coming but I still loved it

[@Shadow81989]

I like the connection of the last mail to your previous video: You were asked to do more things involving trig, and immediately travelled back in time to do a video connected to trig points. That's what I call dedication!

[@dhoyt902]

2^19937 - 1 is my favorite bigger number and is a classic Mersenne prime and 19937 is a prime no matter how cycled, woot.

[@pedronunes3063]

99731 isn't prime.

[@tsawy6]

@@pedronunes3063 Maybe they mean no matter how cycled instead?

[@pedronunes3063]

@@tsawy6 Probably you are right, I've tested them, if do cycle, they are always prime.

[@dhoyt902]

@@pedronunes3063 wrote it not paying attention, meant cycled.

[@pedronunes3063]

@@dhoyt902 Fair enough, it's still a cool fact.

[@HellbladesFFXI]

Honestly, I loved the fact that the 5th line for tan() on the chart showed up while he was explaining tan() while he 'wasn't looking'. Very subtle continuation of the bit lol

[@OwstinGreen]

Was fully expecting our boy tangent to make a surprise appearance at the end :(

[@TonyHammitt]

Yes, revenge of tangent flinging Matt out of frame would have been very funny

[@AlexPinkney]

378163771 because it's the only number that's illegible when you put it in a calculator

[@HagenvonEitzen]

That explanation beats that for 5318008

[@mephystovals]

5:00 "Hey, there it goes!" I didn't see that one coming. XD

[@ecsciguy79]

0118999881999119725.3 That's my favorite. Made me realize that if you sing it to a tune, you can memorize anything.

[@realityChemist]

Aha, it has arrived! Been looking forward to Matt's #MegaFavNumber all day!

[@tmfmxo]

Hi Matt, please keep going with these visual representations. They are not only hilarious but also immensely helpful. Thanks.

[@vycanismelodis]

i continually enjoy the fact that my namesake function is so chaotic

[@Scruffiannat]

I find periodically reading the comments is likewise titillating.

[@justinyoung6342]

That email joke alone earned a like on this video. Thank you for making me laugh while learning a cool bit of maths.

[@PeregrineBF]

"I have no idea how Wolfram Alpha is able to check that a number that big is prime that quickly" The AKS test takes polynomial time (see the paper "Primes is in P"). There are faster probabilistic algorithms, as well.

[@jemmerl]

Hello Matt, I'm enjoying the video! Just wanted to let you know the tan function is very funny and never got old. Cheers, A viewer of a three year old video.

[@1000dots]

My mega-favourite number is a googolplex. When I was in primary school I was one of the only students who didn't go to scripture. After scripture at lunch the kids would bully me and ask stuff like how did the world start if there wasn't a god. I asked my dad about stuff like that and from a fairly early age through necessity gained a basic understanding of science. My dad also taught me the number googolplex. I told the other kids about it and they laughed at me and said it didn't even sound like a real number and I must have made it up. But then in the late 80s Back to the Future 3 came out and had the scene where Doc was saying Clara was "one in a million… one in a billion… one in a googolplex." I love Doc Brown like I love my dad. I gained a lot of credibility when that movie came out :) While everyone else was in scripture I would be in the corridor outside, being 'punished' with doing extra maths.

[@MABfan11]

Doc Brown flexing on your classmates :)

[@Mystery_Biscuits]

Returning to this video, was the bit at the very end a reference to what would become Love Triangle?

[@KittenRaee]

4:57 truer words have never been spoken… …even though tan function is discontinuous

[@looijmansje]

On its domain it's continuous

[@potatoonastick2239]

@@looijmansje isn't everything?

[@rewrose2838]

@@potatoonastick2239 that's the point though

[@vojtechstrnad1]

@@potatoonastick2239 It's not. For example the function that assigns 1 to rationals and 0 to irrationals is discontinuous everywhere, even though its domain is R. I actually think you can prove that "almost all" functions are not continuous at any point.

[@skarrambo1]

6:19 A Shania Twain reference - Matt, you've really outdone yourself

[@108108qwerty]

Favorite number has to be 108109, which is a prime made of consecutive numbers 108 and 109, which completely oppose each other in primality. 108 has factors 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108, while 109 is a prime. Also, if you rotate it upside down you get the number 601801, which is also prime!

[@hstochla]

That is really cool

[@unitrader403]

it is less than a million, but i will give it a pass anyway :D

[@witerabid]

I legit almost fell of my chair from laughing. 😂 I didn't think your entertainment skills could improve any further - I'm glad I was wrong. My mega favorite number is 1000001 btw. It's very interesting, because a) it is also the binary representation of 65, b) it is the same backwards and forwards, and c) by your definition of mega favorite numbers it is the smallest mega favorite number possible. Unfortunately, its tan is less than 1... I guess, I'll keep looking for somenumber more interesting, but for now at least I have one. 😅

[@ashtonsmith1730]

isnt 1÷tan(x)=cot(x)

[@witerabid]

@@ashtonsmith1730 Yes... But even cot(1000001) is not very impressive... It's under 1.5 😅

[@rom65536]

8,675,309 is my mega favorite - couple reasons: A) it's prime B) it gives me something to hold on to C) you can turn to it for the price of a dime D) I got this number from the wall.

[@green0563]

Wait, what? I have no idea what those mean.

[@loganrussell48]

@@green0563 read the digits from left to right :)

[@sergeboisse]

@@loganrussell48 I still don't get it. Maybe because I'm french ☺

[@loganrussell48]

@@sergeboisse it's from a song - an American phone number without an area code +1(xxx)-867-5309

[@matthewziemba7526]

I absolutely loved the tan bit! Also, the doesn't impress me much joke was great!

[@Psychomaniac14]

My mega favourite number is that one prime that's like 500 9s but one of them is an 8

[@sudheerthunga2155]

Glitch

[@Psychomaniac14]

@@sudheerthunga2155 yes that one

[@wild-radio7373]

You are my hero right now. Thank you for the memories 🤜🏻👍🤛🏻♡♡♡

[@javspirata]

That numer blow mi mind

[@tobybartels8426]

99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999989999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

[@boiledham]

Fantastic video and a great community playlist. Love seeing the maths KZbinrs creating content like this.

[@nataliarodriguez3740]

THE BEST TANGENT EXPERIENCE I'VE EVER HAD

[@BinaryReader]

So happy to know there are communities of math minded folk :) This does make me happy simply in the knowledge these communities exist. Good work on the video :) Would be cool to see a general video on Hyperbolic functions maybe one day.

[@MegaMinerd]

"It's in the OEIS, it's technically interesting." So are the Brady numbers, and they're a Fibonacci clone. Don't get me wrong though, I love Numberphile. In fact, my MagaFavNumber, which I decided probably years ago now (though not when its video first came out), is 381654729.

[@angelmendez-rivera351]

Calling them "Fibonacci" clone is not quite correct. The Brady numbers, the Lucas numbers, and the Fibonacci numbers are all example of what is known as a Lucas sequence.

[@joeyhardin5903]

awh dammit i was gonna do that one! when moved up to my current school a few years ago, our math teacher asked us all what our favourite number was. i said 3,816,547,290 and gave a really nerdy explanation as to why, thus solidifying my position as unpopular geeky kid. ive grown out of that since though

[@atmosphericSkull]

Getting emails from viewers during the video was a good bit, but having his emails printed out and handed to him wasn't part of the joke. That's how Matt reads all his email.

[@bsharpmajorscale]

Yes, but what about Micro-Favorite Numbers? :P Also, I have a page on OEIS, so that makes me an expert now. Wew lads.

[@somniad]

as a programmer, the only number greater than a million I regularly have to deal with is 2^32-1 (a little over 4 billion) and 2^31-1; between them, I think I like 2^31-1 better merely for its unshakeable aesthetics - it's a common arbitrary breaking point in games, even for things that are logically unsigned and the programmer just didn't really care

[@121dan121]

Can we call the list of primes that fulfill tan(p)>p Jacob's Ladder?

[@lavneetjanagal]

There is a beautiful relationship between the numbers satisfying tan(n)>n and the two sequences A002485 and A002486 , which are the numerators and denominators for the best rational approximations for Pi. This section is too small to explain the full relationship, but one can figure it out by expanding tan(x) around x= (n+1/2) \pi and keep only the leading term, because for large integers its only the values of x which are very near to this value which can give a solution. The inequality becomes -1 < 1/2 ((-1 + 2 n) \pi - 2 x) x < 1 This clearly can only be satisfied if Abs[ ((-1 + 2 n) \pi - 2 x)] is very small, or in other words 2 x/(2n-1) is a very good approximation for pi. The claim that only prime number which satisfies this inequality boils down to the fact that in A002485 the term is 2*Prime and the corresponding term in A002486 must be odd. All the terms that Matt is listing can be found using this method, e.g., a_{7} = 2*52174. In fact this big number shown at 1:10 in video is 2*a_{86} in A002485 . From the listing of 1000 terms of the sequence A002485 , none other is 2*Prime. So the claim holds up to 10^{512}. One can proceed to prove this analytically if it is the unique solution with the the help of identities which A002485 and A002486 follow.

[@almightyhydra]

The primality check is probably the Miller-Rabin test to a few bases.

[@hammerth1421]

Also, that number isn't at all "big" in prime terms, modern computers are expected to give the primality very quickly.

[@darjanator]

I love your videos, Matt, and I tried, I really did. But this video made me feel like I'm back in primary school. I cried.

[@kamatikos]

Wolfram Alpha, seeing as it's basically the "do-math" version of Google, probably just keeps a table of known prime numbers, so when someone asks if a number is prime, it just checks the table.

[@sbyrstall]

Works for me.

[@frogstereighteeng5499]

Of primes in the 10s of digits? That's probably terabytes of info... (ie x/ln(x)) for P < x)

[@SharienGaming]

@@frogstereighteeng5499 assuming i didnt make a mistake in my rough estimations - storing all primes up to 10^20 as 64 bit integers (which wouldnt work for the largest of them, but im just estimating roughly anyway) would need about 10 exabytes - for reference an exabyte is 10^6 times a terabyte - and 10 exabytes would apparently be 1% of all data stored globally according to wolfram alpha so this might not be feasible (even the lookup on that table would probably be pretty slow due to its size)^^

[@frogstereighteeng5499]

@@SharienGaming that level would take cities worth of power plants to store lmao...

[@rbaillie6647]

According to Wikipedia, Wolfram and Mathematica use what is known as the "Baillie-PSW primality test".

[@Pa_Nic]

1:35 probably using something like Miller-Rabin. MR can check the primality of a 200-bit number in under a second with a really high confidence level (and is only a few lines of python).

[@marklonergan3898]

When Matt said he wasn't sure if prime-ness was related to trig-ness, was anybody else expecting another mid-video email to come in or was it just me? 😀

[@danielbontorin4783]

I watched this video while having breakfast, and I gotta admit the graph part made my day...hilarious

[@elementalsheep2672]

Double Parker videos in one day! you could say it's... double parked

[@aasyjepale5210]

Parker squared?

[@EsperantistoVolulo]

@@aasyjepale5210 parker square jokes are everywhere!

[@michaelzarkos9057]

This video is my first impression. That Tan bit got me subscribe.

[@astropgn]

My mega favorite number is ζ, and I just defined ζ as the next prime number that is greater than its tan, after the one showed in this video. I am having trouble calculating it, but this shouldn't prohibit it from being my fav!!

[@proloycodes]

i saw another comment claiming to have revealed your zeta

[@rickseiden1]

I put this on Brady's video. My mega-favorite is 52! It's the number of arrangements of the cards in a standard deck of cards (minus the Jokers). If you were to have everyone who ever lived make a different arrangement of cards (shuffle them) once a second, and no one ever made the same arrangement twice or the same arrangement as anyone else, it would take roughly 1.77e39 times the age of the universe to make every possible arrangement. That's 107 billion people shuffling cards once a second, every second, without stopping for 2.39e49 years. For a stupid deck of cards! (PS: Matt, I loved the tan gag! Also, the Numberphile cards aren't stupid.)

[@billcook4768]

Notice how Matt instinctively said 2pi when he meant to say pi. Shows how ubiquitous 2pi is and why we need to embrace tau :)

[@harrygenderson6847]

Double check; sin(n*Pi) = 0, cos((n + 1/2)*Pi) = 0 for all integers n, as Matt stated. Both these functions go through 0 twice per period of 2*Pi.

[@freelancer42]

KZbin suggested this even though I'd already seen it. Watched it again. Can confirm the tan bit is still hilarious.

[@jasonremy1627]

Did we just get two videos in less than a day? What did we do to deserve this abundance?

[@blusham4629]

Ive got to admit i would never have thought to find such amazing acting in a maths video on youtube

[@lumipakkanen3510]

This is why mathematicians don't like to get tanned.

[@nianyiwang]

That's ra- (no)

[@morgansearle3912]

It's such a quintessential mathematical idea to have a favourite number you don't even know. Not even learning the digits of it (because it'd be impractical), so you just have 'my favourite number is - well, if you take the tangent of a prime integer in radians...' It's not even that weird if you've been messing around with basic number theory and stuff for a bit, but to a general audience it's such a bizarre concept.

[@MasterHigure]

7:50 N/log N is the number of primes in the interval [0,N], so given an interval, you can use that approximation to find the probability that a random number in that interval is prime, and we get 1/log N. However, knowing that the primes are concentrated at the 0 end of the interval means 1/log N doesn't describe the prime density at the N end of the interval. If you instead use the (better) approximation that the number of primes below N is the integral from 2 to N of dt/log t, then combining this with the fundamental theorem of calculus, that actually DOES say that the probability of N being prime is 1/log N.

[@jamietus1012]

Since your land area video, I've been explaining to my friends why Australia has an infinitely long border

[@MayawireCH]

This reminds me of another question related to tangents that I was thinking about a while ago: Does the sum from 1 to infinity of tan(n)/n^2 converge? If so, does it have a nice, closed form? I was able to find a proof that the sum of tan(n)/n diverged and the sum of tan(n)/n^8 converged, but I couldn't find anything for n^2. At what point between n and n^8 does the series stop converging? I don't know enough about continued fractions and approximations to pi/2+kpi to try to solve it myself, but I thought you might be able to.

[@tangentofaj]

At 04:55, you got a sincere slow clap from me for that tan bit.

[@Codeaholic1]

I wonder if you some how related the trig functions to Ulams spiral you'd find some interesting relationship

[@baerlauchstal]

Matt, I think Wolfram uses a hybrid of the Miller-Rabin test and the Lucas test, both of which are computationally "easy" (and hence quick to execute), and also not all that hard to show with a bit of number theory (and I speak, as you know, as an Applied bod, who often gets a bit scared around yer Pure). It's a probabilistic test, meaning that in principle it allows false positives, but I believe I'm right in saying that no actual false positives are known. I set implementing Miller-Rabin as a first year Python project sometimes. I could dig out some code if you're interested.

[@rbaillie6647]

Correct. Wolfram uses a combination of a Fermat and a Lucas probabilistic primality test. The Wikipedia page for the "Baillie-PSW primality test" has a link to the Wolfram documentation on this.

[@wompastompa3692]

8675309 is a good meganumber.

[@TheAnimatorsUnite]

Congrats on the shout out from Seth Meyers on the newest closer look!

[@JayTheYggdrasil]

To answer the question on how wolfram alpha was able to "calculate" that the large number was prime so quickly is probably because they already have a collection of prime numbers and only have to check if the number you supplied is in the bounds of what the collection contains and then see if the number is actually in the collection.

[@rbaillie6647]

Wolfram uses a combination of a Fermat and a Lucas probabilistic primality test.

[@112048112048]

Allowing for absolute values? That's a classic Parker Tangent right there.

[@tbg10101]

I'll bet Wolfram Alpha just has a big set of primes so it can return results without having to spend many resources for each query.

[@springpurse]

My favourite number over a million is 16,777,216, which is the amount of colours possible with 3 bytes of information

[@roderickwhitehead]

Is it weird of me that I knew the bit with the tan function and the paper was about to happen exactly like it did?

[@macronencer]

No, I saw that coming as well. It was still funny!

[@jlunde35]

That's amazing that you get E-mail delivered to you on paper. Brilliant.

[@BelovedNL]

1000001 - the smallest palindromic number over 1 million. Although it might be considered trivial.

[@tiberiu_nicolae]

999999 the largest palindromic number under 1 million

[@HagenvonEitzen]

@@tiberiu_nicolae Together these are the twin-palindromes closest to 1 million

[@drostealth]

I enjoyed how you did the paper blowing away. you let go of the paper, scrubbed out the paper falling down, then replaced the paper with cgi and oop there it goes, flying away :D

[@LeoStaley]

I was wondering where you entry to this project was. Any idea if/when mathologer is gonna do his?

[@sebastianjost]

Just be patient

[@Wonders_of_Reality]

Hi Matt! Nice compositing! Next time consider using motion blur to make it even better. If your software doesn’t calculate the motion blur automatically, it’s alright to fake it with directional blur for a couple of frames. Good luck!

[@OrangeC7]

Ok guys based on that last bit I think we're getting a trigonometry marathon soon let's go

[@Xnoob545]

ive seen you before somewhere

[@chonpincher]

My favourite is the Littlewood number L: the least natural number n for which π(n) > li(n). Presently, L is unknown: but lies in the range 10^19 < L < 1.4 × 10^316.

[@AsteroidWrangler]

I'm spoiled for maths videos today, it seems!

[@johnchessant3012]

9:10 It was indeed worth my time pausing the video. I am glad to be your kind of viewer!

[@dima12011201]

Matt seems upset not him found this number

[@NitFlickwick]

You got a shout-out from Seth Meyers tonight on Late Night. Congrats!

[@BernardvonSchulmann]

Why? Why would anyone dislike this video??????

[@JamesCoutie]

I saw these videos starting to come out yesterday, and was thinking that the next MPMP was going to have a huge answer :P