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.
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
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.
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
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
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
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.
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.
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
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.”
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* :)
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".
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.
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.
@@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.
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.
@@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.
@@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.
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!
@@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.
@@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.
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.
@@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
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.
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.
@@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'...
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.
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
"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.
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!
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.
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.
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!
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.
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?
@@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.
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!!
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.
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? 😀
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.
@@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)^^
That single dead pixel thew me hard. Kept trying to figure out what was wrong with my screen before noticing it moved when I scrolled down on the video
"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.
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.
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
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.
dangit matt!! somehow you knew this was the first year I'm not teaching trigonometry. so of course I'm looking forward to the trig videos which no longer apply to my classroom!!
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.
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.
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.
the next one is 230835870782558831561617186504559084198719501221763995608082253627620752053749345488376393822837250198036536001853828659466202612019525543362322174085744303421231446484541625047630462908919109308644634605051209877750956648014568322183373423523622941806761765245932401727973436579786298208782013178059220103271409347616696556052706562092799953175234183483071403726145726928572372071037042523626350312132351311366806233135093893271182587352730075523143635168510803804031460442796778933680674070124730971307185688425634077096234482442639666385695677866015904370207368846631450100939158029908242779848800640038255592227473300237596577845602369215568916732445980431078426390412264603773550384039765410088966381694110344811198325354315338629604946794192217817288101344643511450133142277670683067655250506551517767422160650566385017503208608678491109517443585115317845289832567015746473548492179557935154400719019569904865219030736244089287736334048402066257337090606092966121806567484954460809024219605952851728610326005069
this whole video is just going on a tangent
I literally screamed. Well played.
gottem
I’ll co-sign this
Oh I think the tan function threw your comment to the near top
nobody primed me for such witty comments
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.
Wow this is an excellent point. Thanks you!
Well, you could do it with the Taylor series expansion of tan but that’ll take a LOOOOOOOOOOOOT longer...
@@Djake3tooth But does the Taylor series converge everywhere?
@@poissonsumac7922 googled it and, i didn't know that, no it only converges for x=-pi/2 to x=pi/2
@@Djake3tooth I figured as much. If it did though that'd be epic.
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.
Technically interesting, the best kind of interesting!
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
@@DirkDjently he probably hasn't changed his opinion about that.
Can't blame him much.
Yeah, he seemed quite unenthusiastic this time around.
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.
whoever wrote in like that is very wrong, the tan bit is hilarious
That whole bit was absolutely gold.
I laughed so much. The paper flew away! 😂😂😂😂
I don’t know if I would mess with someone who can heckle via paper note as a video is being made.
I wholeheartedly agree with the statement 4:57
Matt always has the funniest, stupidest bits in his videos
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
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
Mines 6064949221531200
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
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.
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.
@@patrickwienhoft7987 that IS interestingly counterintuitive.
I'd like to see a video on this.
@@Jigkuro and thereby technically interesting
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
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.”
Wow, you know a math! (Amazing story btw)
Sounds like you had a great teacher.
couldn't come up with anything for 11 huh
"It's the square root of a perfect square" I'm definitely gonna use that to justify why any arbitrary number is cool lmao
@@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
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* :)
its at 69 likes sorry i cant
Me: When will Matt get speared by the tan function?
@@efulmer8675 kinky
@@bryanchandler3486 i'll tangent YOU. HAHAHAHAHA!!!
Okay, this should clearly be "favorite mega-number", not "mega-favorite number". It's the number that's mega, not the favorite-ness.
Man, you must be fun at parties
@@ericklucasmenezesdelima570 😄
hmmm yes quite right my old chap - this is maths after all, we cant just move modifiers about willy nilly
Clearly, in Matt's head, adjectives are commutative. 😹
mega-favourite mega-number?
Absolutely love the bit about graphing tan.
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".
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.
@Firstname Lastname what's productlog? are you talking about W?
@@NoNameAtAll2 productlog is a special function that is the inverse of xe^x
@@NoNameAtAll2 yes, w is product log
That guy really created a sequence for tan(p)>p. Must be a hobby
My hobby is searching for positive integers where sin(x)>x. I still haven't found any, but maybe someday.
@@cadekachelmeier7251 pro-tip: try the negatives
@@cadekachelmeier7251 |sin(i)|>|i|, good enough?
@@cadekachelmeier7251 😶
@@cadekachelmeier7251 Good luck !
Using continued fractions as suggested by Moritz Ernst Jacob one can find a bigger prime p with p
But tan(11) is -226, not 226. Signs matter
Genessa He’s talking about absolute value, so signs do not matter.
I’m a chemist, therefore my favorite mega number is Avogadro’s number
My favorite number is 5,318,008. Put it in a calculator and turn the display upside-down.
You must have been excited last year when they finally decided exactly what number it is!
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.
@@jamesjennings3312 , It has been well defined integer since 2019 when the definition changed.
@@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.
Alright but that bit where the paper flew out of your hand made me laugh so much. It was a good bit
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.
using miller rabin to test primality only takes a fraction of a second for numbers with well over a thousand digits
Or it just....checks it against the list of known primes.
@@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.
@@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.
A nice overview of primality/compositeness testing algorithms with various trade-offs: cr.yp.to/primetests.html
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!
nice!
also hi lol
@@SunroseStudios oh hey!
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
@@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.
nice pfp. i think i've seen you before at some point. hi emily
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.
I've checked a few hundred million numbers. Haven't found one yet
Very good, impatiently awaiting a second.
What about tan(n) > n^2
Or tan(n) > exp(n)
@@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.
No spoilers but I can confirm that another one exists and so far two people have found it. Good luck!
Engineers: "wait, doesnt tan(n) = n?"
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.
@@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
@@lakejizzio7777 tan in integer radians, not degrees
It's a very good approximation for small n. For example, tan(0.000000123456) = 0.0000001234560000000006272...
#MegaFavNumbers
1,000,017
The smallest technically uninteresting MegaFavNumber.
That makes it interesting! But then the next highest will be... new theorm, all integers are intergesting.
@@frederf3227 when all numbers are interesting, no numbers are interesting.
@@SimonBuchanNz NO CAPES!
prove it
@@matteovasta2326 Brute force search of OEIS starting at 1,000,001.
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.
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.
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.
That is cotan(n)>n actually (looks more elegant that way).
@@landsgevaer Wouldn't it be cotan(n)< -n ?
@@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'...
Matt I'm already busy with the land area video!
Matt: calls people who stayed until the end the hardcore end of the video gang
Me who just stayed for the music: yeah
378163771 because it's the only number that's illegible when you put it in a calculator
That explanation beats that for 5318008
2^19937 - 1 is my favorite bigger number and is a classic Mersenne prime and 19937 is a prime no matter how cycled, woot.
99731 isn't prime.
@@pedronunes3063 Maybe they mean no matter how cycled instead?
@@tsawy6 Probably you are right, I've tested them, if do cycle, they are always prime.
@@pedronunes3063 wrote it not paying attention, meant cycled.
@@dhoyt902 Fair enough, it's still a cool fact.
0118999881999119725.3 That's my favorite. Made me realize that if you sing it to a tune, you can memorize anything.
i continually enjoy the fact that my namesake function is so chaotic
I find periodically reading the comments is likewise titillating.
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.
I don't think you know what prove means.
@@ngc-fo5te ?
Was fully expecting our boy tangent to make a surprise appearance at the end :(
Yes, revenge of tangent flinging Matt out of frame would have been very funny
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
Hi Matt, please keep going with these visual representations. They are not only hilarious but also immensely helpful. Thanks.
5:00 "Hey, there it goes!" I didn't see that one coming. XD
"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.
Fantastic video and a great community playlist. Love seeing the maths RUclipsrs creating content like this.
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!
Aha, it has arrived! Been looking forward to Matt's #MegaFavNumber all day!
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.
Wait, what? I have no idea what those mean.
@@green0563 read the digits from left to right :)
@@loganrussell48 I still don't get it. Maybe because I'm french ☺
@@sergeboisse it's from a song - an American phone number without an area code +1(xxx)-867-5309
"It's in the OEIS. It's technically interesting."
I LOL'd
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.
Ok that Tan joke was interesting. I'm not entirely sure how it made me laugh but it did. Well done.
I saw it coming but I still loved it
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!
That is really cool
it is less than a million, but i will give it a pass anyway :D
RUclips suggested this even though I'd already seen it. Watched it again. Can confirm the tan bit is still hilarious.
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.
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?
I took a double take for it
THE BEST TANGENT EXPERIENCE I'VE EVER HAD
Allowing for absolute values? That's a classic Parker Tangent right there.
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.
4:57 truer words have never been spoken…
…even though tan function is discontinuous
On its domain it's continuous
@@looijmansje isn't everything?
@@potatoonastick2239 that's the point though
@@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.
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!!
i saw another comment claiming to have revealed your zeta
Double Parker videos in one day!
you could say it's...
double parked
Parker squared?
@@aasyjepale5210 parker square jokes are everywhere!
I absolutely loved the tan bit! Also, the doesn't impress me much joke was great!
My mega favourite number is that one prime that's like 500 9s but one of them is an 8
Glitch
@@sudheerthunga2155 yes that one
You are my hero right now.
Thank you for the memories
🤜🏻👍🤛🏻♡♡♡
That numer blow mi mind
99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999989999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
The primality check is probably the Miller-Rabin test to a few bases.
Also, that number isn't at all "big" in prime terms, modern computers are expected to give the primality very quickly.
This video is my first impression. That Tan bit got me subscribe.
That email joke alone earned a like on this video. Thank you for making me laugh while learning a cool bit of maths.
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.
Doc Brown flexing on your classmates :)
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? 😀
I watched this video while having breakfast, and I gotta admit the graph part made my day...hilarious
Ive got to admit i would never have thought to find such amazing acting in a maths video on youtube
Yes, but what about Micro-Favorite Numbers? :P
Also, I have a page on OEIS, so that makes me an expert now. Wew lads.
1000001 - the smallest palindromic number over 1 million.
Although it might be considered trivial.
999999 the largest palindromic number under 1 million
@@tiberiu_nicolae Together these are the twin-palindromes closest to 1 million
the tan bit was hilarious !!!!! loved it !
Two uploads in one day. Awesome!
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.
Works for me.
Of primes in the 10s of digits? That's probably terabytes of info... (ie x/ln(x)) for P < x)
@@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)^^
@@SharienGaming that level would take cities worth of power plants to store lmao...
According to Wikipedia, Wolfram and Mathematica use what is known as the "Baillie-PSW primality test".
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.
I saw these videos starting to come out yesterday, and was thinking that the next MPMP was going to have a huge answer :P
That's amazing that you get E-mail delivered to you on paper. Brilliant.
8675309 is a good meganumber.
Can we call the list of primes that fulfill tan(p)>p Jacob's Ladder?
The tangent function took that killjoy troll's comment right up to positive infinity for a deep slam dunk into negative infinity where it belongs.
That single dead pixel thew me hard. Kept trying to figure out what was wrong with my screen before noticing it moved when I scrolled down on the video
This is why mathematicians don't like to get tanned.
That's ra- (no)
"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.
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.
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
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.
dangit matt!! somehow you knew this was the first year I'm not teaching trigonometry. so of course I'm looking forward to the trig videos which no longer apply to my classroom!!
Did we just get two videos in less than a day? What did we do to deserve this abundance?
Notice how Matt instinctively said 2pi when he meant to say pi. Shows how ubiquitous 2pi is and why we need to embrace tau :)
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.
My favorite Mega number is Graham's number. I keep on seeing visions of ever higher and higher stacks of 3s
Since your land area video, I've been explaining to my friends why Australia has an infinitely long border
I wonder if you some how related the trig functions to Ulams spiral you'd find some interesting relationship
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.
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.
The isprime function in Sympy returns the primality for numbers that big instantly. They must use some black magic.
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?
No, I saw that coming as well. It was still funny!
Did anyone notice the long hair on his right ear?
6:19 A Shania Twain reference - Matt, you've really outdone yourself
The *Law Of Small Numbers* suggests that the best mega-numbers should be slightly larger than a million.
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.
Wolfram uses a combination of a Fermat and a Lucas probabilistic primality test.
I'm spoiled for maths videos today, it seems!
As a physicist my favorite MegaNum ver is speed of light 299792458
Congrats on the shout out from Seth Meyers on the newest closer look!
I was wondering where you entry to this project was. Any idea if/when mathologer is gonna do his?
Just be patient
the next one is 230835870782558831561617186504559084198719501221763995608082253627620752053749345488376393822837250198036536001853828659466202612019525543362322174085744303421231446484541625047630462908919109308644634605051209877750956648014568322183373423523622941806761765245932401727973436579786298208782013178059220103271409347616696556052706562092799953175234183483071403726145726928572372071037042523626350312132351311366806233135093893271182587352730075523143635168510803804031460442796778933680674070124730971307185688425634077096234482442639666385695677866015904370207368846631450100939158029908242779848800640038255592227473300237596577845602369215568916732445980431078426390412264603773550384039765410088966381694110344811198325354315338629604946794192217817288101344643511450133142277670683067655250506551517767422160650566385017503208608678491109517443585115317845289832567015746473548492179557935154400719019569904865219030736244089287736334048402066257337090606092966121806567484954460809024219605952851728610326005069
My mega-favourite number is the Feferman-Schütte ordinal.
It is monstrous moonshine to believe that randomly mashing together two unrelated areas of mathematics could produce an interesting result.
You mean like, say, algebra and topology?
Matt seems upset not him found this number
Ok guys based on that last bit I think we're getting a trigonometry marathon soon let's go
ive seen you before somewhere
I love the dramatic plotting of tan(x)
Must be a weird feeling thinking you’ve found the only prime tangential number anyone will ever find