7 factorials you probably didn't know 

Link to my IG notes instagram.com/p/CSMu_IJnzik/?

If that is the Hyper factorial, the Pickover factorial should be named UltraMegaBlaster factorial instead of merely Super.

Two observations: 1. The double factorial is also known as the semifactorial, which I personally think makes more sense, since you are only multiplying half of the numbers less than or equal to n. 2. All this super-duper-mega-hyper factorial stuff reminds me of when we were kids, and got into an argument about things like whose car was faster, or whose daddy earned more money, like little boys often do. It usually went something like this: - A hundred. - Two hundred. - A thousand! - A thousand thousand! - Ten times more than you can say!!! (And no, that's not a triple factorial. It's just three exclamation marks.)

My TI-nspire sadly passed away calculating the 24 Power Tower... Rest in Pieces

The real question is: "How do you seamlessly switch between pens?!"

"Multiple exclamation marks are a sure sign of a diseased mind." Sir Terry Pratchett

"Five times three times one. You can do that by yourself." *Finally* he gets to a level of mathematics I can do!

For the primorial, ig 1# = 1 makes the most sense to me. Ways to arrive at this conclusion: 1: You also multiply by 1 even if it isn't a prime. 2: Since 2 is a prime number, (2-1)# must be 2#/2, which in this case is 1. 3: An empty multiplication is 1.

Usually the empty product is defined as 1 and empty sum as 0. So if the set of primes equal or lower than 1 is empty the product should be 1 by convention

10:14 I calculated it, but RU-vid doesn't allow posting comments so large they physically create black holes in the server. I've submitted a bug report, when it's fixed I'll get back to you.

6:46 There's no primes less than or equal to one. Therefore, the solution is the product of the empty set, which is 1: the multiplicative identity.

The exponential factorial should use the euro (€) symbol. It's still a monetary symbol so it would remind us of the dollar symbol, and it symbolizes a E, just like 'exponential'.

It's a shame how most math students are never introduced to the double factorial and/or subfactorial during Calc 2. I feel that knowing these concepts would make comprehending series a little easier.

It's funny how the Hyper factorial gives way smaller numbers then the Super factorials (Pickover)

I am surprised no one has come up with a Super Hyper Factorial

What new factorial will you define next?

Still waiting for the five-star-super-deluxe-premium-factorial.

Wow this is pretty fascinating - I didn't know some these existed, and their uses are also interesting! Videos like yours inspire me to share my own maths content as well!

Some crazy stuff! And some not-so-crazy. I chuckled silently when you asked for calculator help with the power-tower, 24^(24^(24^(...^24)...)). I was picturing some poor cuss actually trying to work this out on a calculator. Even taking the log will only "reduce" the tower by 1 "level." And you didn't even crack a smile when you said that. Incidentally, I would say that 1# = 1, because it's a vacuous product - there are no primes ≤ 1. Fred

In the end my takeaway is: -the first 3 are useful notations -number 5 allows to write the biggest numbers with only few symbols -I don't see what 4 is good for but I have a feeling I could run into it naturally -I don't see what 6 is good for and have no idea when I'll ever need it -7 is bigger than 4

7:06 I don’t know the actual answer but I would guess 1#=1 for a reason similar to why 0!=1 We can define (n+1)# as =n# if n+1 isn’t prime and =(n+1) x n# if n+1 is prime 2 is prime and we know that 2#=2 so 2#=2=2 x 1# so 1#=1

It's crazy that the number of derangements !n == the closest integer to n! / e. We looked at the formula for derangements on the first day of my combinatorics lecture because the formula was so cool.

I remember watching your videos about the subfactorial, double, super and hyper factorials. Thank you for always giving me new information!

I wish I saw this video before😅 I still remember when I was trying to solve an Olympiad combo problem and concluded that the answer was the multiplication of the odd numbers from 1 to 2n+1 Then I opened the solution and I was shocked when I saw the answer (2n+1)!! Only then to realise later "they are the same" 😂

Yikes, I thought I was in-the-know because I was familiar with the double factorial; I had no idea about the other factorial variants you showed. Very cool, thank you BPRP!!

I was always curious about those ever since I met the subfactorial on another video - thanks a lot for feeding mine and probably others' curiosities!

this is a great video. thank you for making a video explaining all of these in one place :)

I love him for how reluctantly he called it a hashtag and not a pound sign.

I think I've seen all of these before. I play Four 4s a lot, so factorial extensions are key operations for me. Nice that he included both versions of the super factorial!

Le Giraffe: Calculates all difficult factorials and leaves us the easiest(24 power tower) to solve

I didn't knew how math can be interesting and fun before, thank you for teaching me these new factorials.

whenever I see a new function, I try to graph it on desmos. I'd be very interested to see a video on how you would try to graph these

Here’s a question. Why does everyone solve factorial problems by multiplying integers from greatest to least. For example if a teacher teaches you how to solve for 4! they will likely tell you to multiply 4 by 3 by 2 by 1. Why not 1 by 2 by 3 by 4? You get the same result and it’s much more natural.

10:14 I'm pretty sure future Casio fx calculators will give the answer as 24^24^24.....^24. As the current ones are only limited in giving small answers like you enter 3/2 and press = button to see the answer 3/2.

Would be great if you went over some of the applications of these functions! At least a couple of those weird ones i used to encounter when solving some complex combinatorics tasks

Today I have learned something new which that the most some of them I have not know before,thank you sir

8:50 man you're making me laugh throughout the video 😂

1# according to your definition is an empty product (there is no p

Thank you for this list. This is an interesting set of operations.

I have seen most of them! Pretty fascinating! Thanks for inspiring me so much!

I worked out the Pickover super factorial for 24. It's exactly equal to ERR.

funfact: sf(n) * H(n) = (n!)^(n+1) it is very intuitive, but to prove it nicely you might have to use product of a product formula for switching indexes (if that's what it's called)

Excellent presentation!!

Amazing! Ironically I Heard them when I was 4th grade but I had no Idea Of the applications. Thank you BPRP!

7:05 by logic you need to define that if it’s less than or equal to 1. So you need to goes only +1 to the next p but in 0 (empty set) = 1 in a multiplicative way

I've seen the first three. I remember asking if there was a name for products of all primes up to a given number, and someone told me about primorial. I was (...and actually still am, kinda) messing around with prime generation, and so I had generalized the trick of ignoring even numbers (after 2), and was using what I found to be primorials for that (it's one of several projects that I've never finished, or quit, but just got distracted from.). To define primorial recursively, I'd say `n# = { isPrime(n) : n * (n-1)#, (n-1)# }` (or, just `n# = n^isPrime(n) * (n-1)#`), and we can start off with a base case of 2# = 2. But if we apply recursion to that anyway, 2# = 2 * 1#, but we know that 2# = 2, so 2 * 1# = 2 => 1# = 1... and 1# = 1 * 0# = 1 => 0# = 1. So we have 1, 1, 2, 6, 6, 30, etc.

This is why I love this channel ,I got surprised.

Great video! I didn't know most of these!

10:16 my calculator says "Timed out. Value may be infinite or undefined."

5:12 "Don't be too crazy" 5:15 Puts factorial on n's head

I learned about the super factorial right after this year’s Euclid Math Contest because one problem required a proof that involved the product of factorials

You should cover the rising factorial, it’s used in the hypergeometric function

If there's a superfactorial and a hyperfactorial, does that imply the existence of the maxfactorial? (Pokemon games reference)

Did you know that lim n goes to infinity of !n/n! is 1/e? That's one of my favorite results in all of math!

Enjoyed and learnt a great deal of calculus, hence far aside from the Algebra ,trigonometry i think there's still less content with regards to probability and statistics hop we expect such content in the near future

thanks so much for inspiring me!

Aqui no Brasil o subfatorial é conhecido como permutação caótica.

Hello bprp, small doubt..... Is subfactorial the same as the number of dearrangements??

Excelente video que resumen las ideas

Great stuff

Do mathematicians secretly hate humanity

By trying to calculate the 4! tower (4$), I got back a memory error. However, I was much luckier calculating the 3! tower (3$): 3$=801905114177186421268233247183671872285611243790287670326429840266965276859090994232722804099071308208566642345342525473839197857922206826881247686613054597643639074114299814658910570299338387275018144418060451356204425587436618355894265899469206493496576567060902508216857234809659411883436856907262181406555792173257484458552977375606894392453200909034506894234184478236418421979962663479216120643800922939369420248674473362609602187661563551041157505739642033306712744000213561038789775549335115383195493100990320977797431849066454349854112351669394350351724119648421429675482501486302736500144621886523347992629826999974724330860189653089828532182794794248240477416274638167362282413526807854514320952096682617889397115584667137201322422937457729214489407907405518444344340089061930346769872400573045001311080100230425970533942745847972064970363330555794582550644070075448682407064391762605241178885977478172470439245614352782718873090563810918058676016196022517960964002392982148152622058158104958518830487349863461522737045419079805176828913337987237167998461268815906214056666240308532663321889986375962262141989078341225419274892934633471601337630145021177561682163361588301146273292029772181095793682371661321565671179250200873481397054591452273317157196303425228704984654767851075710532634534940796785677558890950799401875263511992661902169258890278086716291023843497372147231848593552275703330179333395157137953888601584226588131426100524625525615311244683340215525755193173697123985498932994880224661923242660863038692352636818818091446575100518750311622740988660944192795623802082203241025300988864720691114284336174884722725160551906710564699824148484730470707902578930619626494023221095499047958286617225276486876179287677463797214957475199592111410409161111024724320181524607190511675442364059199832339531178389332438871670894278123643702026198922090184989766828514386825218944751917133528352820304932965893847129193929732262192111912880919222840357641983028044015106742642713134002917504796175868158080020653346101062376128143166925008124162624778493310053821947745097837762493928482536937358487491224793636348213860230948090092608071270697036421316013417589210684049327427491895567716870540159334726003182535675968082210912512117117036411988561552555424135025992192431252311247070107037564320408519913415791972361428643569407291782230769633403762980911951260235335468415654697223881790965348650156255150470465709634202169556242801373930782315697735699489821418879261442079714412155375949060050935369523298480393127780154774697206538820578852481294171389639340821243198793285107034663451816584313178509573270340714717653972268811979935455568659825920079977104240044757023571324964943766412817014787831726000431239296277568149403379174685366513529096824121631549336050517240784764044158530092410468898790882906726991168235676755052595083949405892993514487989629327303507999701858400364951812663411243218524311814960565403396906101566037518454582866326674740652656967374738643546913572072027015270654024870872914125274032777679768834616330289620042855458464404935752253141307743949799679373788177021131263060724194551523232678825949835712984835004658258078967038721817894573819554326478723879110512134676175579870238496958283594595247111635504199858696576767040558179086446871276735764539552108394244368401906598270272523213985019325867597404117299522896174182781347656228133260501669599573840643828131130837868317552037425215982186057658406291543623646877113038178380490129752610988187060310837787799219303381539699528293723206372177059719935531506073859021197524406579643039883039728628836461474751067864431977032358675848360773708387211420116787599737621317224241346875009176863639530452676627730931378159457365569487241901935734071637648678771531953675914311001534496147038332750307708867979198279698026903039770263012642154401276299002427289117685602673262358039948743624480371236137632544504304823818957992107773203870105130812284336956828027729321903579499814164578180299915045407689667530374597860119037107839602699845102433609954824008871263055281424268092422912559273889700924995226448267306343535545322900135542162984089368300143981387952516535890373585769044768270079232745085310534780379433679641764412570375902770137404074177820073270088260988742823688892707845709507869126201853287365775198969687579436875786108977542040269149258582213880806730504418248217557255761673402533058045211820437282641288015597565632574887136806808091337017274509640585947630061378243713693613162003445998800513844020356593674967439236032719297765887804559453426094291753338337320872533167029618779345490908355556740326053560776376448793273729369475913183616635968036303958961312252848799884953039291437629677310491001983631561495387558374254249597009726836978531354929462178177642763033790164067445673502415866746505721852575827258860644876762985518399443861444129789611155823260748613960983738802730799807870324833863673572794179621716686213597175126065963043765314408250036111188043650982973774434447477841745166609106376305766597815630308332278922332012868449774553692733717992022275716188668002733820424048869010692647287753683032329124547512690629495028349649028761229072342231520826626527689967862367744521152658974319063649327835030970627742864238920810668385925185216817124523427167003892110153204070727224612710173873389921936290442205620640819677053163599111244195701659784290628033387794423384897379043640715550904349542341988051448696644729119321923974170788984946987136512729765351867471308995876186529082842949528120694579172451660355612447630749890773691802401321948599241617171873740187460875541452669196018430458379320978910452677708740121149389289049260368909671797571587872574361576403325458450829959641703568470576948819313050657979060435743564740553565911085870118497098825973672356583186516354715506718750007325734787689281138147193205163931032061943134231140199543095420684425751639787908398865190601747112700042196582032481766506799648617686643106868998527331337192639617847034473260672095881810378587492712587519328256 If I'm correct, that should be 3!↑↑3!↑↑3!↑↑3!↑↑3!↑↑3!.

Really love your videos sir🙂

CORRECTION: The primorial n# does NOT multiply all the primes

I never liked any of the arguments for 1 not being prime. Somebody needs to show me what breaks if 1 is prime. As such 1#=1.

Neat, I never saw 2 and 5 before, I wonder what they could be used for

Wow, I'm actually reading a Ken Wilber book and I was about to search something related to his work and this video popped up

Every time you raise the level, I get goosebumps!

My maths profesor always told us about the person that invented the subfactorial, or the left factorial because he is Serbian. The name of the mathematician is Đuro Kurepa ( Ђуро Курепа ), but never took the time to explain what it actually does. I finally remembered by myself and found a video about it, thanks.

Awsome !!!made my day.

Intersting thing to point out: Sloane's super factorial and the hyper factorial are very similar in pi notation! Let's take the example of sf(4) and H(4). sf(4) = 4!*3!*2!*1! = (4^1)*(3^2)*(2^3)*(1^4) H(4) = (4^4)*(3^3)*(2^2)*(1^1) and more generally, sf(n) is the product from k=1 to n of: k^(n-k+1) Whereas H(n) is the product from k=1 to n of: k^k (I'll try to write it in pi notation like this Π(index; upper bound; expression) ) sf(n) = Π(k=1; n; k^(n-k+1)) H(n) = Π(k=1; n; k^(k)) Neat!

love the kobe shout out big respect

Great video! Where, in which field, are those factorials used? What shall be the math problem, so I would need those super and hyper factorials?

Wow! I only knew factorial but i didn't knew that there are types of factorial also🤔

The double factorial of wikipedia was so complicated, you e plained it so easily

I love how much it takes me to notice the pokeball, it gets me in every video, I'm so focused that I just dont notice

Merci,très utile !

Do a video on difference between antiderivative and integral please

Very useful video

Good video! It would be interesting to talk about Fibonacci factorial or fibonorial, too.

There is also the subrecursive factorial: srf(n)=n * product(k

@2:51 The way he stopped while saying, "yeah" and then "I am not gonna do it" ! Bro are you reading my mind?!! xD

Video is interesting and ur same that ... thanks 🌹

I like how you ask everyone to try 4 super factorial on the calculators when you know that they won't be able to display the answer

I’m now letting my iPad calculator calculate the power tower ²⁴24 as you asked but spitting out the digits my iPad is developing into a black hole. Cool, I always wanted to see a singula

My mind is blowing up of seeing so many factorials.

Re the question on the Primorial, I would assume that for all integers smaller than 2, the primorial would be defined as 1. This ties into the concept of the null product and null sum, where the null sum is 0 (i.e. adding and subtracting 0 does not change the value) and the null product is 1 (i.e. multiplying and dividing by 1 does not change the value). The null sum and null product are actually really helpful (even if one doesn't know the names), especially when solving equations utliising techniques like completing the square and clearing surds from the denominator of a complex fraction.

Figure this out while playing around with Gamma function :) Here's the formulas for double factorials : (2x)!! = 2^x ×Gamma(X+1) (2x-1)!! = 2^(1-x) ×[Gamma(2x)/Gamma(x)] Anyways, thanks for the useful video!

I feel like the pickover one is impossible to apply. Like, you'd literally probably only be able to go to number 4, as with 5 you'd have 120 exponents which is just ridiculous

Can you do a video of exponents? With diferent types

They're really crazy factorials😱

make a video on Collatz Conjecture.

This video made me excited for math! I can see these operations being useful, which they must be because they exist, but still. I could see ME using them, which is awesome.

Thanks to you I knew all of them😄😂

7:07 We can find this answer with the same logic as why 0!=1. Let n be any integer and p be any prime. We can then define n# as p#/p, with p>n. Following this, we can see that 5#=7#/7=30, 4#=5#/5=6, 3#=5#/5=6, 2#=3#/3=2, and by consequence, 1#=2#/2=1. Following this same logic, not only can we deduce that 1#=1, but that, in fact, for any n

Very useful obviously

Man you holding Pokémon ball and introduce these cute factorials it’s like your catching some Pokémon lmao I love this idea! Fan from Taiwan

The subfactorial can also be calculated by dividing the factorial by e and rounding it to the nearest whole number

I am a simple man. Too dumb for maths. But I LOVE your content and your style of teaching/presenting, so I watch your videos anyways. Oh, and I have subscribed as well. Thank you for your work!

Super and hyper version!!