OK, Pactis right away. I want you a Picture, a number. È una roba in normal big number light the National dad or the number Stars the Galaxy thusere cute this are franky. Zero in. The Grand Scheme. A number so catastrofica mind bandingly collasso there if you try to hold just the bidgets of it in your mind. Just the budget. If you try to fisica in code that much information in the nurance of you Brain, your head with liraly collapse into a. Blackhole.
On the Beckenstein Bound whitch basic dictatis the maximum of information you can Cream into a fine region of Space givenut Energy if you pack too much information, too much entrope entro the volume of the Human School the. Density exeedes the Swarce Child Radios. Exactly. You become a sinulary. Pooh, so we're talking, but Numbers there physicly dangerous to think about about in Air antirity we are.
Talking about hyper large Numbers, the Absolute heavy weights of the mothworld, specificly things like grams number. Grams number is the poster Child for this Hall concept. It's a number so large de the intere observable Universe far forse molto right it down even if you wate a single digit on every single plank volume. Witch is a smalles Misurable unite of Space. Makes this Deep dice Today. So? Fascinating eventual this number is too big for the Universe.
Contain Eventhing now the definize all standard Scientific notation we action know exactly. How itance we do we know the lest, we know the lest, we know the last Billy and digits action. There. The very, very bom of this infinite Towers of Sher, math, Medical caos there. There is a rigido totali deterministic structure. A paterna è giusto lax in the Place. It's a phenomen Matmeticians Call of Frozen digits, or sometimes the years stable digits. So Welcome to the Deep dife.
Today we're going on a serious Journey into the infinite. We're un packing a stack of Pretty Dance Research Papers very heavy on the number Theory primary focassing on the incredible collaborative work of enigmate and mathmation Marco Ripa. Matics of Tetration. Tetrational definer second todays to solve a very specific Mystery. How can a math Medical operation
there? Explodes Tours Infinity ad in Trillias of Digits millisecondi somehouse down into a predictable Frozen pattern at the very. Tailanding can we miser the Speed of a Stability because than is the key contribution of the source imperiale. Ripa and his colleghs EN in matematica dingus nodues the parent dei file spedomater dei developt metric cold congruence Speed. Wheeve. This metric basicaly teles the Speed limit of the number system itself. It's a story.
The takeses from basic aritmetic wet we alnow straight into the weard underground World of Paiatic Analysis. Majler aritmetic ed eventualmente. A really Deep filosofico questionmark Weather we are inventing Numbers come we do this math or just discovering locations on a map that's literally always been. There, the gap. Between Tols and Absolute order. So buckle up we'ere looking for order in the infinite.
OK. So to understand the end the Universe realisti and this Monster Numbers, we have to understand how the your build in the first place. We need to talk about the later of operations. Right moseless probabile no the first three runs of the this later perfettaly you learn the man elemento. Counting THREE Plus Three. Simple enough than room two is moltiplication what is just repeated Edition three Times fours religios three addio. It's of War.
Times THREE Plus Three, Plus Three Plus Three. The run three is expanentiation force repeat moltiplication three de the forth Power is. Three Time, Three Times Three Times Three. Most people get off the buss today. We are the forth wrong. Tetration, tetration it, softcomes from tetra mining Four and iteration it is iterated exponentiation. Visuali. If you reading a math paper this is usaly ritene as a Little superscript number to the left
the bass. So a smallbeefol by a big a we re that is tetris of hight Bee with base a or sometemes beattred to a. For you list if i SEE a. Little Four to the left of the three that is not three tens for it's night event three to the fourth Power what is eightyone Four tour three is. The tour of three Race to the Power three, three Race to the Power three, Race to the Power of three. And crucialmente in Israele Trip Up on i try to do this on African.
You have to a. Valuhead from the tampdown. If you wheels collapse into a sportly larger standard exponent to get the True scale of tetration, you start at the penthouse and work your way down to the lobi. Let's work through the this to give the listener two Sense of the very go involution. For tetrade to three, high Four, base three the very tap the Tower is three to the. Third Power. Seven iseno Stone the Towers One step Shorder naw we take the next three down e wese it to
the. Twentyseventh Power. Leaving the Realm of Human Intuition, three three to the tween seventh Power isley Seven point six trillia specifica Seven trillion sixdred twenty-five billion five hundred ninety Seven, Four hundred eighty Four thousand Nine hundred eighty Seven. OK, Seven point six trileon. That is a massive number. That's the National DAD of a mediam size country writer. But i can still somewth all that concept in my head. Boh, we are.
We sill have One more step we. Hat the boom three. Calculate three Race to the Power of Seven point six trillant. Even is a number with. Refly three point six trilly indigence not the value, the number. Of diject. Yes, if you print the number in a standard twelve point font on normal paper, the stack of paper REACH from the Earth to the Moon and back moltible Times. Orf, you lad the pagers out flat side by side e will rapper on the equator. Hundreds of Times over. The.
Site for this is a baby. Tower, That's. Nothing's Numbers is define usingus upper annotation witch intervices a all other Beast of recursion fondamentaly underneath of the notation it represent part are of threees where the high itself is a number so big the Universe breaks toying to Express the high to. We're effetiva dealing with pow terres of infinite high. Practical. Perpaces, Yes, infinite. South Tour the just never stops grawing it's reaching up Forever. No?
Normaly when we think of things grawing that fast a candid, explosive runsway growth, we take a pure caos. Youth of Enterpey. Exactly, you resume. That is, you Hat another Power three to the very top, the digit at the beat charning like a blender, everything shifting left, everything changing constantling. The intuition, but. Math is Fall of completamente
contorintutive surprises. What? The Research and this is SOS Central to Marco ripus work with eny banca is that as the tour get haire and ire the digit on the Four right Inside the smalls valus this stat moving? Whatsement. Beenpord by The Truck. Building. Iron Hiro floors, It's fluid, It's concentaly changing, but the boom. The Foundation, IT cures it, Arts it set and the Absolute Stone. And once it set, it never changes again nome. Tower, the Botton logic is
completa locting let's. Look at a Simple Exsample Street from the source material. The Hall visualizes pace, five pace. Five is a Classic starting point because it's stabilizes very visibile and very quickly. Let's look at the sequence. OK, High want is just five. Simple. High-too is five to the fifth Power that three thousand One. Hundred twentyfive. High three is five to the Power of three thousand One hundred twentyfive. That is a huge number.
We can't rete the hold thing, but you calculate the tailand of it in Eight Seven. Four to One, Eight, Seven, five. It. Before the five in the previo step, Three Thousand One hundred twentyfive it two. In this New massive step, it is stilla. Two. As the highght the be value increces a. Wave of Frozen Stability creepes from the right. Side of the number to the left want to digit freeses it is permanent it is etc. Into the fabbric of the math. Itself.
Us to know the last digital dream's number beghes greams Numbers essenziale poweretower of threees there is so un immaginabile high there a huge chunk of the boom digits have just Frozen solet. Lost in the clows of Infinity, but we can stand right there in the lobby and look at the serial number card entro the quornerstone e we can read perfettamente. OK, so we know the frees Thephenant's beone for wild this interrally.
New News, the structure to sum extense centuries a. Goal, but the much deoper questionmark and the Real focus, the endingc collaboration is how fast? One DIGIT per step, Tended per step Doest excelerate does it? Stall out. Is it random? Todaytoday a Metric Cold Congruent Speed. Denoted in all the Papers as vievè.
Vie It's a velocity, it is. A velocy represence the Little wate of Freezing V away telles exactly home Andy New stable digital we gain for every single step we add to the high of the Power Tower. Sovie base One we gain One Frozen digit per step. IV It's five we game five Frozen digits per step. Ruffle, Yes, that's Practical Translation and Major Discovery Papers. The. Fluxuating Wildly ed è settles into a specific predectable Inter value. So we are just guessing anymore we can action.
Calculate the Speed Headtime per howl. Whats Determines of a number free Slow? Why Space five supportly, fasterd and based three? Wor is it? The hard we have look at the Engine of the number system itself, we hath look at the
factors. We are the greek letter no sub p. OK, Let's On Pack Pyc Valuation perché onestly it semestre something you need a Phd Advanced topology to understand, but reading through the source material I have a feeling you can explane it to us fai in English. It action very intuitive want you force yourself to change your prospective on what Numbers action are. Usally want we do moth magnitude. We kared how big a Numbers fifty is begger than. Fosse matters in standard.
DNA in are standard dessimo numer system base TEN. The fondamental building blocks are the prime Numbers two and five because two Times five ten. So AP setic valuation Simply as a based question, how many Times can i divide the specific numero by a prime? Puoi take the number forty
forse. Forty is Eight Times five or Britney with primes, it's two to the third Power Times five to the first Power. So the fiber e valuation of forty is One because there is esattamente One five in the prime factorization, and the two Act valutation of forty is three because two to the third powers Eight and Eight divide forty. Perfectly OK, Simple 9 It's Reality counting prime Factor.
It's like Doing a Generic test under number forty and counting home Andy to Chrome zones and five Chrome. Infinite. Tower, the Congruant Speed è. Vve comes directly from this valuations, specifically Marco repast formula show the Speed depens on Homey Times. You can divide the nabors of the base by this. Primes the Nabors One and a plus One. Yes, trivil Manupulation of the base. Or sometimes even a squirre plus One.
This is genuole fassinate me certifid out how stable a huge poweretower of aspecific number is gonna be. You don't action look at the number self. You look at the guys living next door on the number. Line you look maniation exactly catagonis this in to what he calls congruence Classes hebreaks the Hall system down based on what the based number is magilo twenty. Module twenty, Witch for anyone resti under mogel aritmetic is basily just asking W. Suppose your base a and One.
Twenty-One or fourtyone or hundred One. Ok, if you sub track to One Saw a minus One, what happens to those Numbers? One because twenty, fortyone because forty thy all and in zero. Witch become moltipliles of ten dei sedenly have thus Magic Factor of two and five repat found there. This Numbers the congruent Speed Largely depens on how the visible that a minus One nable is by the specific primes.
So therect. The secret, the infinite Stability of twenty-one is actly hidden Inside the number. Twenty. It's Highing. In Planet site. OK, about the trick, the. Tricky matmetically speaking here are Numbers hending in three or Seven, like three, thirteen, Seven, seventeen. This is classified. Is Peace three in the literature OK for this? Baces Semply Subtracting One or add in One doesn't help us all. Take Three, three. Minus One is two that is not visible by five, no good.
Three. Plus One is Four alton not visible by five, still no good. We find the hidden factors of ten for a number like. Three. What is three dry and what is the immediate? Right napor of Nine. Nine? Bing for Baces hding Inter Seven the Congruence Speed Depents on the five eat valuation of a squareplus One the. Square Plus One. Why that specific formula?
The why squared? Because the any number and three or Seven it's Square will inevitabile ent Nine three three Square is Nine Seven squares fourtynine thirteen squares One hundred sixty Nine. And. Hating One to any number ending in Nine. Su the DNA Stability the. Blueprint for healthfest freeses is hidden in how easy you can mathematically turn the base World Square into a moltipli of ten.
Ere essenziali looking for the path of least Math Medical Resistance to the prime Factor to and five OK. Libro in Math Grams number. Is fondamentali a Tower of Threees, so ar base is a ecules Tree se wee define the congruence Speed of three. We need V three repust formula live right now. Sense three is congru to three moodle o ten and moodle twenty we Fall squirry Internet three we wed about se we look a squareplus One. OK, so Three Square Plus One, that's Nine plus One, what is
ten? No, we need the five Arc valutation of ten, remember? Five divide. Tende. Tennis two Times, five to. The first Power corrects. Valuratis V Three One. Precisay One. So a powerpower threees essenziale the core Architecture of grams number gains exactly One New stable digit for every single step you go up the. Towers a very Slow, very stay March One digitati a time it is perfettay. Linear.
Precision for something soft so grams never is World the tuppler of semplificare sixty Four tetrade the three ruffleys the stage of the recution we talking about. Ruffley YES, to give a working moth. So ripeteres work formally prooves the grants number shares exactly White termine. The paperly Slug base, Three of g Mines, One stable digis, the infinite Power Tower Forgbase. Three the Slog or superlogym? È basabile di inverse of Tetration accounts the highght of the Tower.
Since the Concorrent Speed VF three is exactly One, the number of frase and digits is essenziale quial to the high of the Towers self minus One. But for base three? It's a one-to-one ratio, the source material. Something else, but the telegrams number it Mens a pheshift. Ah Yes, the asanth phyeshift. This is the DIGIT havent Frozen years. If I have a Frozen Tele of, say, five hundred complete
stabledigits there. Is a five hundred first digit just the left that Wall the still changing is the Tower growse it's on the borderline. Correct. It's The Act Frontier, and the ovvios questionmark is the frontier. Interly random is the mathis Rolling atside die every single time we Hat. A. Power. Is abolte Ripa an enigma d'America found the digit gesto, the left the Frozen tale action. Cycles for base.
Three specificat the difference between the non stable digits of consecutive haits flips back and forth. Between What vales? Between Four and six. Forensics, back and fours. Forever Mogiler Pattern the Paper descrives this ask faeshift in detail for base three, the sequence of the Shi. Four Dance six than Four dance, six to Infinity, it just baunces back and forth like a metronome, thinking outside the Frozen Wall.
Is wild to think about, so even the socal Keys, just the outside the Frozen. Digits has a heartbeat. We Little grimms number down in a physical Universe, but descrive it Absolute mathmetical Precision and descrive the behavior of the digits tost Beyond the tall, we have a Perfect map of the very edge of the. Number. Love about this Research IT terrificing Monster number feel almost tame like we have successivo captured a Cage of margel. Aritmetics behavior with prouvenist still haste.
To follow usando the Speed of based three, it's Slow. V of Three is One per the Papers dedicato a lot time to what the collant inverse problem. Mining inserit taking a random number entro find it Speed, can we Act build a number that a specific Speed? Like if I want a base number that frees One hundred digits per step, can i engineer One? Perfect, Perfect squares. Perfect cubes. Yes, Numbers of the General Form X to the Power of Sea.
The Research House è completamente fastina in Relationship Twein Net Exponent Sea and the result incongruent Speed. Go One. Ripa matmetical proves you can find infinitamente many based Numbers dapperpowers and have a congrurant Speed exactly. Equal to Sea. Wanna Speed of Five for Perfect fifth Powers. Essere. Yes, a very specific algebra Construction, mentre in the paper to garantied it look something like ten to the Power of K plus T plus ten to the Power T plus One.
All of art rayse to the Power of c. Wow OK let's dico da formula bit for the Listner tendo a Power Plus Tend a Power Plus One all Race to see e Luke scarim paper but what is it action doing meccanicaly? Uno structure Inside, thus fantasy. It's a number constructed interrally of once and zeroes powers of TEN plus One at the end.
This, the specific Illy engineer structure ensures that when you look at the thus palatic valuations, the scompcyl prime Factor we talk about there are perfettaly aline. Se you are basicly handcrafting a base number, this is perfecly divisible by exactly the prime Factor. You need to be divisible your stacking the deck and when you findly raise thatire prendere structure to the Power of c you
effettivamente. So you are matmatically rigorosally garanting che the Final congrurant Speed esact sì. So we Willy Clean Engineer Numbers to freeze whatever rate we want. We can building custom high Speed Numbers. Within SER and boundes, yes it's Perfect highlights at the freesing Effect isn? Some magical isolated property of the number three of the number five. It is a fondamental structural property of how the based interx with the numer System Excel.
Speaking the Numer System system this to. Writing is World the most mindbending parts of this intere, Deep dife. The Environment matters. The. Retix who Space TEN Human have tenders of evolute ten e una base TEN tetration semestre to behave Pretty well stabilizes it. Frees IT have specifically because based tennis World Square free. Square free to find out, forse.
A number free if it is notvisible by any Perfect Square other then One se look it's prime Factor are two and five, two Times five are any Square primis. One to One five distinct primes nel lockup base x to temps. Three again distinct PRIMES. It's Square free. OK of a non Square free number. For is litteraly tosquare it's a
Square or lockabas twelve. Twelve is Four Times three witch is two Square. Times Tree contains a Square Factor. Or considerare sixteen witch computer Scientist love exidesmo sixteen is two. The fourth Power it is a Square of a Square. Password. This distinction between Square free baces and non Square free baces mex a massive difference
in how teatration behaves. A critical fondamental difference in a Square Free System everyday base TEN Stability is robust almost every single based number you trial eventual digital a Constant Speed. It is a stable math Medical. Universe System. Retaling with is specific rule in the. Paper isoling the radical. The radical the number are denoted is ratevar è sempre the product of it distinct prime factors ignorang any exponts. Software base sixteen wit is
two. The Forth, the only distinct prime Factor is two se. The radical sixteen is just two. Correct. Now here is the instability criterian ladyscover in any non Square free ratex are if you tetris base a happens to be thevisible by the radical. The Congruant Speed never stabilies never it. Never freesen to a Constant. Rate never IT Excel complete out of control. Woo, let's Trace that's imagine we are Alien beans for sixteen finger e we nebbiable counting
exadesimal base sixteen. OKA base Sixteen Universe an I. Pick a base for my teatration lets a tools devisible by the radical of sixteen whit we service two. Soctor building a powter of tools in this base sixteen Alien World, What action happens to the digis? The number of stable digits whitful be the trailing zeros, in questo caso crowes esplosivamente. It doesn't set on to a nice Constant rate like One digit per. Step. E Mike V One Zero on the First and Jump to Four 0, 0 and.
Hundred zero. Caos in the congreens calculations, the zeros accumulate so unbelevable fast the paiac valuation never sels down to a flatline jumping expoly. So grams number witch is a Tower based on three. World the Frees is nicely, but if we try to calculare the Tower two in B sixteen. Productivable Youd find a runaway Train faster investment the needle Brooks of. Enry Parmix a Pretty Bold conjecture about this in the material.
Disney he Does he Esplicitaly conception that square-frinness is the soul criterian he suspects that only in Square free numero Systems do you get this ironc guarantee of Constant Speed for all baces. There is profund e basicly suggeste are entere ability producted hail of a Monster like grams. Nember realize everyly onde tolely arbitrary. Sixteen, the. Math of. Infinite Numbers would look like a completaly different, much more.
Karic landskake. Implies the this Stability is purally a property of the number self, but a property of the specific lands we happen to be viewing it OK. Speaking of the philosophical applications, we spen a live time talking about the how the formules, the congruent speeds, the baces the ratises ma we have reality tesh on the. Deoper, wy the wis. I know would because the White is the number, freeze it all. White does it just Channing out random digits Forever even. What is physical?
Your matmetically. Stopping as quantities the Groabs physical location. Thing about. It weally. Think of the poweretowers garing bigger and begger rushing endless letteras positive Infinity. It's moving constantly to the right on the standard numberline. Sense in the realm of Factor and valuations, rushing Tours a very specific point. We need introdusse the concept of the PC manager specificat denoted as Z subscript in for a
based ten. How is AP edic innerture different from a regular eneater use everyday? A regular induture is a fine night string of ditests. One, two, three, or five. It Four it's stops, but the payat integer can extende infinitamente. To the left, to the left, usely only thing Infinity is thingor the right, like pie p point One Four One five Nine under the right. That's True for Real number. Yes, Real Numbers fill the infinite microscopic gaping indags, but. C Numbers go the other
interrally. Thy are sequences of digits that stret infinitamente into the larger place valerioes like da da da Nine, Nine, Nine, Nine, Nine, Nine, or data Eight to One. Eight Seven fivec intetration. That specific string of stable thegis of the. Last hours, a fine night Life of One of this infinite pyc Numbers. Power Tower of Threees and The Street Mostics frees in the place we are essenziale, just revealing more more of this preesistenti infinite leftard
number. Yes, we are waping the dirt of fuscal and this infinite number string of noise it a precise solution to a specific algebrac equation. The equation is. Why? To the Power of teen caolse. Why? But in base TEN specifically, the stable tale of disdetion Tares converges to a solution. Quasion White to the fifth. Power Y White second. If i take a normal number like two to two, the fifth thirty-the-o that is definita not to vita three three to the fifth to hundred fourthree
alfont three. The only normal Numbers were wyd the thirtyckles Wire zero One e maybe negative One. Regular integers. Yes you are completa right. But in the World of Tenc integrars this is infinite left Word strings. There are altre solutions. There are what i like to col Ghost Numbers leaorking Deep and the arctecture number number solutions of eurous Works and Infinity string of zeros ending in a One Works ones. There is an infinite tenetic number that ens in Nine to zero.
Five Seven. If you Take That Inter infinite number and raise to the fifth Power, you get the exact same infinite number back. Il ministero? Stationar number system a tractors. A number. Iterating expanentiation over e over euro, passing a based number into a Illy Dynamic system and the Gravity the mathmecal Gravity of this patacle fixed points start polling the number in the magnetic pollsing. On we see the DIGIT freesing.
Yes, the DIGIT lock Inter. Place semplice because the number in Closer and Closer to that solution and for base TEN. There only a Handfull of the Solutions available. The paperlist every. Single time your in base ten you are inevitabilmente sliding down the gravy wealth words, One of witch specifica when you hit depens on your starting base and this underlind structure. Changes if you change the base. Yes in base. Six, forse the equation isn?
White to the fifth, White to the third equest why the intere invisible structure changes base on the prime Factor of the base Your operating. Massive. Tale stable digits, we know exets it is essenziali a Shadow. The Shadow ha completamente deterministic Shadow cast by One of the pinatic fixpoints number three been rollentasi toord a specific infinite solution of wy to the fith eackles. Why Congruence Speed of One the V Three Walls One.
Suring the exact velocity with false Internet Gravity well, the infinite feel. Surprising Structured the Ultimate. Beauty of it finding order in the. Infine we normaly think of Infinity is asmeless che hotic terrifiing expense beh math Marcon logic actly imposes a very rigid skeleton underneather event the Monsters Hat follow the. Rules of the House speciale the Monsters souls. Recaptis. Incredibley Deep churney we.
Started with the conceptiuly simple operation tetration, the Tower of Power. Physics of. The Universe, the farraight Tales of this Numbers freees complete Solid we learnd about congruent. Speed VVAE Now Marco Repa and is collaborator enig matematica manager to quantify exactly how fast at freesing happens, and i did it using the DNA of the Numbers prime Factor through patic. Valuations. We found VF Three One Slow stady linearly feeling in the digit
One step of a time itself. The retics determinent it is Eni Stability at all base tennis safe because it's perfectly Square free bayxxteen is pure cats Beats it contense Square prime Square freeness is the Ultimate. Key to garante in order freasing process is. Reality a number falling enevabley tour the piac fix point a go solution to White to the feat equose why the exysted
Wong before Human even road. Number Theory and Advanced Patic Analysis into One Beautiful cohern Picture it ready changes how you. Look at a simple Member like three. It's not a quantity three things anymore, it is a seed. What a highly specific predeterment project through the infinit specific Speed. Travels so cheers of Final. Provocative, a molover as we up. We canzone talk about calculato in groms number. We talk about super Computers charning through algoritms to
find news digits. Pre-exysting paix solutions. Waiting in the invisibles structure of the paigenagers, are we trully creading this Numbers when we do the moth or we semply discovering locations on a mathus aree ritme on the sealing of the Universe, Discovery invention. It's the Eternal Questionmark of Mathematics, I think we. Sorti from enigment the maplice Orly. We're just finely building tune of Lights to see iTunes. The structure is always. There, you just have to know
exakry next. Next Time you look at the number. Three. Or five for eventus TEN. Remember the Ghost Numbers waiting Patient at the end. Other infinite hours lock at the minus One in a Square Plus One differenty for now. One One Problem there. The actional Keys to unlacking the Speed of Infinity there is always Deep. Structure. Thanks for taking this incredible. Deep 10.