First of all, i am not a math geek. That's why i am asking you math genies .

In school i invented a very simple equation that has a single solution. The equation is:

x ^ x = 100

I can approximate the value for x with a computer program, but i wonder if there is a mathematical solution for this. My teacher didn't know at that time.

1
4
27
256

So it's between 3 and 4.

Wouldn't it be CRAZY if it worked out to 3.1415...?!

Edit: pi^pi = ~35

DoubleEdit: 3.6^3.6 = 100.6blahblahblah, so it's HELLA close to 3.6

3.5973 is closer

Edit: I presume to do it mathmatically requires manipulation of logs.

I can't remember my log rules right now and I don't have time to google it and find out, sorry.

HughMcB wrote:3.5975 is closer

Well, 22/7 is closer to pi than 3.14 is. so there.

According to excel:
3.59728546389372000000000

babel wrote:According to excel:
3.59728546389372000000000

wow, cool.

I would start by taking the natural log from both sides

Mononoke wrote:I would start by taking the natural log from both sides

Indeed, and then you would have the equation x log(x) = log(100).

I doubt if the solution to this equation is rational, as the Excel solution appears to imply. I think Excel is simply using limited precision and giving an answer that exaggerates the true precision. I'll run it through Mathematica and see what happens.

(Two minutes later). Mathematica gives 50 decimal places as

3.5972850235404175054976522517822860691355430548866.

Let me know if you want a thousand or a million decimal places. I'm guessing that I can prove fairly easily that x isn't rational. And people who know number theory well can probably invoke a standard theorem to prove that x is transcendental.

Wait a minute...it's coming back to me. If a is algebraic and not zero and b is an irrational algebraic number, then a^b is transcendental. So, either x is rational, or it's transcendental. It can't be an irrational algebraic number. OK, so now all I have to do is prove that it's not rational. I'll let you know how that goes.

Roger Cooke wrote:

Mononoke wrote:I would start by taking the natural log from both sides

Indeed, and then you would have the equation x log(x) = log(100).

I doubt if the solution to this equation is rational, as the Excel solution appears to imply. I think Excel is simply using limited precision and giving an answer that exaggerates the true precision. I'll run it through Mathematica and see what happens.

(Two minutes later). Mathematica gives 50 decimal places as

3.5972850235404175054976522517822860691355430548866.

Let me know if you want a thousand or a million decimal places. I'm guessing that I can prove fairly easily that x isn't rational. And people who know number theory well can probably invoke one of Vinogradov's theorems to prove that x is transcendental.

Awwww, I thought it was cool that it suddenly dropped to a bunch of zeros.

CURSE YOU EXCEL!!!!!!

Cool, can you please tell me what the millionth digit is?

num1cubfn wrote:

Roger Cooke wrote:

Mononoke wrote:I would start by taking the natural log from both sides

Indeed, and then you would have the equation x log(x) = log(100).

I doubt if the solution to this equation is rational, as the Excel solution appears to imply. I think Excel is simply using limited precision and giving an answer that exaggerates the true precision. I'll run it through Mathematica and see what happens.

(Two minutes later). Mathematica gives 50 decimal places as

3.5972850235404175054976522517822860691355430548866.

Let me know if you want a thousand or a million decimal places. I'm guessing that I can prove fairly easily that x isn't rational. And people who know number theory well can probably invoke one of Vinogradov's theorems to prove that x is transcendental.

Awwww, I thought it was cool that it suddenly dropped to a bunch of zeros.

CURSE YOU EXCEL!!!!!!

Cool, can you please tell me what the millionth digit is?

I finally aborted the evaluation after Mathematica ran for 10 minutes. It would eventually get the answer, but I'm not sure how long it would take. I'll let it run overnight tonight and see what it produces.

Meanwhile, if you want to compute the number yourself, I have found it easy to do. Since this is a convex function, Newton's method is guaranteed to work. I programmed it in Mathematica and had it run 10 steps of the approximation with 1000 digits of accuracy. Since the last two steps were identical, I'm assuming I actually have 1000-place accuracy here. Here's what the input and output looked like:

f[x_]:=(x+Log[100])/(1+Log[x])

Table[N[Nest[f,7/2,k],1000],{k,1,10}]

{3.597879714527434874419322351070673233760002738054671459168194193265202137470008234490837549130587617519958354336843152769674582024289134126366297240420590710425481510271362235671059928940825156156347415477393592735605592680497903208995157477269061456981969612866232060903613749397423272194997022212704451995696293277422154920745064182959534436928918879925354621486342761226470846901298873276879846307573929057083865973020917892092813818262653847048979267473239267321921106136482240169633218672327152178094156832197011014487632696953625135986239195715186083978092751036191188438015530812406382904740841131208826148585359179994947299722580151059179894145040766778848905252010982278805627063044103590641280534041144985198923522345039941085328669258274760420892733816918577019178914144210058654266976131451014303983777973857085972414705421513525356335076448756838046654485937290463730320180286954269306184485612325783503267087010263002159801427044877252613951985218779739412123787965683936843042618280740,
3.597285045094498263223757661882020536815158568443951645929620889874673879520693007258821845742552320356447221211252551620255081991924843794350373134543095737139908302410835032383245223410922965704850076581299209213289573119362118343747839535417063340255733986412168085761865892219430385078158551252710084803442878523561456990539250470123218521505957617948965248921241051711967113602142108378373584597187030655632497449215170042237555946525278265295710079051689330025410186049650555893791117453871506892793925917409783252515279037393790018129229049414672170100219038957135149564187578563585291381191480355922805324484929207939832799835521774504255975635194434946783989833662601469980753348015951122888504427090664622430695376271296244083011218433613087023503391450506822339908529947063830019875359601505625146113346295444949576283487534132071698276888031982126204870046851066928013758531032674966157828641141222109183355159133188417250120522137699195356799032644654646396813418842792059859598057300491,
3.597285023540417533817123822582891478097198507933629058291394533673075764259830332505774726444814016930952076334328052659942603859395862805223923852481207472421038232510522599980671234026592198804959069689774579625633787561469152046877599647617239995511443577326318056881995659979806666044321157132976905245070029213240086534200237119415826179377420946016233473465744479226778267762122844788038265216090397686605762702353136347356527334742937125734151138248151331955542572220294911999934677312924275021653541280623575820827175975916461721806278818844875934134515960450279959524711229341696231745317814053873167632039801787322673820322981800475869056669179542954620246285369451810120750441297766656371385691616526905631337494859365319912453861081242675818108282246437870748802722177360165250003894505421363587468443118294822319562510449624778152525198250178486496842503040219528071816171339000237705157980371216633643692039259335568535729686823421103835696930557646896944512158601684799406765072063236,
3.597285023540417505497652251782286118022885293448071975498356757394646441099209888867706681516367249026755244034574647589375437428509412539588917661525146534089666229958307064287186510773944180137072727937750640080618888756477289809035913107405569113134727463074705343843297908113617261735926618305472039650930848375396575024700856229991441391549692075017978537336652653261166320363050978433700337260717270948821378225581624339880746892223449809585917477640957709941500364700792478931926594410552131954081041453338253993148864678287532188114059269071267252859362088964191036318712781078422378570407429909781250528509619229356083748784952780691608263233374626304495622301442580752880306053751946262361478109984319881901908206106890949831813544145402454088801376134161314820348006347797257083635694148698197482425855853617846279327104889096979395032949379811214394891978854487130088834567199793880491987732891829043085356032408396913663047721682211518061270448478173644484876984180310144611917930163977,
3.597285023540417505497652251782286069135543054886576783720252127972957653242367218909117506138019309787699075175737888060981171759542249685920746259333583642571977612222184426363562408994037086598756787780555280577054872047696242596013673688933149506660361112603686751356857548443528844198878930502277349282502820319751544838065494868735538975801377281703115354295129550504206555421288137065321379648687820387719624291530085482098781309396575220331927612327393643588456904813338356549215905699726362914202592707637900053391624276817452525428717857877575525023648257098333996320243343828968741424410355500384246291967497238810035617978188779827516761848421699766324354393817326746216890579418377264560348085970930773019613554498322217346052540595247406324306257460890541411381047839162251893950316354791027769767839166239560515585527824750997440235577848799034586115974278650623801359769267685457896776036408593799689021049070512575505077198744907879138302170241907403296239383935127408212301774637851,
3.597285023540417505497652251782286069135543054886576783720252127972957507555973931818035224037059178330204976748439827655104077645025252558986485154257375407589700778316215797508001047976522480317713806344377028505509971869044870199749218918620566342457273648914828988476203838911936114782907546415347888459307637236170570040087886854641667906688531308101054774703302343229145963607299371655512196128601729276853487059227876487450137297356505885673907526716488648504529775513403137496875865038189213401112398982525894639748679809722008261479876445202252265766342087361486383823997321564215331776659788118324324232225397673741152837629197644022508102091929865110973671915127699729214950226446241567946836555579985811833965397732687270614654375288108807942634285848706951345948448048120216665314529575483515916493409537614384160894970494451295072395095125095841108376157946600945379840092140230740598647399655726757759549369148877202397435762850582777652803703491637999127438687972936010356847335680370,
3.597285023540417505497652251782286069135543054886576783720252127972957507555973931818035224037059178330204976748439827655104077645025252558985191363273649877468862127787139218891217674698017207725879103061299416755295482258129215690094787197199355531366322623109200371547499618906616530070232102394251442860106639068110454636178587217989641576371757673675807442441690827891776005695491402140757000046787995799530828950802713362379899163485131393899063187249445280947027599545384541731846329113406758324466034452084213941027198633011295208673357439308162380960104584397876429540110574743559456830211747100000029851176663991131394068244877627465300460028865474375169754552055569440495860109595624073488106665485302428516577405633396397626788261422868816733453533993168185695811238514933909317914228013608496346367517406992099115987411063383482610339887297275305300111321296094629838891272220595703078968448948635711506991961143260251799986922740847983053735670810046421509516299703732795746545584499346,
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Oops, looks like it doesn't all display. However, if you quote this post, you'll be able to see what the output was. The thousandth digit is 4 (possibly rounded up, however).

Cali just showed a paper on how to calculate the millionth digit WITHOUT calculating the first 999,999 digits. But I'm not a math wizard or I'd say how. I thought maybe you could do it that way. Oh well

num1cubfn wrote:Cali just showed a paper on how to calculate the millionth digit WITHOUT calculating the first 999,999 digits. But I'm not a math wizard or I'd say how. I thought maybe you could do it that way. Oh well

Likely. I'm back, having just proved that x is irrational. Turns out to be easy. The assumption that x = m/n in lowest terms leads to the equation

m^m = 100^n n^m,

which contradicts the unique factorizability of integers into products of primes. (The right-hand side is divisible by 5^(2n) but not by 5^(2n+1), which means the left-hand side is divisible by 5^m. Since m > 3n, this is a contradiction.)

Hence, from what I said above, x is a transcendental number. That's probably why it's so easy to approximate quickly with Newton's algorithm.

OK, main questions exhausted. Now, is x a Liouville number? Probably, because, as I said, it's easy to approximate with rational numbers.

ETA: No, on second thought, the rapidity with which Newton's method converges is irrelevant, since it would converge equally quickly at a number that is hard to approximate with rationals, like the Golden number, which is algebraic.

okay this has piqued my curiosity. Wonder if there is a reasonable way to solve this analytically as opposed to computationally.

Mononoke wrote:okay this has piqued my curiosity. Wonder if there is a reasonable way to solve this analytically as opposed to computationally.

Not with algebra. The function x^x isn't an algebraic function of x. It is, however, monotonically increasing on the interval from 1/e to infinity, so it has a unique inverse on that interval. What usually happens in that case is that a new name is given to the function. But it's not even an elementary function or the integral of an elementary function, so the new name would be just a shorthand telling us nothing new.

By the way, I had Mathematica run the numerical solution again, asking for 1,000,001 digits (so as to include the 3 ahead of the decimal point). The last two digits were 30, so the millionth digit after the decimal point is 0 (and obviously it couldn't have been rounded up). It took about 15 minutes to run, so I was simply too impatient yesterday.

Mononoke wrote:okay this has piqued my curiosity. Wonder if there is a reasonable way to solve this analytically as opposed to computationally.

Roger's proof is impressive, simple and does not appeal to computation. What's the problem?

EDIT: Just occurred to me. Are you asking why transcendental numbers cannot be derived algebraically? That's a contradiction in terms, and has already be answered. Maybe you meant something else.

Roger Cooke wrote:which contradicts the unique factorizability of integers into products of primes. (The right-hand side is divisible by 5^(2n) but not by 5^(2n+1), which means the left-hand side is divisible by 5^m. Since m > 3n, this is a contradiction.)

Sorry Roger, I'm a bit tired (I know you posted this days ago). I followed this up to "Since m > 3n" - did you derive that, or is it just observed from the approximation for x above (since x > 3)?

Edit: Ahh never mind. Stupid question, it's quite obvious x > 3 after all...

Cynergy wrote:

Mononoke wrote:okay this has piqued my curiosity. Wonder if there is a reasonable way to solve this analytically as opposed to computationally.

Roger's proof is impressive, simple and does not appeal to computation. What's the problem?

EDIT: Just occurred to me. Are you asking why transcendental numbers cannot be derived algebraically? That's a contradiction in terms, and has already be answered. Maybe you meant something else.

I was thinking about solving X^X=100.

Mononoke wrote:

Cynergy wrote:

Mononoke wrote:okay this has piqued my curiosity. Wonder if there is a reasonable way to solve this analytically as opposed to computationally.

Roger's proof is impressive, simple and does not appeal to computation. What's the problem?

EDIT: Just occurred to me. Are you asking why transcendental numbers cannot be derived algebraically? That's a contradiction in terms, and has already be answered. Maybe you meant something else.

I was thinking about solving X^X=100.

Yes, and that's been done (above). x = 3.598 (to 3 d.p.). See above for values of increasing precision.

Cynergy wrote:

Mononoke wrote:

Cynergy wrote:

Mononoke wrote:okay this has piqued my curiosity. Wonder if there is a reasonable way to solve this analytically as opposed to computationally.

Roger's proof is impressive, simple and does not appeal to computation. What's the problem?

EDIT: Just occurred to me. Are you asking why transcendental numbers cannot be derived algebraically? That's a contradiction in terms, and has already be answered. Maybe you meant something else.

I was thinking about solving X^X=100.

Yes, and that's been done (above). x = 3.598 (to 3 d.p.). See above for values of increasing precision.

no no you're not getting it. I'm not talking about computationally solving it. that's pretty simple. MATLAB will do it with a couple of lines. I was talking about an analytical solution. The obvious problem there is that X^X does not have a standard polynomial representation. So it's transcendental function, hense a little impervious to algebra.

no no you're not getting it. I'm not talking about computationally solving it. that's pretty simple. MATLAB will do it with a couple of lines. I was talking about an analytical solution. The obvious problem there is that X^X does not have a standard polynomial representation. So it's transcendental function, hense a little impervious to algebra.

. Yeah. You got it.