9^(9^9)

[LaurV]

Originally Posted by Batalov

9!^9!^9! seems to be slightly larger.

(9!^9!^9!)!

[WraithX]

Originally Posted by LaurV

(9!^9!^9!)!

Or the still slightly larger:

(9! 9! 9!)!

Using Knuth's up arrow notation as described here

[Batalov]

I am channelling V.I."electron is as inexhaustible as the atom, nature is infinite"Lenin who would remark that one "9" is all you need:
9!!!!!!!!!!!!!!!!!!!!!!...!!!!!!!!!!!!!!!!!!!!!!!!!!...!!!!!!!!!!!!!!!!!!!

[retina]

Originally Posted by Batalov

I am channelling V.I.Lenin ("electron is as inexaustible as the universe") who would remark that one "9" is all you need:
9!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

I think you will find you need brackets for that: eg. ((((9!)!)!)!)! (and other larger variants that I am too lazy to type in).

[LaurV]

Originally Posted by Batalov

I am channelling V.I."electron is as inexhaustible as the atom, nature is infinite"Lenin who would remark that one "9" is all you need:
9!!!!!!!!!!!!!!!!!!!!!!...!!!!!!!!!!!!!!!!!!!!!!!!!!...!!!!!!!!!!!!!!!!!!!

well... following the rule that the !! is not double factorial, but semifactorial, I imagine your Lenin's... stuff is quite small... :gotcha:

[Batalov]

What rule?

If the rule you followed brought you to this, of what use was the rule?

GP> 
? 3!!
%1 = 720
? 3!!!
%2 = 2601218943565795100204903227081043611191521875016945785727541837850835631156947382240678577958130457082619920575892247259536641565162052015873791984587740832529105244690388811884123764341191951045505346658616243271940197113909845536727278537099345629855586719369774070003700430783758997420676784016967207846280629229032107161669867260548988445514257193985499448939594496064045132362140265986193073249369770477606067680670176491669403034819961881455625195592566918830825514942947596537274845624628824234526597789737740896466553992435928786212515967483220976029505696699927284670563747137533019248313587076125412683415860129447566011455420749589952563543068288634631084965650682771552996256790845235702552186222358130016700834523443236821935793184701956510729781804354173890560727428048583995919729021726612291298420516067579036232337699453964191475175567557695392233803056825308599977441675784352815913461340394604901269542028838347101363733824484506660093348484440711931292537694657354337375724772230181534032647177531984537341478674327048457983786618703257405938924215709695994630557521063203263493209220738320923356309923267504401701760572026010829288042335606643089888710297380797578013056049576342838683057190662205291174822510536697756603029574043387983471518552602805333866357139101046336419769097397432285994219837046979109956303389604675889865795711176566670039156748153115943980043625399399731203066490601325311304719028898491856203766669164468791125249193754425845895000311561682974304641142538074897281723375955380661719801404677935614793635266265683339509760000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

[Raman]

Originally Posted by CRGreathouse

Puzzles, probably. Better fit than Math or Homework Help.

I was supposed to put it into the Programming forum
The main intention was being to make the program to store the 369693100 digit number into the text file.
I didn't post there due to the fact that most of the library was being used from the apfloat.org, I did few minor modifications, changes

http://apfloat.org/apfloat_java/applet/calculator.html

http://mathforum.org/library/drmath/view/58235.html
http://mathforum.org/library/drmath/view/59172.html
http://mathforum.org/library/drmath/view/61451.html

Originally Posted by LaurV

... and what do you need the parenthesis for?
it should be the same 9^9^9 and you save two keypresses

Bitchy LaurV

No, it is not. Don't confuse yourself.
9^9[SUP]9[/SUP] is not being the same as (9⁹)⁹ at all, as such
the latter is being merely 9⁸¹ = 3¹⁶² =
196627050475552913618075908526912116283103450944214766927315415537966391196809

9^9^9 is rather being quite ambiguous, in my opinion

Originally Posted by Batalov

9!^9!^9! seems to be slightly larger.

Tuch: What's the largest number that can be made with 3 threes?
Tay: 3^3[SUP]3[/SUP] = 3²⁷ = 7625597484987
Tuch: Wrong. 3³³ = 5559060566555523 is being much more greater
Tay: What about it when we can add one more factorial?
Tuch: We could add two factorials to it at the end, as such
Tay: We could add any number of factorials, in order to make the result to be diverging to infinity
Tuch: What's the largest number that can be made with 4 ones?
Tay: 11¹¹ = 285311670611
Tuch: If I ask the largest number, then it is ambiguous, not being well defined at all, I will rather ask something to make it into a fixed thing, as such
Tay: Yes, please go ahead
Tuch: How do you make 127 by using the digits 1, 2, 7
Tay: A = -1+2⁷ digits in the same order, as such!
Tuch: How do you make 127 by using the digits 1, 2, 7 twice
Tay: B = (71*2)-(17-2)
Tuch: How do you make 127 by using the digits 1, 2, 7 n times
Tay: n = 1 mod 2 => (n-1)/2 A - (n-1)/2 A + A
n = 0 mod 2 => (n-2)/2 A - (n-2)/2 A + B

[Dubslow]

Originally Posted by Raman

9^9^9 is rather being quite ambiguous, in my opinion

It's a fact that the ^ is right associative, so 9^9^9 is un-ambiguously the same as 9^(9^9).
http://en.wikipedia.org/wiki/Exponentiation

Exponentiation is not associative either. Addition and multiplication are. For example, (2+3)+4 = 2+(3+4) = 9 and (2·3)·4 = 2·(3·4) = 24, but 2^3 to the 4 is 8^4 or 4096, whereas 2 to the 3^4 is 2^81 or 2,417,851,639,229,258,349,412,352. Without parentheses to modify the order of calculation, by convention the order is top-down, not bottom-up:

[Raman]

Originally Posted by Dubslow

It's a fact that the ^ is right associative, so 9^9^9 is un-ambiguously the same as 9^(9^9).
http://en.wikipedia.org/wiki/Exponentiation

It is being certainly a different thing when you write it 9^9[SUP]9[/SUP], (9⁹)^[SUP]9[/SUP]

Try typing out, feeding off 9^9^9 into FX-4800P calculator and then see what it says rather

Originally Posted by Dubslow

Exponentiation is not associative either. Addition and multiplication are. For example, (2+3)+4 = 2+(3+4) = 9 and (2·3)·4 = 2·(3·4) = 24, but 23 to the 4 is 84 or 4096, whereas 2 to the 34 is 281 or 2,417,851,639,229,258,349,412,352. Without parentheses to modify the order of calculation, by convention the order is top-down, not bottom-up:

Certainly, the exponentiation notation is being missing clearly
You should have to be meant

Exponentiation is not associative either. Addition and multiplication  are. For example, (2+3)+4 = 2+(3+4) = 9 and (2·3)·4 = 2·(3·4) = 24, but  23 to the 4 is 84 or 4096, whereas 2 to the 34 is 281 or  2417851639229258349412352. Without parentheses to modify the  order of calculation, by convention the order is top-down, not  bottom-up:

[LaurV]

Originally Posted by Raman

It is being certainly a different thing when you write it 9^9[SUP]9[/SUP], (9⁹)^[SUP]9[/SUP]

Certainly, 9^9^9 is different of 3^457. And my dog is different of my piano, may both have tails, but one has 4 legs and one has only 3. And we weren't talking about that. We were telling apart 9^9^9, the CORRECT writing, from the 9^(9^9) as it appears in the thread title and the first post, which is computationally identical (i.e. it leads to the SAME number when it is expanded, WITHOUT any ambiguity) and insults the intelligence of the reader, implying he is so stupid he doesn't know how to associate the exponentiation...

My former post was a joke, but if you insist...

[m_f_h]

Well, in a case like this one, it is not *so* bad to write 9^(9^9) to make things 100% unambiguous. (Though I concede that a^b^c should mean a^(b^c), in particular because (a^b)^c could (and thus should and would) at no more cost rather be written as a^(bc), while there's no such simplification for a^(b^c).

OTOH, calculation of 9^9^9 is indeed no particular exploit, /any/ arbitrary precision calculator can do this [provided enough memory], and about any computer algebra package will calculate the last N digits of this in a fraction of a second (via binary exponentiation in Z/nZ, n=10^N) (and even faster the first N digits, for any arbitrary precision calculator).

(It is more challenging to implement a powmod(x,m) function (almost never available by default), which would (efficiently and accurately) calculate
x[1]^....^x[n] mod m
for any n-component (integer) vector x.)

And obviously "the largest number which can be written using 3 digits" depends on the operations you allow between these digits, as well as the digits you allow. It has been mentioned that we are to consider the decimal representation of the result, but I did not see any specification about the numbering system used for the "input" - e.g. z^z^z is quite a bit larger, using three times the largest base-36 digit.