You might like to look also at Arie Groeneveld's message,
[Jprogramming] Crash calculating large hyper operations
http://www.jsoftware.com/pipermail/programming/2015-November/043351.html
Unfortunately, running on the latest stable release,
JVERSION
Engine: j806/j64nonavx/windows
Release: commercial/2017-11-06T10:01:33
Library: 8.06.09
Qt IDE: 1.6.2/5.6.3
Platform: Win 64
Installer: J806 install
InstallPath: j:/program files/j
Contact: www.jsoftware.com
the following,
S=: 1 :0
:
if. 0= y do. 0
else. 'q r'=. (0,m) #: y
( r *(1+m) ^ (0 (m S) x)) + (1+x) m S q
end.
)
G=: (>:@[ $: _1+ 4 : '0 (x S)y')`[@.(0=])
2 G"0 i.4
now produces a crash ("J has stopped working") instead of,
2 3 5 7
As far as I can see, the code should run on J804 but I do not know if it
runs on J805 (or on the latest and greatest j807).
Anyway, in theory, 2 G"0 i.5 should include the additional number,
_1 + 3 * 2 ^ 402653211x
in practice, of course, it cannot (even trying to execute the sentence _1 +
3 * 2 ^ 402653211x produces a limit error).
On Wed, Apr 11, 2018 at 9:15 AM, 'Jon Hough' via Programming <
[email protected]> wrote:
> My first answer was actually comletely wrong, and only works for the
> simplest cases. This is a more robust and correct solution
>
> goodstein=: 4 : 0"0 0
> if. y = x do.
> x+1 return.
> elseif. y = 0 do.
> 0 return.
> end.
> s=. I. x (|.@:((>:@:>.@:^. # [) #: ])) y
> d=. (x+1) ^ (x:x) goodstein x: s
> +/d
> )
>
> G=: <:@:goodstein
>
> NB. generates sequence
> genSeq=: 3 : 0"1
> 'base val its'=. y
> c=. 0
> vals=. val
> whilst. its > c=. c+1 do.
> val=. base G val
> vals=. vals,val
> base=. base+1
> end.
> vals
> )
>
> genSeq 2 4 10
> 4 26 41 60 83 109 139 173 211 253 299
>
> genSeq 2 19 3
> 19 7625597484990 134078079299425970995740249982
> 058461274793658205923933777235614437217640300735469768018742
> 98166903427690031858186486050853753882811946569946433649006084099
> 191101259794547752035640455970396459919808104899009433713951
> 27892465205302426158030...
>
>
> --------------------------------------------
> On Wed, 4/11/18, 'Jon Hough' via Programming <[email protected]>
> wrote:
>
> Subject: [Jprogramming] Goodstein Sequences and Hereditary base-n notation
> To: "Programming Forum" <[email protected]>
> Date: Wednesday, April 11, 2018, 5:14 PM
>
> Goodstein's theorem: https://en.wikipedia.org/wiki/Goodstein%27s_theorem
> This states that every Goodstein
> sequence eventually terminates at 0.
> The wikipedia page defines Goodstein
> sequences in terms of Hereditary base-n notation
> one such sequence is 4,26,41,60...
>
> Copying verbatim from wikipedia:
> ==============
> The Goodstein sequence G(m) of a number
> m is a sequence of natural numbers. The first element in the
> sequence G(m) is m itself. To get the second, G(m)(2), write
> m in hereditary base-2 notation, change all the 2s to 3s,
> and then subtract 1 from the result. In general, the
> (n + 1)-st term G(m)(n + 1) of the Goodstein
> sequence of m is as follows:
>
> Take the hereditary base-n + 1
> representation of G(m)(n).
> Replace each occurrence of the
> base-n + 1 with n + 2.
> Subtract one. (Note that the next term
> depends both on the previous term and on the index n.)
> Continue until the result is zero, at
> which point the sequence terminates.
> ===============
>
> The sequences take an impossibly long
> time to terminate for most inputs, so there is no use
> writing a verb that iterates until convergance. This is my
> verb that will calculate the first N elements of the
> sequence,
> starting with a given base and val. the
> base should start at 2, to conform to the above definition.
>
>
> goodstein=: 3 : 0
> 'base val its'=. y
> vals=. ''
> c=: 0
> whilst.its> c=: c+1 do.
> if. val = 0 do. vals return.
> else.
> t=: ((1+
> >.base^.val)#base) #: val
> if. base < # t do.
> if. 0 < base {
> |.t do.
> tr=: 0
> (base}) |.t
> if.
> (base+1) < # tr do.
> ts=:
> (1+(base+1){tr) ((x+1)}) tr
> ts=:
> |.ts
> else.
> ts=:
> tr,1
> ts=:
> |.ts
> end.
> else. ts=: t end.
> else.
> ts=: t
> end.
> val=. <:(base+1) #.
> ts
> vals=. vals,val
> base=. base+1
> end.
> end.
> vals
> )
>
>
>
> NB. example
> goodstein 2 4 9
> 26 41 60 83 109 139 173 211 253
> NB. continues to very large numbers.
>
> goodstein 2 3 5
> 3 3 2 1 0 NB. terminates after 6
> iterations
>
> Is was hoping the goodstein verb could
> be defined tacitly, but my verb is clearly a bit of a mess.
> Just for fun, any elegant solutions?
>
> Thanks,
> Jon
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