Taxicab numbers: Difference between revisions

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(lowed the extra credit taxicab number.)
(→‎{{header|REXX}}: updated the taxicab numbers for the extra credit portion of the task.)
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'''output'''   using the default input:
'''output'''   using the default input:
<pre>
<pre>
1 1729 ────► 10**3 + 9**3, 12**3 + 1**3
1 1729 10**3 + 9**3, 12**3 + 1**3
2 4104 ────► 15**3 + 9**3, 16**3 + 2**3
2 4104 15**3 + 9**3, 16**3 + 2**3
3 13832 ────► 20**3 + 18**3, 24**3 + 2**3
3 13832 20**3 + 18**3, 24**3 + 2**3
4 20683 ────► 24**3 + 19**3, 27**3 + 10**3
4 20683 24**3 + 19**3, 27**3 + 10**3
5 32832 ────► 30**3 + 18**3, 32**3 + 4**3
5 32832 30**3 + 18**3, 32**3 + 4**3
6 39312 ────► 33**3 + 15**3, 34**3 + 2**3
6 39312 33**3 + 15**3, 34**3 + 2**3
7 40033 ────► 33**3 + 16**3, 34**3 + 9**3
7 40033 33**3 + 16**3, 34**3 + 9**3
8 46683 ────► 30**3 + 27**3, 36**3 + 3**3
8 46683 30**3 + 27**3, 36**3 + 3**3
9 64232 ────► 36**3 + 26**3, 39**3 + 17**3
9 64232 36**3 + 26**3, 39**3 + 17**3
10 65728 ────► 33**3 + 31**3, 40**3 + 12**3
10 65728 33**3 + 31**3, 40**3 + 12**3
11 110656 ────► 40**3 + 36**3, 48**3 + 4**3
11 110656 40**3 + 36**3, 48**3 + 4**3
12 110808 ────► 45**3 + 27**3, 48**3 + 6**3
12 110808 45**3 + 27**3, 48**3 + 6**3
13 134379 ────► 43**3 + 38**3, 51**3 + 12**3
13 134379 43**3 + 38**3, 51**3 + 12**3
14 149389 ────► 50**3 + 29**3, 53**3 + 8**3
14 149389 50**3 + 29**3, 53**3 + 8**3
15 165464 ────► 48**3 + 38**3, 54**3 + 20**3
15 165464 48**3 + 38**3, 54**3 + 20**3
16 171288 ────► 54**3 + 24**3, 55**3 + 17**3
16 171288 54**3 + 24**3, 55**3 + 17**3
17 195841 ────► 57**3 + 22**3, 58**3 + 9**3
17 195841 57**3 + 22**3, 58**3 + 9**3
18 216027 ────► 59**3 + 22**3, 60**3 + 3**3
18 216027 59**3 + 22**3, 60**3 + 3**3
19 216125 ────► 50**3 + 45**3, 60**3 + 5**3
19 216125 50**3 + 45**3, 60**3 + 5**3
20 262656 ────► 60**3 + 36**3, 64**3 + 8**3
20 262656 60**3 + 36**3, 64**3 + 8**3
21 314496 ────► 66**3 + 30**3, 68**3 + 4**3
21 314496 66**3 + 30**3, 68**3 + 4**3
22 320264 ────► 66**3 + 32**3, 68**3 + 18**3
22 320264 66**3 + 32**3, 68**3 + 18**3
23 327763 ────► 58**3 + 51**3, 67**3 + 30**3
23 327763 58**3 + 51**3, 67**3 + 30**3
24 373464 ────► 60**3 + 54**3, 72**3 + 6**3
24 373464 60**3 + 54**3, 72**3 + 6**3
25 402597 ────► 61**3 + 56**3, 69**3 + 42**3
25 402597 61**3 + 56**3, 69**3 + 42**3


4000 365841702037 5806**3 + 5541**3, 6274**3 + 4917**3
2000 1802609991 1002**3 + 927**3, 1056**3 + 855**3
4001 381281044228 6137**3 + 5315**3, 6581**3 + 4583**3
2001 1836207171 1134**3 + 723**3, 1155**3 + 666**3
4002 389845801000 6340**3 + 5130**3, 6555**3 + 4765**3
2002 1876948136 1050**3 + 896**3, 1148**3 + 714**3
4003 403718156352 6244**3 + 5432**3, 6478**3 + 5090**3
2003 1898380125 1090**3 + 845**3, 1125**3 + 780**3
4004 423414336247 6119**3 + 5792**3, 6230**3 + 5663**3
2004 1948631048 1010**3 + 972**3, 1154**3 + 744**3
4005 435912538698 6353**3 + 5641**3, 6547**3 + 5375**3
2005 2027722248 1074**3 + 924**3, 1153**3 + 791**3
4006 438857254017 6217**3 + 5834**3, 6462**3 + 5529**3
2006 2080128456 1058**3 + 964**3, 1136**3 + 850**3
4007 476650386296 6353**3 + 6039**3, 6524**3 + 5838**3
2007 2134551608 1052**3 + 990**3, 1124**3 + 894**3
</pre>
</pre>

Revision as of 06:19, 14 March 2014

Template:Draft

definition

A taxicab number (the definition that is being used here) is a positive integer that can be expressed as the sum of two positive cubes in more than one way.

The first taxicab number is   1729,   which is:

13   +   123       and
93   +   103.

Taxicab numbers are also known as:

  • taxi numbers
  • taxicab numbers
  • taxi cab numbers
  • Hardy-Ramanujan numbers

task requirements

The task requirements are:

  • compute the lowest 25 taxicab numbers.
  • exactly 25 taxicab numbers are to be shown (in numeric order).
  • show the taxicab numbers as well as its constituent cubes.
  • show all numbers in a very readable (aligned) format.
  • show the 2,000th taxicab number + a half dozen more (extra credit)).

See also



J

<lang J> 25 {. ;({."#. <@(0&#`{.@.(1<#))/. ])/:~~.,/(+,/:~@,)"0/~3^~1+i.100 

 1729     1   1728
 4104     8   4096
13832     8  13824
20683  1000  19683
32832    64  32768
39312     8  39304
40033   729  39304
46683    27  46656
64232  4913  59319
65728  1728  64000

110656 64 110592 110808 216 110592 134379 1728 132651 149389 512 148877 165464 8000 157464 171288 4913 166375 195841 729 195112 216027 27 216000 216125 125 216000 262656 512 262144 314496 64 314432 320264 5832 314432 327763 27000 300763 373464 216 373248 402597 74088 328509</lang>

Explanation:

First, generate 100 cubes.

Then, form a 3 column table of unique rows: sum, small cube, large cube

Then, gather rows where the first entry is the same. Keep the ones with at least two such entries.

Note that the cube root of the 25th entry is slightly smaller than 74, so testing against the first 100 cubes is more than sufficient.

REXX

<lang rexx>/*REXX program displays the first (lowest) taxicab numbers. */ parse arg L1 H1 L2 H2 . /*obtain the optional numbers. */ if L1== | L1==',' then L1=1 /*L1 " " " " " */ if H1== | H1==',' then H1=25 /*H1 " " " " " */ if L2== | L2==',' then L2=4000 /*L2 " " " " " */ if H2== | H2==',' then H2=4007 /*H2 " " " " " */ mx=max(L1, H1, L2, H2) /*find how many taxicab #s needed*/ w=length(mx) /*width is used formatting output*/ numeric digits 20 /*prepare to use larger numbers. */

  1. =0; @.=0; $.=; $=; b='■'; t='**3' /*initialize some REXX variables.*/
                                      /* [↓]  generate extra taxicab #s*/
do j=1   until  zz==mx                /*taxicab nums won't be in order.*/
jjj=j**3                              /*might as well calculate a cube.*/
z=;      do k=1  for j-1; s=jjj+k**3  /*define a whole bunch of cubes. */

         if @.s==0  then do           /*if cube not defined, then do it*/
                         @.s = "────►"right(j,6,b)t"■■■+"right(k,6,b)t
                         iterate      /* ··· and then go and do another*/
                         end          /* [↑] define one cube at a time.*/
         z=s                          /*save cube for  taxicab #  list.*/
         @.s=@.s","right(j,9,b)t"■■■+"right(k,6,b)t   /*build the text.*/
         $.s=right(s,15,b)'■■■'@.s    /*define the rest of taxicab #s. */
         $=$ z                        /*define a   #.   taxicab number.*/
         zz=words($)                  /*count the number of taxicab #s.*/
         end   /*k*/                  /* [↑]  build cubes one-at-a-time*/
end   /*j*/                           /* [↑]  complete with overage #s.*/
                                      /*sort sub builds array, sorts it*/

list=esort(zz) /*sort taxicab nums, create list.*/ 

                                      /* [↓]  list  N  taxicab numbers.*/
    do j=L1  to H1;    _=word(list,j) /*get one taxicab num at a time. */
    say right(j,w)  translate($._,,b) /*display taxicab # (with blanks)*/
    end   /*j*/                       /* [↑] ■ are translated to blanks*/

say 

    do j=L2  to H2;    _=word(list,j) /*get one taxicab num at a time. */
    say right(j,w)  translate($._,,b) /*display taxicab # (with blanks)*/
    end   /*j*/                       /* [↑] ■ are translated to blanks*/

exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────ESORT subroutine────────────────────*/ esort: procedure expose #. $; parse arg N 1 h a 

      do j=1  for N;   #.j=word($,j);  end   /*j*/
      do  while  h>1;  h=h%2;          do i=1  for N-h; j=i; k=h+i
      do  while  #.k<#.j;     t=#.j;   #.j=#.k;  #.k=t
          if h>=j  then leave;  j=j-h;   k=k-h;  end;  end;  end;
      do m=1  for  N;  a=a #.m;  end  /*m*/;                     return a</lang>

output   using the default input: 

 
   1            1729            10**3   +     9**3,       12**3   +     1**3
   2            4104            15**3   +     9**3,       16**3   +     2**3
   3           13832            20**3   +    18**3,       24**3   +     2**3
   4           20683            24**3   +    19**3,       27**3   +    10**3
   5           32832            30**3   +    18**3,       32**3   +     4**3
   6           39312            33**3   +    15**3,       34**3   +     2**3
   7           40033            33**3   +    16**3,       34**3   +     9**3
   8           46683            30**3   +    27**3,       36**3   +     3**3
   9           64232            36**3   +    26**3,       39**3   +    17**3
  10           65728            33**3   +    31**3,       40**3   +    12**3
  11          110656            40**3   +    36**3,       48**3   +     4**3
  12          110808            45**3   +    27**3,       48**3   +     6**3
  13          134379            43**3   +    38**3,       51**3   +    12**3
  14          149389            50**3   +    29**3,       53**3   +     8**3
  15          165464            48**3   +    38**3,       54**3   +    20**3
  16          171288            54**3   +    24**3,       55**3   +    17**3
  17          195841            57**3   +    22**3,       58**3   +     9**3
  18          216027            59**3   +    22**3,       60**3   +     3**3
  19          216125            50**3   +    45**3,       60**3   +     5**3
  20          262656            60**3   +    36**3,       64**3   +     8**3
  21          314496            66**3   +    30**3,       68**3   +     4**3
  22          320264            66**3   +    32**3,       68**3   +    18**3
  23          327763            58**3   +    51**3,       67**3   +    30**3
  24          373464            60**3   +    54**3,       72**3   +     6**3
  25          402597            61**3   +    56**3,       69**3   +    42**3

2000      1802609991          1002**3   +   927**3,     1056**3   +   855**3
2001      1836207171          1134**3   +   723**3,     1155**3   +   666**3
2002      1876948136          1050**3   +   896**3,     1148**3   +   714**3
2003      1898380125          1090**3   +   845**3,     1125**3   +   780**3
2004      1948631048          1010**3   +   972**3,     1154**3   +   744**3
2005      2027722248          1074**3   +   924**3,     1153**3   +   791**3
2006      2080128456          1058**3   +   964**3,     1136**3   +   850**3
2007      2134551608          1052**3   +   990**3,     1124**3   +   894**3
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