1.Introduction

We show integer solutions of x^3+dy^3+d^2z^3-3dxyz = 1.
This equation is related to unit of cubic field as follows.

Norm(x + d^(1/3)y + d^(2/3)z) = x^3+dy^3+d^2z^3-3dxyz = 1.

Hence, to obtain the unit of x + d^(1/3)y + d^(2/3)z, we have to solve for x^3+dy^3+d^2z^3-3dxyz = 1.

We used the lattice reduction method to obtain integer solutions for x^3+dy^3+d^2z^3-3dxyz = 1.


2. Result 

   
    
[  d                           x                          y                          z]

[  2                           1                          1                          1]
[  3                          -2                          0                          1]
[  4                           1                          1                         -1]
[  5                           1                         -4                          2]
[  6                           1                         -6                          3]
[  7                           2                         -1                          0]
[  9                          -2                          1                          0]
[ 10                           1                          6                         -3]
[ 11                           1                          4                         -2]
[ 12                        -107                         33                          6]
[ 13                          -4                         -3                          2]
[ 14                           1                          2                         -1]
[ 15                           1                        -30                         12]
[ 16                           1                         50                        -20]
[ 17                          18                         -7                          0]
[ 18                           1                         -3                          1]
[ 19                          -8                          3                          0]
[ 20                         -19                          7                          0]
[ 21                         -47                          6                          4]
[ 22                          23                          3                         -4]
[ 23                      -41399                      -3160                       6230]
[ 24                           1                         -9                          3]
[ 25                       70001                      13850                     -12924]
[ 26                           3                         -1                          0]
[ 28                          -3                          1                          0]
[ 29                  -322461439                  103819462                     370284]
[ 30                           1                          9                         -3]
[ 31                        -367                         54                         20]
[ 32                      -31999                       4920                       1625]
[ 33                     3742201                      97392                    -394098]
[ 34                         613                        -24                        -51]
[ 35                         -24                        -45                         16]
[ 36                           1                          3                         -1]
[ 37                          10                         -3                          0]
[ 38                        -151                         55                         -3]
[ 39                         -23                          0                          2]
[ 40                         -79                          6                          5]
[ 41               -211991370839               305478475184               -70761183382]
[ 42                           1                        -42                         12]
[ 43                          -7                          2                          0]
[ 44                        1585                        303                       -213]
[ 45                        1081                         66                       -104]
[ 46                       -4139                         48                        309]
[ 47                     -592199                     -69704                      64786]
[ 48                       67393                      -3738                      -4074]
[ 49                       -3106                         84                        209]
[ 50                        4001                       -795                        -79]
[ 51                      -11015                       2592                        102]
[ 52                           1                         -4                          1]
[ 53                     -344340                       4517                      23202]
[ 54                        -161                        -33                         20]
[ 55                     6597361                   -1012254                    -189996]
[ 56                       -1007                        114                         39]
[ 57                        1084                         57                        -88]
[ 58                           1                         -8                          2]
[ 59                 46334227393               -42285555004                 7804684934]
[ 60                           1                        -12                          3]
[ 61                           1                        -16                          4]
[ 62                           1                        -24                          6]
[ 63                           4                         -1                          0]
[ 65                          -4                          1                          0]
[ 66                           1                         24                         -6]
[ 67                           1                         16                         -4]
[ 68                           1                         12                         -3]
[ 69     13753611475894008059401    -5630668308465438120720      555253697459615284770]
[ 70                           1                          8                         -2]
[ 71                  1386226224                 -309672529                   -6061266]
[ 72                       -1295                        174                         33]
[ 73                         154                        -87                         12]
[ 74                        -961                        -23                         60]
[ 75          -26202397175198999           4869795112154880            318574403874066]
[ 76                           1                          4                         -1]
[ 77                   -40232807                   -7113592                    3894986]
[ 78                    -2134079                     841944                     -80154]
[ 79                         292                         95                        -38]
[ 80                      190081                     -32118                      -2784]
[ 81                       -5831                      12078                      -2480]
[ 82               1039923106201               874904383770              -256479590595]
[ 83            1146072952401913            454318149188752           -164383888363874]
[ 84                       98281                      51621                     -16911]
[ 85                   445707361                 -213091974                   25409628]
[ 86                          -7                          6                         -1]
[ 87        -2025487074437153495        2295582499903407744        -414906405399771342]
[ 88                        -175                         26                          3]
[ 89                   555597720                   35328569                  -35784720]
[ 90                           1                        -54                         12]
[ 91                           9                         -2                          0]
[ 92                    -8686915                   42041629                   -8886612]
[ 93                      -16022                     -64428                      15001]
[ 94                    -1128751                     107457                      30965]
[ 95                     1867321                    -419488                       2246]
[ 96         -107501957683948799         -54417516859416360          17011979169593655]
[ 97                -12891251368                 7987833890                -1127854887]
[ 98                      -60729                       -378                       2939]
[ 99                -50708057399                10025456082                  202315384]