Problem of the Month (June 2011)

The matrix below has the property that, ignoring spaces, every row and column is a power of 2.

16
65536
512
32
6 4

What is the smallest symmetric matrix of digits and spaces that contains a row of bn where every row and column is some power of b? (Here "smallest" means fewest entries, or smallest sum of powers for the same sized matrices.) What if the matrices do not need to be symmetric? What if the matrices are not allowed to use any power of b more than once?

We can ask the same question of any sequence of numbers that grows at least exponentially. What about Fibonacci numbers? What about factorials?

Are there non-trivial matrices where every row is a power of b and every column is a power of c? (The trivial examples are 1×1 matrices, when b is a power of c, or "disconnected" matrices that are a combination of smaller results.)


ANSWERS

Solvers this month include Joe DeVincentis, Bryce Herdt, Jon Palin, and Berend van der Zwaag.

Joe DeVincentis pointed out that there are many powers that use only two digits, such as 816192 = 6661661161, but here we only consider small bases.

Jon Palin proved that there are solutions for all Fibonacci numbers, factorials, and powers of 2, 3, 4, and 5. Bryce Herdt proved it for all powers of 6. Berend van der Zwaag proved it for powers of 8, 11, 12, 16, and 21. Joe DeVincentis proved it for powers of 51.

Joe DeVincentis conjectured that there are no other solutions for powers of 7 and 9, but was not quite able to prove this.

Here are the best-known solutions for symmetric matrices:

Powers of 2
1
1
2
2
4
4
8
8
16
1
16
32
32
2
64
64
4
128
128
2
8
256
2
256
64
512
512
1
2
1024
1
1024
2
4
2048
2
2048
4
8
4096
4
10 24
4096
64
2
4
8192
8
1
8192
2
16384 (JP)
1
16
16384
8
4
32768
3 2
2
32768
64
8
2

65536
16
65536
512
32
6 4
131072 (BH)
1
3 2
1
10 24
131072
2
2
4
2
262144 (BH)
2
262144
2
1
4
4
524288 (BH)
524288
2
4
2
8
8
1048576 (JD)
     1
 1   0  24
     4
     8
    25 6
1048576
     64
    6  4
 2
 4
2097152 (JD)
   8
    2
   10   24
8 1 9 2
 2097152
    1
   25 6
    2
  2
  4
4194304 (JD)
      1
   4
   1
 4194304
   4
   3  2
1  0 24
   4
8388608 (BZ)
     8
     3 2
     8
     8
    16
8388608
     8
 2
16777216 (BZ)
      3     2
     32
    12     8
   16
  16777216
 32 7 6   8
32  768
    2
    1
    6    4
     8
  8
2

33554432 (BZ)
 3 2
33554432
 51   2
25 6
 4
 4
 32
 2
67108864 (BZ)
    1
 16
 67108864
  1
1 0 2  4
  8
  8
  6 4
  4
134217728 (BZ)
       3   2
      1     
      32    
      4     
      2     
      1     
 134217728  
3 2   76  8 
      2     
      8     
       8    
2           
268435456 (BH)
     2   
 1   6   
     8   
     4   
     3 2 
268435456
     4   
    25 6 
     6  4
536870912 (BZ)
       8    
     5 1   2
     3    2 
     64     
     8      
 536870912  
   4 09 6   
81   9 2    
     16     
     2      
  2         
 2          
1073741824 (BH)
     3      2
   1
  10 2     4
 1073741824
   3   2
3 27 6 8
   4
   128
   8
   2
   4
  4
2
2147483648 (BZ)
   2      
   1      
   4      
2147483648
   4      
   8      
   3  2   
   6   4  
   4      
   8      
4294967296 (BH)
       4     
 1     2 8   
         1   
     819    2
       4     
   8 1 9   2 
   1   6     
42 94967296  
       2     
 81    9 2   
       6  4  
     2       
   2         

8589934592 (BZ)
         8  
 1 2 8      
     5   1 2
 2 048      
   409 6    
 8589934592 
     32     
    64      
     5  12  
8 1  9  2   
     2      
  2         
17179869184 (BZ)
       1      
   3  27  68  
  5    12     
 3 2   76   8 
    81 92     
    12 8      
 2    56      
17179869184   
  262  14    4
       8      
 6     4      
 8            
   8          
        4     
34359738368 (BZ)
1     28      
        3     
  5    12     
   34359738368
   4          
   3 2        
2  5    6     
8 19   2      
 327  6 8     
   3     2    
   8          
   3       2  
   6        4 
   8          
68719476736 (BZ)
         4     
       1  6    
   1  63  8   4
  13   1 07  2 
          1    
     4 0  9  6 
  6       4    
 131 0    7  2 
             1 
4  0     96    
 6871947 6736  
          32   
          6  4 
   2 6 21   44 
  4            
137438953472 (BZ)
        4       
            3 2 
    1   0   2  4
    3    2      
  137438953472  
    4           
    3     2     
    8           
4 0 9   6       
   25       6   
    3 2         
    4           
 32 7    6  8   
    2           
 2              
  4             
274877906944 (BZ)
         1       
       32        
      327  68    
            1    
        4        
        8        
  3   2 7     6 8
 32     7    6 8 
 27 4877906 944  
1       02 4     
        6 4      
  6      4       
  81    9   2    
       64        
      6 4        
       8         
      8          
549755813888 (BH)
    512        
    4
  819  2      
  1
549 7 55813888
1
2   5 6
  2 5  6  
    8
    1
    3     2
    8
    8
    8

Powers of 3
1
1
3
3
9
9
27
27
729
9
81
81
1
243 (JP)
72 9
243
3
9
729
729
27
9
2187
2187
1
8 1
7 29
9
6561 (BH)
1
9
1
196 83
6561
19 68 3
81
3
3
19683 (JD)
1
9
19683
81
3

59049 (JD)
1
9
7 2 9
6 56 1
9
590 49
19 6 83
2 43
9
1
9
177147 (JD)
1
2 7
2 7
1
243
177147
3
531441 (BZ)
      2  7
     2  7
  531441
  3
  1
 24  3
2 4   3
  1
 7       29
7       29
        9
1594323 (BH)
    2 7 
  1     
 1594323
  9     
2 4 3   
  3     
7 2   9 
  3     
4782969 (BZ)
2    4   3
 2   7
     8  1
   7 2 9
     9
4782969
     9
   9
  1
3
14348907 (BZ)
      1
 2    4    3
      3
   2  4   3
      8  1
      9
14348907
      729
       9
    1
   3
 3

Powers of 4
1
1
4
4
16
1
16
64
64
4
256
1
102 4
256
64
4
1024
1
1024
25 6
4
6 4
4096
4
10 2 4
4096
64
2 56
64
4
16384 (JD)
1
1 6
256
65536
1 6384
64

65536 (JD)
1
1 6
256
65536
1 6384
64
262144 (JD)
262144
64
2 56
1 6
4
4
1048576 (BZ)
      1    
    1 02  4
     2 5 6 
    6 4    
 1 63 8  4 
  2   5 6  
10 48576   
 25   6    
     6  4  
  6 4      
 4         
4194304 (BZ)
    4     
       1  
      4   
      1   
4   0 9 6 
      4   
  4194304 
 1    02 4
    6 4   
       4  
16777216 (BZ)
      1       
     16       
    16        
   16         
  16777  2  16
 16 777 2  16 
16  7772  16  
      2  56   
     2  56    
    2  56     
      16      
     16       
    16        
    6        4

Powers of 5
1
1
5
5
25
25
5
125
1
125
5
625
625
25
5
3125
3125
1
2 5
5
15625
1
5
15 625
25
5
78125 (JD)
97 65625
781 25
1
6 2 5
5
62 5
25
5
390625 (BZ)
   31   25
 1 953125 
     1    
39 062 5  
15 625    
 3125     
 1        
 2 5      
25        
5         
1953125 (JD)
1
1953125
5
3 125
1 25
2 5
5
9765625 (JD)
97 65625
781 25
1
6 2 5
5
62 5
25
5

48828125 (JD)
9         765   625
 9   7  65     625
  9 7 65      625
   488  2 8  125
  78      1  25
 7 8 1     2 5
  6       25   
  5
 6 2      5
 5
7  81 2 5
6    25
5
   125
  625
 625
625
25
5
244140625 (BZ)
           31      25
 1         95 3  125 
  2 44  14 06    25  
           62    5   
  4 88 2 8    125    
  4 88  28   125     
         2    5      
    2  5             
  1  2  5            
  4 882  8125        
         1           
3906     2 5         
1562     5           
     1               
 3  125              
    25               
    5                
 125                 
 25                  
25                   
5                    
1220703125 (JD)
      3    12  5
    1 9 5312  5
    2        5
    2       5
 1220703 125
    78 1 25
39  0 62 5
    3125
 5
 3  125
 1  25
12  5
2  5
  5
 5
5
6103515625 (BZ)
  6   2   5
  1        
6103515625 
  31   25  
  5        
  1        
2 5        
  62   5   
  25       
  5        
5          

Powers of 6
1
1
6
6
36
36
6
216
216
1
6
1296
1
2 16
1296
6
1
6
7776
777 6
777 6
7776
6
6
6
46656 (BH)
77        7  6
77        7 6 
   6          
  60  466176  
      6       
      6       
   4665   6   
   6          
   6          
   1          
77 7  6       
   6          
 6            
6             
279936 (JD)
21 6
21 6
2 16
2 79936
12 96
129 6
3 6
6
6
6
1
6

1679616 (BH)
    1     
 7776     
 777    6 
 77 7    6
16 79616  
    6     
    1     
    6     
  6       
   6      
10077696 (BZ)
           1  
 2 1       6  
  60   46617 6
 10077 6   96 
   777     6  
   777 6      
           1  
  46 6 5   6  
  6           
  6           
  1           
16796 16      
   6          
  6           
60466176 (BH)
77        7  6
77        7 6 
   6          
  60  466176  
      6       
      6       
   4665   6   
   6          
   6          
   1          
77 7  6       
   6          
 6            
6             
362797056 (BZ)
       1           
        3         6
  2  1  6          
        2    1   6 
    7   7  7    6  
  1  2  9    6     
      7 7    7 6   
1      00 77696    
 36279705    6     
             1     
       7  77 6     
    7  7  76       
       6           
   1 679616        
       6           
      6            
    6              
   6               
 6                 
2176782336 (BZ)
     21     6
     1       
  7 77     6 
     6       
  7 77    6  
2176782336   
1    29 6    
     3 6     
     36      
     6       
    6        
  6          
6            

Powers of 7
1
1
7
7

Powers of 8
1
1
8
8
64 (JD)
  64
    5           12
6   4
4  09            6
 549755813888
    51           2
    5 1  2
    8
    1
    3 2  7    68
    8
    8
    8
                 1
         6       4 
         8
 1
 2 6 2       14  4
512 (JD)
  64
    5           12
6   4
4  09            6
 549755813888
    51           2
    5 1  2
    8
    1
    3 2  7    68
    8
    8
    8
                 1
         6       4 
         8
 1
 2 6 2       14  4
4096 (JD)
  64
    5           12
6   4
4  09            6
 549755813888
    51           2
    5 1  2
    8
    1
    3 2  7    68
    8
    8
    8
                 1
         6       4 
         8
 1
 2 6 2       14  4
32768 (JD)
  64
    5           12
6   4
4  09            6
 549755813888
    51           2
    5 1  2
    8
    1
    3 2  7    68
    8
    8
    8
                 1
         6       4 
         8
 1
 2 6 2       14  4

262144 (JD)
  64
    5           12
6   4
4  09            6
 549755813888
    51           2
    5 1  2
    8
    1
    3 2  7    68
    8
    8
    8
                 1
         6       4 
         8
 1
 2 6 2       14  4
2097152 (JD)
           51 2
 5   1 2             
     6     4        
    4  0   9  6
   40  9          6
 16  7 7   7   216
               6  4
 2 097 1   5  2    
           5  12   
           8      
           1       
5 49 7 55813       888
1                 
               1
2  6   21      4 4
     26 2    144
     1             
     6        4 
    6 4           
           8
           8
           8
16777216 (JD)
           51 2
 5   1 2             
     6     4        
    4  0   9  6
   40  9          6
 16  7 7   7   216
               6  4
 2 097 1   5  2    
           5  12   
           8      
           1       
5 49 7 55813       888
1                 
               1
2  6   21      4 4
     26 2    144
     1             
     6        4 
    6 4           
           8
           8
           8
134217728
?
1073741824
?

8589934592 (JD)
            5    1    2
 6 4
  6         4
 4 0        9         6
    6      4
     6    4
      6             4
                     8
                     5 1 2
                     8
     4    0          9   6
    4      0         9    6
5 49        75581    3      888
            5 1       2
            512
            8
            1
1                       
                  6  4
                     51  2
      4             09     6
       858993     4592
2  6         2     1  4    4
        1           
                         1
        2 6        2    144
           6             4
                    6 4
            8
            8
            8   
68719476736 (JD)
       5        1     2
 6     4
  6   4
     409              6
    409         6
   409         6
  409         6
54 9   755813 8            88
       5          1   2
       5 1          2
       8
       1
       3    2 7   6       8
              1
      68    719476736
     6        4      
1   6         7 7 7 2 1 6
              6       4
        1   6 7 7 721  6
              3   27 6   8
         2    6 2 1    44
                   6  4
2  6    2       14   4
                  6 4
                6   4
                   8
            8
       8
       8
549755813888 (JD)
  64
    5           12
6   4
4  09            6
 549755813888
    51           2
    5 1  2
    8
    1
    3 2  7    68
    8
    8
    8
                 1
         6       4 
         8
 1
 2 6 2       14  4

Powers of 9
1
1
9
9
81
81
1

Powers of 10
1 (JD)
1
10 (JD)
1
10
100 (JD)
1
10
100
1000 (BH)
1
10
1 0
1000
10000 (BH)
1
10
1 0
1 0
10000
Clearly this trend continues.

Powers of 11
1 (JD)
1
11 (JD)
1
11
121 (JD)
1
121
1
1331 (JD)
11
1331
1331
11
14641 (BZ)
        11         
      1   21       
   1      1        
  14  6   4  1     
     11            
    13    3   1    
 1 61 0   5    1   
       1948717  1  
1      94 8717   1 
1      4 6    4   1
 214 3588 8  1     
 1     77  1 56  1 
       11          
   1   77 15 6  1  
     1   4 6  41   
      1       1    
       1     1     
        1  1       
         1         
161051 (BZ)
        11         
      1   21       
   1      1        
  14  6   4  1     
     11            
    13    3   1    
 1 61 0   5    1   
       1948717  1  
1      94 8717   1 
1      4 6    4   1
 214 3588 8  1     
 1     77  1 56  1 
       11          
   1   77 15 6  1  
     1   4 6  41   
      1       1    
       1     1     
        1  1       
         1         
1771561 (BZ)
        11         
      1   21       
   1      1        
  14  6   4  1     
     11            
    13    3   1    
 1 61 0   5    1   
       1948717  1  
1      94 8717   1 
1      4 6    4   1
 214 3588 8  1     
 1     77  1 56  1 
       11          
   1   77 15 6  1  
     1   4 6  41   
      1       1    
       1     1     
        1  1       
         1         
19487171 (BZ)
        11         
      1   21       
   1      1        
  14  6   4  1     
     11            
    13    3   1    
 1 61 0   5    1   
       1948717  1  
1      94 8717   1 
1      4 6    4   1
 214 3588 8  1     
 1     77  1 56  1 
       11          
   1   77 15 6  1  
     1   4 6  41   
      1       1    
       1     1     
        1  1       
         1         
214358881 (BZ)
        11         
      1   21       
   1      1        
  14  6   4  1     
     11            
    13    3   1    
 1 61 0   5    1   
       1948717  1  
1      94 8717   1 
1      4 6    4   1
 214 3588 8  1     
 1     77  1 56  1 
       11          
   1   77 15 6  1  
     1   4 6  41   
      1       1    
       1     1     
        1  1       
         1         

Powers of 12
1 (JP)
1
12 (JP)
1
12
144 (JP)
1
1
144
1 44
1728 (BZ)
         1
       1  
      1   
    1   2 
   14   4 
     1728 
  1  72 8 
 1   2    
   2488 32
1       2 
20736 (BZ)
                      1
                    1  
                   1   
              1      2 
           1     2     
          1      4   4 
         1    72 8     
        14     4       
       1  7 2  8       
      14    4          
     1  7 2 8          
    1      7  28       
        248 83    2    
            3583180  8 
   1  7    2 8         
      248  8 3 2       
             1    7 28 
    248      8    32   
            20  736    
  1              2     
 1              2      
   2 4       8  8    32
1                    2 
248832 (BZ)
         1
       1  
      1   
    1   2 
   14   4 
     1728 
  1  72 8 
 1   2    
   2488 32
1       2 
2985984 (BZ)
                                   1
                                 1  
                                1   
                               1    
                          1       2 
                         1   2      
                     1       4    4 
                    1      2        
             1     2                
            1      4       4        
           1     2                  
          1    2                    
         1    2                     
        1     4  4                  
            248    8  3 2           
           2   9 8    5 9  8     4  
                1   728             
          2  4 8   8  3       2     
                      1  728        
        24    8  8 3     2          
       1        7   2 8             
      1         2                   
              35831 8 0 8           
                       358 3180   8 
              29      859    8   4  
     1            72   8            
    1             2                 
       2 4     8  8    3   2        
                       1      7 2 8 
     24                88     32    
                 2     0    736     
   1                         2      
  1                         2       
 1             4        4           
    2 4                8    8     32
1                                 2 
35831808 (BZ)
                      1
                    1  
                   1   
              1      2 
           1     2     
          1      4   4 
         1    72 8     
        14     4       
       1  7 2  8       
      14    4          
     1  7 2 8          
    1      7  28       
        248 83    2    
            3583180  8 
   1  7    2 8         
      248  8 3 2       
             1    7 28 
    248      8    32   
            20  736    
  1              2     
 1              2      
   2 4       8  8    32
1                    2 

Powers of 15
1 (JP)
1
15 (JP)
1
15
225 (JP)
1
1
22 5
225
1 5
1 5

Powers of 16
1 (JP)
1
16 (JP)
1
16
256 (JP)
         1
        1 
       1  
   1  6   
     25  6
    2 5 6 
   65536
  1   6   
 1   6    
1   6   
65536 (JP)
         1
        1 
       1  
   1  6   
     25  6
    2 5 6 
   65536
  1   6   
 1   6    
1   6   
1048576
?
16777216 (JD)
              1
       1
      16
     16
    16
   16777  2  16
  16 777 2  16
 16  7772  16
       2  56
      2  56
     2  56
       16 
      16    
     16
1    6  

Powers of 21
1 (JD)
1
21 (JD)
21
1
441 (JD)
44 1
441
1
1
9261 (JD)
                 1
        1
  1     944      8        1
       44                1
        4         4     1
     4 08 4 1    0     1
                 1
   4 0 8 4 1     0    1
 19448  1
  4    4 1
  4  4    1
       1   9   448   1
     1        9448  1
              2 1
            9261
           44 1
           441
1 8  010   88    541
    4            41
                 1
            1
           1
       1
     1
    1   
   1 
  1
194481 (JD)
               1
        1
  1     944    8        1
       44              1
        4       4     1
     4 08 4 1  0     1
               1
   4 0 8 4 1   0    1
 19448  1
  4    4 1
  4  4    1
       1    9448   1
     1     9 448  1
           441
           44 1
1 8  010   88  541
    4          41
               1
            1
           1
       1
     1
    1
   1   
  1
4084101 (JD)
               1
        1
  1     944    8        1
       44              1
        4       4     1
     4 08 4 1  0     1
               1
   4 0 8 4 1   0    1
 19448  1
  4    4 1
  4  4    1
       1    9448   1
     1     9 448  1
           441
           44 1
1 8  010   88  541
    4          41
               1
            1
           1
       1
     1
    1
   1   
  1

85766121 (JD)
             1
       1 
  1    9   4481
       4                    4            1
       4                     4          1
      4                       4        1
     408 41  0  1                       
 1944 8      1
        4    0     8    4 1 0         1
      4  0   8          4  1 0       1
      1   9448                      1
  4       41
  4       4 1
1 8   01088        5          4    1
  1                  94  4   8    1
                     2           1
      1             9 4  4  8   1
                            44 1
                    2        1
        8    5     76612    1
                9 26     1
              92   6    1
              4 4  1
                   2   1  
        44           1
              4 4   1   
        1
         1
   4    0       84 1         01
    4    0    8  41         01
     4       4              1
                 1
                1
               1
              1
             1
          1
         1
        1
     1
    1
   1
1801088541 (JD)
               1
        1
  1     944    8        1
       44              1
        4       4     1
     4 08 4 1  0     1
               1
   4 0 8 4 1   0    1
 19448  1
  4    4 1
  4  4    1
       1    9448   1
     1     9 448  1
           441
           44 1
1 8  010   88  541
    4          41
               1
            1
           1
       1
     1
    1
   1   
  1

Powers of 38
1 (JD)
1
1444 (JD)
     1
    1
   1
  1444
 1 444
1  444

Powers of 51
1 (JD)
1
51 (JD)
51
1
2601 (JD)
 13    2   6           51
13    26              51
3 45   02         52 51
  5 1
   132  65        1
    2 6 01
 2   60       1
260       1 
  2 60  1
    51
       1  3 2      651
6          765  2  01
          2601
           51
      1       3265 1
              2601
           2  601
              51
  5 1         
  2       60  1
          51 
  5       1
 51  
51
1
132651 (JD)
 13    2   6           51
13    26              51
3 45   02         52 51
  5 1
   132  65        1
    2 6 01
 2   60       1
260       1 
  2 60  1
    51
       1  3 2      651
6          765  2  01
          2601
           51
      1       3265 1
              2601
           2  601
              51
  5 1         
  2       60  1
          51 
  5       1
 51  
51
1
6765201 (JD)
 13    2   6           51
13    26              51
3 45   02         52 51
  5 1
   132  65        1
    2 6 01
 2   60       1
260       1 
  2 60  1
    51
       1  3 2      651
6          765  2  01
          2601
           51
      1       3265 1
              2601
           2  601
              51
  5 1         
  2       60  1
          51 
  5       1
 51  
51
1
345025251 (JD)
 13    2   6           51
13    26              51
3 45   02         52 51
  5 1
   132  65        1
    2 6 01
 2   60       1
260       1 
  2 60  1
    51
       1  3 2      651
6          765  2  01
          2601
           51
      1       3265 1
              2601
           2  601
              51
  5 1         
  2       60  1
          51 
  5       1
 51  
51
1

Fibonacci Numbers
1
1
2
2
3
3
5
5
8
8
13
1
13
21 (BH)
21
1
34
3
34
55
5
55
89
8
89
144
1
34
144
233
2
3
233
377 (BH)
233
377
377
610 (BH)
1
6 10
8 9
1 44
10946
987
8 9
8
987
1597 (BH)
1
5
89
1597

2584
2
5
8
2584
4181
4181
1
8
1
6765
3
6765
37 7
6765
5 5
10946 (BH)
1
6 10
8 9
1 44
10946
17711 (JD)
13
17711
377
1
1
28657
3
2
8
28657
5
3 7 7
46368
34
463 68
3
1
6 10
8
75025 (JP)
3
37 7
5
75025
2
5
121393 (JP)
1
2
1
3
121393
3
196418 (JP)
13
89
196418
3 4
1
8

317811 (JP)
3
1
317811
8
1
1
514229 (JP)
5
1
3 4
2
2
514229
832040 (BH)
8
3
2
610
144
832040
1346269 (BZ)
   1   
   3   
  14 4 
1346269
   233 
  46368
   9 87
2178309 (BZ)
       8
   2  1 
   1    
 2178309
   8  9 
   3  4 
 1 0946 
8  9    
3524578 (JP)
3
5
2
3 4
5
3524578
8
5702887 (JP)
3
5
37 7
5702887
2
8
8
3 77
9227465 (BZ)
8 9    
  233  
9227465
 37  7 
 34    
  67 65
  5  5 

Factorials
1
1
2
2
6
6
24
2
24
120
1
2
120
720
1
2
720
2
120
5040
1
2
2
1
2
5040
120
2 4
12 0
40320
1
2
1
2 4
1 20
40320
24
12 0
362880 (BH)
                  1
                 1 
                  2
                 2 
                1  
               1   
      2 4          
       40     3 2 0
      40     3 2 0 
              6    
             6     
              2    
             2     
        3 6 288 0  
       3 6 2 880   
     1  2     0    
    1  2     0     
 1 2    0          
1 2    0           

3628800 (JD)
                     1
                     2
                    1 
                    2 
                   1  
                   2  
                  1   
                 1    
        2 4           
         40     3 2  0
        40     3 2  0 
                6     
               6      
                2     
               2      
          3 6 288  0  
         3 6 2 8800   
       1  2     0     
      1  2      0     
    12         0      
  12      0           
12       0            
     
39916800 (BZ)
                              1
                              2
                             1 
                             2 
                            1  
                            2  
                           1   
                           2   
                          1    
                          2    
                         1     
                        1      
            2   4              
               40      3 2    0
                       6       
             4 0  3     2    0 
            40   3     2    0  
                399 1 68  00   
               3 991 6 800     
                  1            
                 1             
                  6            
                 6             
             36 288    0       
           1   2  0            
          1  2    0            
        12       0             
      12         0             
    12          0              
  12           0               
12           0                 
479001600 (BZ)
                  1
                  2
                 1 
                 2 
                1  
                2  
             1     
             2     
            1      
         2 4       
           72     0
         479001600 
        1 20       
      12   0       
           1       
           6       
    12     0       
  12       0       
12        0        

Mersenne Primes
3 (JD)
3
7 (JD)
7
31 (JD)
3
31
127 (JD)
31
127
7
131071 (JD)
      3
 3  1
    3
   31
 131071
    7
3   1


Here are the best-known solutions for non-symmetric matrices when they are smaller:

Powers of 2
64
1
64
128
128
2048
1
2048
2
4

Powers of 4
64
1
64

Powers of 5
125
125
5

Powers of 11
11 (JD)
11

Fibonacci Numbers
13
13
21
21
55
55
144 (BH)
233
144
233
233
2584
3
2584
4181
3
4181
17711 (BH)
3
3
17711
377


Here are the best-known solutions for matrices that are not allowed to repeat a row or column:

Powers of 2
1, 4, 16, 64
1
64
2, 8, 128
128

Powers of 4
1, 4, 16, 64
1
64

Fibonacci Numbers
1, 3, 13
13
2, 21
21
144 (BH)
1
34
258 4
233
1
233
5, 8, 34, 2584
3
2584
55 (BH)
3
2584
5
4181
3
2
4181


Joe DeVincentis proved that 5 does not have any solutions with even bases.

Here are the best-known solutions for matrices whose rows are powers of b and whose columns are powers of c:

Powers of 2 and 3
(BH)
81

Powers of 2 and 6
(JD)
16
(BH)
1
216
(BH)
1
1
1
216
46656
6
(JD)
       1
      1
     1
    1
  1
 1
1
  6
 6
6
777    6
777   6
777  6
  216
 216
216
 6
1
6       

Powers of 2 and 9
(BH)
81

Powers of 2 and 11
(JD)
121
(JD)
1331
1 2 1
1 2 1
(JD)
1
14641
(BH)
     1      
   1  21    
  16105 1   
 1   2   1  
1    46   41

Powers of 2 and 12
(JD)
12
(JD)
144
(BH)
               1
           24883   2
               1
     358318    0  8
        20     736
      1        2
    1    2
   1     4       4
  1             2
 1    2
1    2
(JD)
248832
2
          1
    35831808
     1 2
   1 2
  1 2
 1        2
1         4 4

Powers of 2 and 21
(BH)
21
(BH)
441
(BH)
     1
 1
4084101
     2   1
 2      1
 4   4 1

Powers of 2 and 38
(BH)
1444

Powers of 3 and 8
(BH)
81

Powers of 3 and 11
(JD)
1331
(JD)
1         2     1
 1      2      1
  1   2       1
    19487171
   1  3     31
       1
(JD)
                 12 1
                1  2 1           
               1  77  1   561
                                        121
                                    121
                               121
              1           9     4  8 7 1 7 1
                              1 3 31
                       16105 1
        12 1
       1  2 1
      1  77  1          5  6                1
   121
  1 4                   6 4                  1
                                                    121
                           1                     21
                        1                     21
 1                        9                   4 87 1 7 1
1   3                                         3 1

Powers of 4 and 38
(BH)
1444

Powers of 5 and 15
(JD) (BH)
225
1 5
1 5

Powers of 4 and 6
(JD)
16
(BH)
1
1
1
216
46656
6
(JD)
       1
      1
     1
    1
  1
 1
1
  6
 6
6
777    6
777   6
777  6
  216
 216
216
 6
1
6       

Powers of 8 and 9
(BH)
81

Powers of 11 and anything
11

Powers of 12 and 38
(JD)
    1
   1
  144
 1 44
1  44

Powers of 21 and 38
(BH)
 1
1
44  1
44 1
441


If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/6/13.