# Positive ternary quadratic forms in the same genus that represent the same numbers

There are three genera of positive, integral, ternary quadratic forms in which both forms (classes...) are regular, so the paired forms represent the same numbers. These pairs (complete genera) are:

$$x^2 + y^2 + 9 z^2 + xy, \; \; \; x^2 + 3 y^2 + 3 z^2 + 3 y z,$$

$$x^2 + y^2 + 7 z^2 + zx, \; \; \; x^2 + 2 y^2 + 4 z^2 + y z + xy,$$

$$x^2 + 4y^2 + 7 z^2 + zx, \; \; \; x^2 + 5 y^2 + 7 z^2 + 5 y z +zx + xy.$$

In these three cases the relationship of represented numbers is proven.

In response to a question by a master's student in New Zealand, originating with his adviser, Steven Galbraith, I have found, so far, two pairs of irregular forms, in the same genus, that represent the same numbers up to $10^6;$ these are

$$3x^2 + 3y^2 + 7 z^2 + yz +2zx +xy, \; \; \; 3x^2 + 5 y^2 + 5 z^2 + 3 y z +zx + 3xy,$$

$$5x^2 + 5y^2 + 8 z^2 +4zx +3xy, \; \; \; 5x^2 + 7 y^2 + 7 z^2 + 6 y z +zx + 5xy.$$ Cannot prove these relationships, of course. In my accounting, these are discriminant 232 and 648, respectively. The 232 genus has three classes, one more than I displayed, and the 648 genus has 6 classes. I'm just saying. For 232, the genus, collectively, misses $4n+2$ and $4^k (16n+6).$ These are traditionally called the "progressions." The two forms also miss $4^k \{1\}.$ For 648, the progressions are $9n \pm 3, \; \; 81n \pm 27, \;\; 4n+2, \; \; 4^k (16 n + 14).$ The two forms indicated above also miss $4^k \cdot \{1,40\}.$

NOTE, September 2017: I also checked the two pairs of forms I found for primitive representations. In both cases, the pair of forms in each genus primitively represent the same numbers up to a large bound. If ever proved, these examples violate Conjecture A on page 237, Regular Positive Ternary Quadratic Forms, J. S. Hsia, Mathematika, volume 28, (1981), pages 231-238.

Anyway, i have written a very fast program and am searching for other pairs. i am disappointed at finding so few.

Note that there are finiteness results about the number of exceptions missed by a ternary form, see Chan and Oh (2004), item 12 at OH_HOME. And, if we count number of representations, no two distinct forms match, i.e. the theta series determines the positive ternary form, proof A. Schiemann.

Meanwhile, note that, if we allow the discriminant to change, there are infinitely many pairs that represent the same numbers, elementary proofs:

$$A(x^2 + xy+ y^2 ) + B z^2, \; \; \; A(x^2 + 3 y^2 ) + B z^2,$$

$$A(x^2 + y^2 + z^2 ) + B (yz + z x + x y), \; \; \; Ax^2 + (2A-B) y^2 + (2A+B) z^2 + 2 B zx.$$ For this one, noticed by Irving Kaplansky, we have discriminant $\Delta = 4A^3 - 3 A B^2 + b^3 = (2A-B)^2 (A+B);$ for this to be positive definite, we need $A > 0$ and $2A > B > -A.$ Given that, the form still might not be "reduced," so it took a while to realize that the different appearances of this were really all the same.

Well, there is the QUESTION, are there only finitely many pairs of forms in the same genus that represent the same numbers?

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=====Discriminant  232  ==Genus Size==   3
Discriminant   232
Spinor genus misses     no exceptions
232:    1     4         15      2    1    0 vs. s.g.   3  7  11  31  43
232:    3     3          7      1    2    1 vs. s.g.   1  single squareclass
232:    3     5          5      3    1    3 vs. s.g.   1  single squareclass
--------------------------size 3
The 150 smallest numbers represented by full genus
1     3     4     5     7     8     9    11    12    13
15    16    17    19    20    21    23    25    27    28
29    31    32    33    35    36    37    39    40    41
43    44    45    47    48    49    51    52    53    55
56    57    59    60    61    63    64    65    67    68
69    71    72    73    75    76    77    79    80    81
83    84    85    87    89    91    92    93    95    97
99   100   101   103   104   105   107   108   109   111
112   113   115   116   117   119   120   121   123   124
125   127   128   129   131   132   133   135   136   137
139   140   141   143   144   145   147   148   149   151
153   155   156   157   159   160   161   163   164   165
167   168   169   171   172   173   175   176   177   179
180   181   183   184   185   187   188   189   191   192
193   195   196   197   199   200   201   203   204   205

The 150 smallest numbers NOT represented by full genus
2     6    10    14    18    22    24    26    30    34
38    42    46    50    54    58    62    66    70    74
78    82    86    88    90    94    96    98   102   106
110   114   118   122   126   130   134   138   142   146
150   152   154   158   162   166   170   174   178   182
186   190   194   198   202   206   210   214   216   218
222   226   230   234   238   242   246   250   254   258
262   266   270   274   278   280   282   286   290   294
298   302   306   310   314   318   322   326   330   334
338   342   344   346   350   352   354   358   362   366
370   374   378   382   384   386   390   394   398   402
406   408   410   414   418   422   426   430   434   438
442   446   450   454   458   462   466   470   472   474
478   482   486   490   494   498   502   506   510   514
518   522   526   530   534   536   538   542   546   550

ALL ODD
Disc: 232
==================================

232:    1     4         15      2    1    0
misses, compared with full genus
3            7           11           12           28
31           43           44           48           56
79          112          115          120          124
141          165          168          172          176
184          192          224          295          301
309          316          448          456          460
471          480          487          496          555
564          568          589          616          660
672          688          704          736          760
768          805          840          896

232:    3     3          7      1    2    1
misses, compared with full genus
1:      1    2:      4    4:     16    8:     64   16:    256

ONE ODD

232:    3     5          5      3    1    3
misses, compared with full genus
1:      1    2:      4    4:     16    8:     64   16:    256

ONE ODD

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=====Discriminant  648  ==Genus Size==   6
Discriminant   648
Spinor genus misses     no exceptions
648:    1     4         41      2    1    0 vs. s.g.   7  11  19  23  31
648:    1     5         35      4    1    1 vs. s.g.   8  13  29  31
648:    1    11         17      9    1    1 vs. s.g.   5  7  8  65  179
648:    4     5          9      3    0    2 vs. s.g.   1  8
648:    5     5          8      0    4    3 vs. s.g.   1  40
648:    5     7          7      6    1    5 vs. s.g.   1  40
--------------------------size 6
The 150 smallest numbers represented by full genus
1     4     5     7     8     9    11    13    16    17
19    20    23    25    28    29    31    32    35    36
37    40    41    43    44    45    47    49    52    53
55    59    61    63    64    65    67    68    71    72
73    76    77    79    80    81    83    85    88    89
91    92    95    97    99   100   101   103   104   107
109   112   113   115   116   117   119   121   124   125
127   128   131   133   136   137   139   140   143   144
145   148   149   151   152   153   155   157   160   161
163   164   167   169   171   172   173   175   176   179
180   181   185   187   188   191   193   196   197   199
200   203   205   207   208   209   211   212   215   217
220   221   223   225   227   229   232   233   235   236
239   241   243   244   245   247   251   252   253   256
257   259   260   261   263   265   268   269   271   272

The 150 smallest numbers NOT represented by full genus
2     3     6    10    12    14    15    18    21    22
24    26    27    30    33    34    38    39    42    46
48    50    51    54    56    57    58    60    62    66
69    70    74    75    78    82    84    86    87    90
93    94    96    98   102   105   106   108   110   111
114   118   120   122   123   126   129   130   132   134
135   138   141   142   146   147   150   154   156   158
159   162   165   166   168   170   174   177   178   182
183   184   186   189   190   192   194   195   198   201
202   204   206   210   213   214   216   218   219   222
224   226   228   230   231   234   237   238   240   242
246   248   249   250   254   255   258   262   264   266
267   270   273   274   276   278   282   285   286   290
291   294   297   298   300   302   303   306   309   310
312   314   318   321   322   326   327   330   334   336

Disc: 648
==================================

648:    1     4         41      2    1    0
misses, compared with full genus
7           11           19           23           28
31           35           44           76           79
88           92          107          112          124
140          152          176          280          304
316          344          352          368          428
448          472          496          536          560
608          616          704          811

648:    1     5         35      4    1    1
misses, compared with full genus
8           13           29           31           32
128          512

648:    1    11         17      9    1    1
misses, compared with full genus
5            7            8           32           65
128          179          512

648:    4     5          9      3    0    2
misses, compared with full genus
1:      1            8           32          128          512

648:    5     5          8      0    4    3
misses, compared with full genus
1:      1    2:      4    4:     16           40    8:     64
160   16:    256          640

648:    5     7          7      6    1    5
misses, compared with full genus
1:      1    2:      4    4:     16           40    8:     64
160   16:    256          640

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• Bit of a relief to get a false positive overnight. Both discriminant $\Delta = 6804,$ not actually the same genus, one classically integral and the other not, and agreement up to my bound, $$32 \lfloor \sqrt[3] \Delta \rfloor,$$ partly because one form really does dominate the other, an observation by Kap in "Three Theorems" unpublished. This confirms that the software does recognize when two forms agree up to my bound. I was getting worried. – Will Jagy Dec 28 '13 at 22:53
• AHA! for some infinite families of pairs of forms, it is necessary to check up to, at least, something along the lines of $\Delta/12.$ So, i am just keeping the first test as it was, and then testing up to $100 + \Delta$ before having the computer record the pair. Also started a file of "false positives," pairs that agree up to the lower bound but not the higher. Onward and upward. – Will Jagy Dec 29 '13 at 1:37

Spent a month checking, this is what I suspect is the complete list of 'sporadic' or 'exceptional' pairs. No restriction that they be in the same genus or have the same discriminant. I was able to check discriminant ratio $4$ and discriminant ratio $1$ very high. The other ones just seem sort of random little sets, two quadruples (see the repeated forms). Also some patterns that dry up, discriminants $111,333,999$ but not $2997,$ also on the left $24, 72, 216, 648, 1944,$ but not $5832.$ This last begins with three pair of regular forms, I typed those in. The list of pairs of regular forms that agree is enormous.

---------------------------------------------------------------
54 :     2    2    4    1    2    0      216 :     2    4    8    4    1    1
75 :     1    4    5    1    1    0      111 :     1    4    7    1    0    0
78 :     3    3    3    1    1    3      142 :     3    3    5    2    3    1
78 :     3    3    3    1    1    3      158 :     3    3    5   -1    2    1
78 :     3    3    3    1    1    3      190 :     3    5    5    5    2    3
142 :     3    3    5    2    3    1      158 :     3    3    5   -1    2    1
142 :     3    3    5    2    3    1      190 :     3    5    5    5    2    3
156 :     3    3    5    2    2    0      284 :     3    5    6    4    2    2
156 :     3    3    5    2    2    0      316 :     3    5    6    0    2    2
156 :     3    3    5    2    2    0      380 :     3    5    7    2    0    2
158 :     3    3    5   -1    2    1      190 :     3    5    5    5    2    3
162 :     2    2   14    1    2    2      648 :     2    6   14    3    1    0
177 :     2    4    7    4    2    1      213 :     2    4    7    0    1    1
225 :     3    4    7    4    3    3      333 :     3    4    7    1    0    0
232 :     3    3    7    1    2    1      232 :     3    5    5    3    1    3
284 :     3    5    6    4    2    2      316 :     3    5    6    0    2    2
284 :     3    5    6    4    2    2      380 :     3    5    7    2    0    2
316 :     3    5    6    0    2    2      380 :     3    5    7    2    0    2
324 :     4    4    6    0    3    2      567 :     4    6    7    3    2    3
486 :     2    2   41    1    2    2     1944 :     2    6   41    3    1    0
531 :     5    5    6    0    3    2      639 :     5    5    8   -1    2    4
648 :     4    7    7    5    2    2     2592 :     4    7   25   -4    2    2
648 :     5    5    8    0    4    3      648 :     5    7    7    6    1    5
675 :     5    5    8   -1    4    2      999 :     5    8    8   -5    1    4
These are pairs of positive quadratic forms that represent the
same numbers, and violate a Kaplansky conjecture.

Delta : A B C R S T     means

f(x,y,z) = A x^2 + B y^2 + C z^2 + R y z + S z x + T x y,

and Delta = 4ABC + RST - A R^2 - B S^2 - C T^2.

The two pair within a genus each are
232 :   3   5     5   3  1  3      232 :  3   3     7   1  2  1
648 :   5   7     7   6  1  5      648 :  5   5     8   0  4  3

The most productive discriminant ratio  is 4,
which includes Kap's two infinite families, also
24 :   1   2     4   2  1  1        6 :  1   1     2   1  1  0
72 :   2   2     5   1  1  1       18 :  2   2     2   1  2  2
216 :   2   5     6   3  0  1       54 :  2   2     5   1  2  2
648 :   2   6    14   3  1  0      162 :  2   2    14   1  2  2
1944 :  2   6    41   3  1  0      486 :  2   2    41   1  2  2
or
48N-24: 2   6    N    3  1  0     12N-6:  2   2     N   1  2  2
where N = (1+ 3^k)/2, and the pairs for N = 1,2,5 are regular
and have been Schiemann reduced.
------------------------------------------------------------
Reminder: Kap's two infinite families are equivalent to those
below, which need not be "reduced." For the first, require
gcd(A,C) = 1 and 0 <A,C. For the second, gcd(A,R) = 1, with
A > 0 and  -A < R < 2 A.

4D : A    3A    C   0   0  0    D :   A   A   C    0    0    A

4D:  A  2A-R  2A+R  0  2R  0    D :   A   A   A    R    R    R

For the first, D = 3 A^2 C, for the second D = (A+R)(2A-R)^2 .


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Why not. The complete list of regular pairs, evidently 182 pairs among which various triples, quadruples and so on may be found. Smaller discriminant put first on the line.

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    2 :     1    1    1    1    1    1       18 :     1    2    3    2    1    0
2 :     1    1    1    1    1    1       32 :     1    2    4    0    0    0
2 :     1    1    1    1    1    1        8 :     1    1    2    0    0    0
3 :     1    1    1    0    0    1       12 :     1    1    3    0    0    0
3 :     1    1    1    0    0    1       12 :     1    2    2    1    1    1
3 :     1    1    1    0    0    1       21 :     1    2    3    0    0    1
4 :     1    1    1    0    0    0       16 :     1    2    2    0    0    0
4 :     1    1    1    0    0    0       36 :     1    2    5    2    0    0
5 :     1    1    2    1    1    1       20 :     1    2    3    1    0    1
5 :     1    1    2    1    1    1       20 :     1    2    3    2    0    0
6 :     1    1    2    0    0    1       22 :     1    2    3    0    1    0
6 :     1    1    2    0    0    1       24 :     1    2    3    0    0    0
6 :     1    1    2    1    1    0       15 :     1    1    4    0    1    0
6 :     1    1    2    1    1    0       24 :     1    1    6    0    0    0
6 :     1    1    2    1    1    0       24 :     1    2    4    2    1    1
6 :     1    1    2    1    1    0       33 :     1    2    5    1    1    1
8 :     1    1    2    0    0    0       18 :     1    2    3    2    1    0
8 :     1    1    2    0    0    0       32 :     1    2    4    0    0    0
8 :     1    1    3    1    1    1       32 :     1    3    3    2    0    0
8 :     1    1    3    1    1    1       72 :     1    3    7    2    1    1
9 :     1    1    3    0    0    1       36 :     1    3    3    0    0    0
9 :     1    1    3    0    0    1       36 :     1    3    4    3    1    0
9 :     1    1    3    0    0    1       63 :     1    3    6    3    0    0
10 :     1    2    2    2    1    1       15 :     1    2    2    1    0    0
10 :     1    2    2    2    1    1       40 :     1    2    5    0    0    0
12 :     1    1    3    0    0    0       12 :     1    2    2    1    1    1
12 :     1    1    3    0    0    0       21 :     1    2    3    0    0    1
12 :     1    1    4    0    0    1       48 :     1    3    4    0    0    0
12 :     1    2    2    1    1    1       21 :     1    2    3    0    0    1
12 :     1    2    2    2    0    0       44 :     1    2    6    2    0    0
12 :     1    2    2    2    0    0       48 :     1    2    6    0    0    0
14 :     1    1    5    1    1    1       30 :     1    3    3    1    1    1
14 :     1    1    5    1    1    1       46 :     1    3    5    3    1    1
14 :     1    1    5    1    1    1       56 :     1    3    5    2    0    0
15 :     1    1    4    0    1    0       24 :     1    1    6    0    0    0
15 :     1    1    4    0    1    0       24 :     1    2    4    2    1    1
15 :     1    1    4    0    1    0       33 :     1    2    5    1    1    1
15 :     1    2    2    1    0    0       40 :     1    2    5    0    0    0
16 :     1    2    2    0    0    0       36 :     1    2    5    2    0    0
18 :     1    1    6    0    0    1       72 :     1    3    6    0    0    0
18 :     1    2    3    2    1    0       32 :     1    2    4    0    0    0
18 :     2    2    2    1    2    2       45 :     2    2    3    0    0    1
18 :     2    2    2    1    2    2       72 :     2    2    5    1    1    1
18 :     2    2    2    1    2    2       72 :     2    3    3    0    0    0
18 :     2    2    2    1    2    2       99 :     2    3    5    3    1    0
20 :     1    1    7    1    1    1       80 :     1    3    7    2    0    0
20 :     1    2    3    1    0    1       20 :     1    2    3    2    0    0
22 :     1    2    3    0    1    0       24 :     1    2    3    0    0    0
24 :     1    1    6    0    0    0       24 :     1    2    4    2    1    1
24 :     1    1    6    0    0    0       33 :     1    2    5    1    1    1
24 :     1    2    4    2    1    1       33 :     1    2    5    1    1    1
25 :     2    2    2   -1    1    1      100 :     2    2    7   -1    1    1
25 :     2    2    2   -1    1    1      100 :     2    3    5    0    0    2
27 :     1    1    7    0    1    0       27 :     1    2    4    1    0    1
27 :     1    1    9    0    0    1      108 :     1    3   10    3    1    0
27 :     1    1    9    0    0    1      108 :     1    3    9    0    0    0
27 :     1    1    9    0    0    1       27 :     1    3    3    3    0    0
27 :     1    3    3    3    0    0      108 :     1    3   10    3    1    0
27 :     1    3    3    3    0    0      108 :     1    3    9    0    0    0
27 :     2    2    2    1    1    1      108 :     2    2    8    2    2    1
27 :     2    2    2    1    1    1      108 :     2    3    5    0    2    0
27 :     2    2    2    1    1    1      189 :     2    3    8    0    1    0
28 :     2    2    3    2    2    2      112 :     2    3    5    2    0    0
28 :     2    2    3    2    2    2       60 :     2    3    3    0    0    2
28 :     2    2    3    2    2    2       92 :     2    3    5    2    0    2
30 :     1    1   10    0    0    1      120 :     1    3   10    0    0    0
30 :     1    3    3    1    1    1       46 :     1    3    5    3    1    1
30 :     1    3    3    1    1    1       56 :     1    3    5    2    0    0
32 :     1    3    3    2    0    0       72 :     1    3    7    2    1    1
36 :     1    1   12    0    0    1      144 :     1    3   12    0    0    0
36 :     1    3    3    0    0    0       36 :     1    3    4    3    1    0
36 :     1    3    3    0    0    0       63 :     1    3    6    3    0    0
36 :     1    3    4    3    1    0       63 :     1    3    6    3    0    0
36 :     2    2    3    0    0    2      144 :     2    3    6    0    0    0
44 :     1    2    6    2    0    0       48 :     1    2    6    0    0    0
45 :     2    2    3    0    0    1       72 :     2    2    5    1    1    1
45 :     2    2    3    0    0    1       72 :     2    3    3    0    0    0
45 :     2    2    3    0    0    1       99 :     2    3    5    3    1    0
46 :     1    3    5    3    1    1       56 :     1    3    5    2    0    0
48 :     1    4    4    4    0    0      192 :     1    4   12    0    0    0
50 :     1    4    4    3    1    1      200 :     1    5   10    0    0    0
50 :     1    4    4    3    1    1       75 :     1    4    5    0    0    1
54 :     1    1   18    0    0    1      216 :     1    3   18    0    0    0
54 :     1    4    4    2    1    1      216 :     1    6    9    0    0    0
54 :     1    4    4    2    1    1      297 :     1    6   13    3    1    0
54 :     2    2    5    1    2    2      135 :     2    5    5    5    1    2
54 :     2    2    5    1    2    2      216 :     2    5    6    0    0    2
54 :     2    2    5    1    2    2      216 :     2    5    6    3    0    1
54 :     2    2    5    1    2    2      297 :     2    5    8   -2    1    1
54 :     2    3    3    3    0    0      216 :     2    3    9    0    0    0
60 :     1    4    5    4    1    0      132 :     1    5    7    1    0    1
60 :     1    4    5    4    1    0       96 :     1    4    7    4    0    0
60 :     2    2    5    0    0    2      240 :     2    5    6    0    0    0
60 :     2    3    3    0    0    2      112 :     2    3    5    2    0    0
60 :     2    3    3    0    0    2       92 :     2    3    5    2    0    2
64 :     3    3    3   -2    2    2      256 :     3    3    8    0    0    2
64 :     3    3    3   -2    2    2      576 :     3    3   19   -2    2    2
72 :     2    2    5    1    1    1       72 :     2    3    3    0    0    0
72 :     2    2    5    1    1    1       99 :     2    3    5    3    1    0
72 :     2    3    3    0    0    0       99 :     2    3    5    3    1    0
75 :     1    4    5    0    0    1      200 :     1    5   10    0    0    0
80 :     3    3    3    2    2    2      320 :     3    4    7    0    2    0
90 :     1    1   30    0    0    1      360 :     1    3   30    0    0    0
92 :     2    3    5    2    0    2      112 :     2    3    5    2    0    0
96 :     1    4    7    4    0    0      132 :     1    5    7    1    0    1
98 :     3    3    3   -1    1    1      392 :     3    5    7    0    0    2
100 :     2    2    7   -1    1    1      100 :     2    3    5    0    0    2
100 :     3    3    3    1    1    1      400 :     3    5    7    0    2    0
108 :     1    1   36    0    0    1      432 :     1    3   36    0    0    0
108 :     1    3    9    0    0    0      108 :     1    3   10    3    1    0
108 :     1    4    7    0    1    0      108 :     1    5    7    5    1    1
108 :     1    6    6    6    0    0      432 :     1    6   18    0    0    0
108 :     2    2    8    2    2    1      108 :     2    3    5    0    2    0
108 :     2    2    8    2    2    1      189 :     2    3    8    0    1    0
108 :     2    2    9    0    0    2      432 :     2    6    9    0    0    0
108 :     2    3    5    0    2    0      189 :     2    3    8    0    1    0
108 :     3    3    5    3    3    3      432 :     3    5    8    4    0    0
135 :     2    5    5    5    1    2      216 :     2    5    6    0    0    2
135 :     2    5    5    5    1    2      216 :     2    5    6    3    0    1
135 :     2    5    5    5    1    2      297 :     2    5    8   -2    1    1
144 :     3    4    4    4    0    0      576 :     3    4   12    0    0    0
150 :     2    5    5    5    0    0      600 :     2    5   15    0    0    0
180 :     2    2   15    0    0    2      720 :     2    6   15    0    0    0
180 :     3    5    5    5    3    3      288 :     3    5    5    2    0    0
180 :     3    5    5    5    3    3      396 :     3    5    8    2    0    3
192 :     1    8    8    8    0    0      704 :     1    8   24    8    0    0
192 :     1    8    8    8    0    0      768 :     1    8   24    0    0    0
196 :     3    5    5   -4    2    2      784 :     3    5   14    0    0    2
216 :     1    6    9    0    0    0      297 :     1    6   13    3    1    0
216 :     2    5    6    0    0    2      216 :     2    5    6    3    0    1
216 :     2    5    6    0    0    2      297 :     2    5    8   -2    1    1
216 :     2    5    6    3    0    1      297 :     2    5    8   -2    1    1
240 :     1    4   16    4    0    0      384 :     1    4   24    0    0    0
256 :     3    3    8    0    0    2      576 :     3    3   19   -2    2    2
270 :     3    3   11    3    3    3     1080 :     3    9   11    6    0    0
288 :     3    5    5    2    0    0      396 :     3    5    8    2    0    3
300 :     1   10   10   10    0    0     1200 :     1   10   30    0    0    0
384 :     4    4    7    0    4    0      528 :     4    7    7    6    0    4
400 :     3    7    7   -6    2    2     1600 :     3    7   20    0    0    2
432 :     1   12   12   12    0    0     1728 :     1   12   36    0    0    0
432 :     5    5    5   -2    2    2     1728 :     5    8   12    0    0    4
448 :     5    5    5    2    2    2     1472 :     5    8   12    8    4    0
448 :     5    5    5    2    2    2     1792 :     5    8   12    0    4    0
448 :     5    5    5    2    2    2      960 :     5    5   12   -4    4    2
450 :     5    5    6    0    0    5     1800 :     5    6   15    0    0    0
540 :     5    5    8    2    4    5     1188 :     5    8    9    6    3    2
540 :     5    5    8    2    4    5      864 :     5    8    8    8    2    4
540 :     6    6    7    6    6    6     2160 :     6    7   13    2    0    0
576 :     3    8    8    8    0    0     2304 :     3    8   24    0    0    0
704 :     1    8   24    8    0    0      768 :     1    8   24    0    0    0
720 :     3    8    8    4    0    0     1152 :     3    8   12    0    0    0
768 :     1   16   16   16    0    0     3072 :     1   16   48    0    0    0
864 :     5    8    8    8    2    4     1188 :     5    8    9    6    3    2
900 :     3   10   10   10    0    0     3600 :     3   10   30    0    0    0
960 :     5    5   12   -4    4    2     1472 :     5    8   12    8    4    0
960 :     5    5   12   -4    4    2     1792 :     5    8   12    0    4    0
960 :     5    8    8    8    0    0     3840 :     5    8   24    0    0    0
1024 :     3   11   11  -10    2    2     4096 :     3   11   32    0    0    2
1152 :     5    5   12    0    0    2     1584 :     5    5   17   -2    2    2
1280 :     7    7    7   -2    2    2     5120 :     7   12   16    0    0    4
1350 :     7    7    7   -1    1    1     5400 :     7   13   15    0    0    2
1472 :     5    8   12    8    4    0     1792 :     5    8   12    0    4    0
1728 :     1   24   24   24    0    0     6912 :     1   24   72    0    0    0
1728 :     8    8    9    0    0    8     6912 :     8    9   24    0    0    0
2160 :     5    8   17    8    2    4     3456 :     5    8   24    0    0    4
2304 :     3   16   16   16    0    0     9216 :     3   16   48    0    0    0
2700 :     9   11   11   -8    6    6    10800 :     9   11   30    0    0    6
2880 :     8    8   15    0    0    8    11520 :     8   15   24    0    0    0
3136 :     3   19   19  -18    2    2    12544 :     3   19   56    0    0    2
3456 :     8   11   11   -2    4    4     4752 :     8   11   15    6    0    4
4800 :     1   40   40   40    0    0    19200 :     1   40  120    0    0    0
6144 :    11   11   16    8    8    6     8448 :    11   11   19    2    2    6
6144 :     7   15   16    0    0    6     8448 :     7   15   23   -6    2    6
6400 :     3   27   27  -26    2    2    25600 :     3   27   80    0    0    2
6912 :     1   48   48   48    0    0    27648 :     1   48  144    0    0    0
6912 :     9   17   17  -14    6    6    27648 :     9   17   48    0    0    6
8640 :    13   13   13    2    2    2    34560 :    13   24   28    0    4    0
14400 :     3   40   40   40    0    0    57600 :     3   40  120    0    0    0
18432 :    17   17   20   -4    4   14    25344 :    17   20   20   -8    4    4
18432 :     5   20   48    0    0    4    25344 :     5   20   68   -8    4    4
43200 :     9   41   41  -38    6    6   172800 :     9   41  120    0    0    6
55296 :    11   32   44  -16    4    8    76032 :    11   32   59    8   10    8


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• I don't understand your class. What is the meaning of to touch a computer all the decisions? If you find the solution then write the equation. Two forms are equated to each other. For a given condition and solve the equation. Turning the computer figures we do not understand the phenomenon. – individ Sep 25 '17 at 6:46
• @individ It may help you to read zakuski.math.utsa.edu/~kap/Timofeev_1963_Uspekhi.pdf The best explanation of this new material is this entire document zakuski.math.utsa.edu/~kap/Jagy_Encyclopedia.pdf especially pages 36-42. – Will Jagy Sep 25 '17 at 17:42