Which integers are representable by $x^2+2y^2+7z^2$?

Question

I have been trying to characterize integers representable by several ternary forms and reached a roadblock with this particular form: $$x^2+2y^2+7z^2$$

Ideally, I am looking for a characterization of square-classes represented. Computer calculations were unable to find any class number 1 ternary sublattices or an entire ternary genus. Any help with this problem or direction towards useful sources would be very much appreciated.

complete genera are available at http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Brandt_1.html with any mixed term coefficients even. Choices in reduction may vary slightly from my current tables. — Will Jagy, Aug 21 '14 at 21:47
I have been using this table quite a bit, though I wish it marked regular forms. I have been referencing yours, Kaplansky's, and Schiemann's "There are 913 regular ternary forms," but it is difficult to compile these forms directly from the pdf. Do you have the list in a form which is easy to computationally parse? — Tahs, Aug 21 '14 at 22:06
I have everything; all but one are small enough to email. These are text files, just as the output I put below. Anyway, use my gmail address. — Will Jagy, Aug 21 '14 at 22:44
Even the raw list of regular forms goes over the 30,000 character limit for answers here. — Will Jagy, Aug 21 '14 at 23:05

Will Jagy · Accepted Answer · 2014-08-21T21:13:15.100

Very surprising to see a post on this, where someone knows what the words mean. Kaplansky wrote an early article on $x^2 + y^2 + 7 z^2$ in which he found it convenient to include your form. I'm afraid my websites are currently down with the host computer; anyway Math Comp 1995, The First Nontrivial Genus of Positive Definite Ternary Forms.

The entire genus is $$ x^2 + 2 y^2 + 7 z^2, \; \; x^2 + y^2 + 14 z^2. $$

Kaplansky called your form $k.$ He points out that Gordon Pall proved that $k$ represents all eligible numbers that are equivalent to $0,1 \bmod 3. $

alright, up to 1,000,000, and ignoring multiples of 49 (either form represents a number $n$ if and only if it represents $49n$) the comparison of the forms of the genus on eligible numbers comes out to:

exceptions for first form   (   1   1  14   0   0   0) are 

          3          6          7         11         12         28
         33         38         44         47         62         71
         77         83        102        107        110        132
        161        188        203        267        278        284
        308        318        332        335        357        428
        437        470        473        566        644        812
        863        878       1043       1068       1085       1118
       1155       1190       1253       1302       1340       1428
       1631       1673       1748       1892       2870       3017
       3452       4172       4262       4277       4340       4620
       4838       4907       5012       6524       6692       6923
       7118       9518       9758      10367      12068      12257
      13286      17108      19628      21182      27398      27692
      36953      41468      49028     100310     147812 

There were 83  exceptions for first form

exceptions for second form  (   1   2   7   0   0   0) are 

          5         14         20        158        518       3542


There were 6  exceptions for second form

Meanwhile, the numbers that are NOT represented by the full genus are $49^n (49m+21,35,42):$

The 150 smallest numbers NOT represented by full genus
    21    35    42    70    84    91   119   133   140   168
   182   189   217   231   238   266   280   287   315   329
   336   364   378   385   413   427   434   462   476   483
   511   525   532   560   574   581   609   623   630   658
   672   679   707   721   728   756   770   777   805   819
   826   854   868   875   903   917   924   952   966   973

Hey, this is just great. I was hoping you'd find this. I found the same problem with characterizing integers representable by the similar-looking form x^2+2y^2+7z^2+2yz -- no class #1 sublattices and it doesn't contain any ternary genuses. Has this form been handled as well? — Tahs, Aug 21 '14 at 21:18

score 2 · Answer 2 · answered Aug 21 '14 at 21:23

other genus:

=====Discriminant  52  ==Genus Size==   2
   Discriminant   52
  Spinor genus misses     no exceptions
        52:    1     2          7      2    0    0 vs. s.g.   5  10  14  21  46
        52:    2     3          3      2    0    2 vs. s.g.   1  6  26  34  741
--------------------------size 2
The 150 smallest numbers represented by full genus
     1     2     3     4     5     6     7     8     9    10
    11    12    14    15    16    17    18    19    20    21
    22    23    24    25    26    27    28    29    30    31
    32    33    34    35    36    37    38    40    41    42
    43    44    45    46    47    48    49    50    51    53
    54    55    56    57    58    59    60    61    62    63
    64    65    66    67    68    69    70    71    72    73
    74    75    76    77    78    79    80    81    82    83
    84    85    86    87    88    89    90    91    92    93
    94    95    96    97    98    99   100   101   102   103
   104   105   106   107   108   109   110   111   112   113
   114   115   116   118   119   120   121   122   123   124
   125   126   127   128   129   131   132   133   134   135
   136   137   138   139   140   141   142   143   144   145
   146   147   148   149   150   151   152   153   154   155

The 150 smallest numbers NOT represented by full genus
    13    39    52   117   130   156   182   208   221   286
   299   325   351   377   390   455   468   494   520   546
   559   624   637   663   689   715   728   793   806   832
   858   884   897   962   975
Disc: 52
==================================

Comparison up to 1,000,000, ignore multiples of 169:

exceptions for first form   (   1   2   7   2   0   0) are 

          5         10         14         21         46         61
         65         78        141        174        181        210
        229        265        394        429        474        481
        598        874        894        949       1209       1261
       1270       1417       1794       2266       2301       2470
       2561       3406       4030       4966       5446       5486
       5941       6006       6149       6565       7189       8034
      10894      11089      11401      12246      12610      15405
      15574      19409      19669      20254      26169      37609
      44421      51181      51649      68809      85930     109486
     125086     140374 

There were 62  exceptions for first form

exceptions for second form  (   2   3   3   2   0   2) are 

   1:     1          6         26         34        741        754
       2509       3289 

There were 8  exceptions for second form

       1 1,000,000
do you want to do another one?  type y or n: n

Great news, the program I was using had a minor blip. This form does in fact contain an entire genus, [3 7 11 -2 2 2] and [4 7 8 4 0 0]. That should help with the characterization. — Tahs, Aug 22 '14 at 07:22

Which integers are representable by $x^2+2y^2+7z^2$?

2 Answers2