Multivariate Bezout identity

Question

By Bezout's identity, we know that by varying x, y, z, p, q, r over all integers, the expression $ax + by + cz + dp + eq + fr$ would yield all integers if and only if gcd(a,b,c,d,e,f) = 1 (i.e. the coefficients are setwise prime integers).

However, if we have the expression $ax^2 + by^2 + cz^2 + dxy + eyz + fxz$, then we know by Bezout's identity that the expression will evaluate to integers, however the expression will not yield all integers by varying x,y,z since the integers $x^2$, $y^2$, $z^2$, $xy$, $xz$, $yz$ are correlated.

So the problem is to know which integers will the expression $ax^2 + by^2 + cz^2 + dxy + eyz + fxz$ evaluate to?

For example the first 50 positive integer values of $3x^2 + 3y^2 + 8z^2 - 3xy$ are 3, 8, 9, 11, 12, 17, 20, 21, 27, 29, 32, 35, 36, 39, 41, 44, 47, 48, 53, 56, 57, 59, 63, 65, 68, 71, 72, 75, 80, 81, 83, 84, 89, 92, 93, 95, 99, 101, 107, 108, 111, 113, 116, 117, 119, 120, 125, 128, 129

and of $x^2 + y^2 + z^2 + xy + yz + xz$ are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52

Answer 1

I see that you added two forms. One of them, $x^2+y^2+z^2+yz+zx+xy,$ is regular, and represents exactly the same numbers as $u^2 + v^2 + 2 w^2,$ namely all but $4^k (16m+14).$

The other is irregular, although the other form (class) in the same genus is regular. The generic misses for $3x^2 + 3 y^2 + 8 z^2 + 3 xy$ are $$ 3k+1 \; , \; \; \; 9k+6 \; , \; \; \; 4k+2 \; , \; \; \; 4^m(16k+10) \; . \; \; \; $$ The sporadic misses I have found are $$ 5, 23, 45, 77, 173, 207, 377, 495, 647, 693, 1557, 1683, 3393, 5823. $$ There is absolutely no way to prove the above list complete.

From Alexander Schiemann's tables at LATTICES, here we have discriminant 216, where your form is in the first listed genus, of just two forms. Note that 1896: and 1898: are just identification numbers for the computer, they have no mathematical meaning. The regular form is the second, $3x^2 + 5 y^2 + 5 z^2 + 2yz + 3 zx + 3xy \; . \; $

B.-I.discr = -216 = -1 *2^3 *3^3
    1^1 3^-1 9^-1 
        1^-2_II 16^1_1 
             1896: 3 3 8 0 0 3
             1898: 3 5 5 2 3 3
    1^-1 3^1 9^-1 
        1^2_II 16^-1_5 
             1887: 1 4 15 3 0 1
             1899: 4 4 4 -1 2 2
    1^-1 3^-1 9^1 
        1^-2_II 16^1_1 
             1882: 1 1 72 0 0 1
             1884: 1 3 19 3 1 0
    1^1 3^1 9^1 
        1^2_II 16^-1_5 
             1894: 2 5 6 3 0 1
    1^-2 27^-1 
        1^-2_II 16^1_1 
             1885: 1 3 20 2 0 1
             1889: 1 7 8 0 0 1
             1890: 1 7 9 5 1 1
             1891: 1 8 9 8 1 0
             1895: 3 3 7 -1 1 2
             1897: 3 4 5 0 1 2
    1^2 27^1 
        1^2_II 16^-1_5 
             1883: 1 2 31 1 0 1
             1886: 1 4 14 2 1 0
             1888: 1 7 8 1 1 0
             1892: 2 4 8 4 1 1
             1893: 2 4 8 2 1 2
             1900: 4 4 4 0 1 3

===========================================

=====Discriminant  216  ==Genus Size==   2
   Discriminant   216
  Spinor genus misses     no exceptions
       216:    3     3          8      0    0    3 vs. s.g.   5  23  77  173  377
       216:    3     5          5      2    3    3 vs. s.g.   regular candidate
--------------------------size 2
The 150 smallest numbers represented by full genus
     3     5     8     9    11    12    17    20    21    23
    27    29    32    35    36    39    41    44    45    47
    48    53    56    57    59    63    65    68    71    72
    75    77    80    81    83    84    89    92    93    95
    99   101   107   108   111   113   116   117   119   120
   125   128   129   131   135   137   140   143   144   147
   149   152   153   155   156   161   164   165   167   171
   173   176   179   180   183   185   188   189   191   192
   197   200   201   203   207   209   212   215   216   219
   221   224   225   227   228   233   236   237   239   243
   245   248   251   252   255   257   260   261   263   264
   269   272   273   275   279   281   284   287   288   291
   293   297   299   300   305   308   309   311   315   317
   320   323   324   327   329   332   333   335   336   341
   344   345   347   351   353   356   359   363   365   368


The 150 smallest numbers NOT represented by full genus
     1     2     4     6     7    10    13    14    15    16
    18    19    22    24    25    26    28    30    31    33
    34    37    38    40    42    43    46    49    50    51
    52    54    55    58    60    61    62    64    66    67
    69    70    73    74    76    78    79    82    85    86
    87    88    90    91    94    96    97    98   100   102
   103   104   105   106   109   110   112   114   115   118
   121   122   123   124   126   127   130   132   133   134
   136   138   139   141   142   145   146   148   150   151
   154   157   158   159   160   162   163   166   168   169
   170   172   174   175   177   178   181   182   184   186
   187   190   193   194   195   196   198   199   202   204
   205   206   208   210   211   213   214   217   218   220
   222   223   226   229   230   231   232   234   235   238
   240   241   242   244   246   247   249   250   253   254

Disc: 216
==================================


       216:    3     3          8      0    0    3
misses, compared with full genus
            5           23           45           77          173
          207          377          495          647          693



       216:    3     5          5      2    3    3
misses, compared with full genus

=====================================================

In comment, you mention the sum of three squares, and the numbers integrally represented in that manner.

In 1997, Irving Kaplansky and I and Alexander Schiemann found all the (reduced) positive forms that behave in the same way. These forms are "regular," meaning the numbers represented can be described by congruences. Actually, we were able to prove about 891 of these. At this point, there are still 8 forms with no proof.

I guess the thing to do is show the list of 102 diagonal regulars.

Here is a link to the article:

Here is a link to my hundred page introduction

There is a large collection of related material at TERNARY