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Oeis A306970 lists positive integers of the form $8k+3$ which are not reprented by $$f(x,y,z):=2x^2+4y^2+4yz+9z^2$$

over the integers as $3,43,163,907$. It says this list may not be complete and references an article by Kaplansky, which in turn references page 182 of this article by Jones and Pall. Kaplansky gives the stronger result: There are no further integers $10^5> 8k+3>0$ which are not represented by $f$ over the integers.

I'm not very familiar with the theory of quadratic forms. Can anyone prove A306970 is complete, or give a reference to a proof?

Background: Let $d<0$ be a fundamental discriminant, $d\equiv 5\pmod 8$, let $\tau\in\mathcal{H}$ be a point of discriminant $d$ and let $j:=j(\tau)$. Then $2^{-15}j$ is an algebraic integer. If $2^{-16}j$ is an algebraic integer, then $-d$ is not represented by $f$ over the integers. On the other hand, it turns out that $2^{-16}j\in\overline{\mathbb{Z}}$ for all $d\in\{-3,-43,-163,-907\}$.

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  • $\begingroup$ $f(x, y, z) = 2x^2 + (2y+z)^2 + 2(2z)^2$, there might be relevant generalizations of Legendre's three-square theorem. $\endgroup$
    – Daniel Weber
    Commented Nov 3, 2023 at 14:53
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    $\begingroup$ I think this is likely to be very hard. For comparison, there are three positive-definite ternary quadratic forms that are conjectured to represent all positive integers, but for which there is currently no proof. Proving that A306970 is complete assuming GRH is probably doable. (I did something similar in Section 6 of the paper here.) $\endgroup$
    – Jeremy Rouse
    Commented Nov 3, 2023 at 16:34
  • $\begingroup$ What is meant by the discriminant of a point? What is meant by $\overline{\bf Z}$? $\endgroup$
    – Gerry Myerson
    Commented Nov 4, 2023 at 8:51
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    $\begingroup$ @GerryMyerson, I write $\overline{\mathbb{Z}}$ for the ring of all algebraic integers over $\mathbb{Z}$, and a point $\tau\in\mathcal{H}$ has discriminant $d$ if there exist $a,b,c\in\mathbb{Z}$ with $(a,b,c)=1$, $a\tau^2+b\tau+c=0$ and $d=b^2-4ac$. $\endgroup$
    – Mastrem
    Commented Nov 4, 2023 at 12:14
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    $\begingroup$ @CommandMaster, $f(x,y,z)=(x+2z)^2+(x-2z)^2+(2y+z)^2$ as well. $\endgroup$
    – Mastrem
    Commented Nov 4, 2023 at 12:15

1 Answer 1

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First things first, all the relevant numbers really are represented by $x^2 + 2 y^2 + 32 z^2.$ See http://zakuski.math.utsa.edu/~kap/Kap_Jagy_Schiemann_1997.pdf for regular ternaries.

From my giant list of ternary genera with spinor genera and list of sporadic numbers up to some bound I don't remember.

As Jeremy points out, this one has bad features, not spinor regular, more than one squareclass of numbers missing compared with the full genus. Jones and Pall proved good behavior with local restriction $1 \pmod 8,$ but that misses your desired $8k+3$ Let me see what Kap article you've cited.

I put lots of ternary things at http://zakuski.math.utsa.edu/~kap/

including my notes http://zakuski.math.utsa.edu/~kap/Jagy_Encyclopedia.pdf

and there is reason to think that webpage will be on good behavior for the next year.

http://zakuski.math.utsa.edu/~jagy/report_1000.txt

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

=====Discriminant  256  ==Genus Size==   2
   Discriminant   256
  Spinor genus misses     no exceptions  
       256:    1     2         32      0    0    0 vs. s.g.   regular candidate 
       256:    2     4          9      4    0    0 vs. s.g.   1  3  43  163  907
--------------------------size 2 
The 150 smallest numbers represented by full genus
     1     2     3     4     6     8     9    11    12    16
    17    18    19    22    24    25    27    32    33    34
    35    36    38    40    41    43    44    48    49    50
    51    54    56    57    59    64    65    66    67    68
    70    72    73    75    76    80    81    82    83    86
    88    89    91    96    97    98    99   100   102   104
   105   107   108   113   114   115   118   120   121   123
   128   129   130   131   132   134   136   137   139   140
   144   145   146   147   150   152   153   155   160   161
   162   163   164   166   168   169   171   172   176   177
   178   179   182   184   185   187   192   193   194   195
   196   198   200   201   203   204   208   209   210   211
   214   216   217   219   224   225   226   227   228   230
   232   233   235   236   241   242   243   246   248   249
   251   256   257   258   259   260   262   264   265   267

The 150 smallest numbers NOT represented by full genus
     5     7    10    13    14    15    20    21    23    26
    28    29    30    31    37    39    42    45    46    47
    52    53    55    58    60    61    62    63    69    71
    74    77    78    79    84    85    87    90    92    93
    94    95   101   103   106   109   110   111   112   116
   117   119   122   124   125   126   127   133   135   138
   141   142   143   148   149   151   154   156   157   158
   159   165   167   170   173   174   175   180   181   183
   186   188   189   190   191   197   199   202   205   206
   207   212   213   215   218   220   221   222   223   229
   231   234   237   238   239   240   244   245   247   250
   252   253   254   255   261   263   266   269   270   271
   276   277   279   282   284   285   286   287   293   295
   298   301   302   303   308   309   311   314   316   317
   318   319   325   327   330   333   334   335   340   341

Disc: 256
==================================


       256:    1     2         32      0    0    0
misses, compared with full genus 



       256:    2     4          9      4    0    0
misses, compared with full genus 
    1:      1            3           43          163          907

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

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