Why the Little Methuselah form is the "Little Methuselah"s form?

Question

This is my first question on MathStackexchange. Let me know if I am violating rules, or my question is somewhat ugly.

I am reading Conway's book "Sensual (Quadratic) Form". He introduces a tenary quadratic form $ F(x,y,z) = x^2 + 2y^2 + yz + 4z^2$. This form has a name "Little Methuselah form". And he introduces a theorem about this form;

$F$ represents every integer from $1$ to $30$, and fails to represent $31$.
Every integer-valued positive definite quadratic tenary form $G$, not equivalent to $F$, fails to represent some integer between $1$ to $30$.

My question is : Why this form $F$ has such a name? I know that Methuselah is the greatest macrobian in Genesis, but I cannot relate this name and this form. Is there any Big Methuselah form? Or, does this form "live long" in some manner?

Thank you. And, Happy birthday Conway! (26th Dec)

Answer 1

There is a Methuselah form. It gives the extreme behavior in the 290 theorem of Bhargava and Hanke. It is

$$ h(x,y,z,t,u) = x^2 + 2 y^2 + 4 z^2 + y z + z x + 29 (t^2 + t u + u^2). $$ It integrally represents every number from 1 to 289. It does not represent 290. Then it represents every number 291, 292, 293, on forever.

The form is incorrectly typed in Duke's survey article in the AMS Notices, see Duke_1997.pdf at TERNARY. It is given correctly on pages 9 to 10 in Jagy_Encyclopedia.pdf at the same site. It appears I spelled Methuselah incorrectly there.

In my tables below, a (positive) ternary quadratic form is given as $$ \Delta : \; A \; \; B \; \; C \; \; R \; \; S \; \; T $$ which refers to $$ f(x,y,z) = A x^2 + B y^2 + C z^2 + R y z + S z x + T x y, $$ and $$ \Delta = 4ABC + RST - AR^2 - BS^2 - CT^2 $$ is my normalization for the discriminant, same as Lehman 1992.

Let's see, first 31:

---

=====Discriminant  31  ==Genus Size==   3
   Discriminant   31
  Spinor genus misses     no exceptions
        31:    1     1          8      0    1    0 
        31:    1     2          4      1    0    0 
        31:    1     2          5      2    1    1 
--------------------------size 3
The 150 smallest numbers represented by full genus
     1     2     3     4     5     6     7     8     9    10
    11    12    13    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    39    40
    41    42    43    44    45    46    47    48    49    50
    51    52    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    94    95    96    97    98    99   100   101
   102   103   104   105   106   107   108   109   110   111
   112   113   114   115   116   117   118   119   120   121
   122   123   124   125   126   127   128   129   130   131
   132   133   134   135   136   137   138   139   140   141
   142   143   144   145   146   147   148   149   150   151

The 150 smallest numbers NOT represented by full genus
    93   186   341   372   403   465   527   651   682   713
   744   806   837   899   930
Disc: 31
==================================


        31:    1     1          8      0    1    0
misses, compared with full genus
            3            6            7           22           27
           43           55           62          155          166
          187          217          478          558          589



        31:    1     2          4      1    0    0
misses, compared with full genus
           31          279


        31:    1     2          5      2    1    1
misses, compared with full genus
            3           10           13           31           62
          124          310

---

Now 29

---

=====Discriminant  29  ==Genus Size==   3
   Discriminant   29
  Spinor genus misses     no exceptions
        29:    1     1         10      1    1    1 
        29:    1     2          4      1    1    0 
        29:    1     3          3      2    0    1 
--------------------------size 3
The 150 smallest numbers represented by full genus
     1     2     3     4     5     6     7     8     9    10
    11    12    13    14    15    16    17    18    19    20
    21    22    23    24    25    26    27    28    30    31
    32    33    34    35    36    37    38    39    40    41
    42    43    44    45    46    47    48    49    50    51
    52    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   117   118   119   120   121   122
   123   124   125   126   127   128   129   130   131   132
   133   134   135   136   137   138   139   140   141   142
   143   144   146   147   148   149   150   151   152   153

The 150 smallest numbers NOT represented by full genus
    29   116   145   174   203   261   377   464   580   638
   667   696   725   812   870   957   986
Disc: 29
==================================


        29:    1     1         10      1    1    1
misses, compared with full genus
            2            5            6            8           17
           23           33           38           58           77
           78           98          101          137          138
          158          182          232          233          305
          353          406          458          493          551
          557          609          767


        29:    1     2          4      1    1    0
misses, compared with full genus
           87          290


        29:    1     3          3      2    0    1
misses, compared with full genus
            2           58

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