# Numbers represented by a cubic form

EDIT, April 11, 2013: See answer at https://mathoverflow.net/questions/127160/numbers-integrally-represented-by-a-ternary-cubic-form/127295#127295

This is part 2 ( of 25 discriminants of class number exactly three) of primes represented integrally by a homogeneous cubic form I think two will be enough.

Given integers $a,b,c,$ and cubic form
$$f(a,b,c) = a^3 + b^3 + c^3 + a^2 b – a b^2 + 3 a^2 c – a c^2 + b^2 c – b c^2 – 4 a b c$$
$$f(a,b,c) = \det \left( \begin{array}{ccc} a & b & c \\ c & a + c & b + c \\ b + c & b + 2 c & a + b + 2 c \end{array} \right) .$$
what primes $p$ can be integrally represented as
$$p = f(a,b,c)?$$

I think it is all primes $(p| 11) = -1 ,$ and all $p = u^2 + 11 v^2$ in integers, but not any $q = 3 u^2 + 2 u v + 4 v^2.$
Note that, if $-p$ is represented, so is $p.$

I also suspect that if prime $q = 3 u^2 + 2 u v + 4 v^2$ and $f(a,b,c) \equiv 0 \pmod q,$ then all three $a,b,c \equiv 0 \pmod q,$ and $f(a,b,c) \equiv 0 \pmod {q^3}.$

Note that if $f$ integrally represents both $m,n$ then it represents $mn.$ That is because $f(a,b,c) = \det(aI + b X + c X^2),$ where
$$X = \left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \end{array} \right)$$ Then $X^3 = X^2 + X + I$ and $X^4 = 2 X^2 + 2 X + I.$

If all suspicions are correct, we can correctly describe all numbers integrally represented by this polynomial: positive or negative are unimportant, most prime factors are unimportant, all that matters is that every exponent of a prime factor $q = 3 u^2 + 2 u v + 4 v^2$ must be divisible by 3.

I should have done this last time: most of the class field part has already been done, by Hudson and Williams (1991), Theorem 1 and Table 1 on page 134. You get my version of the polynomial by negating their variable $x.$ Note the connection to the “tribonacci” numbers, with precisely the formalism I described at Fibonacci Sequence in $\mathbb Z_n$.

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           p           a           b           c
................................................
2           0           1           1
7           0         -11           6
11           0          -3           2
13           0          -1           2
17          -1           0           2
19           1           2           4
29           0          -7           4
41           0           3           2
43           0           4          -1
47           0           5          -2
53           0           1           4
61           0          46         -25
73           2         -36          19
79           0           3           4
83           0          24         -13
101          -1          12          -6
103           0          15          -8
107           1          -9           5
109           1           2           6
127           1          -2           4
131           1           7          -3
139           1          -6           4
149          -1           4           2
151           0         -20          11
163           0           5           2
167          -1           1           5
173           0           6          -1
193           1         -52          28
197           0           9          -4
199          -1           5           1
211           0         -12           7
227          -2           0           5
233           0         -16           9
239           0          -6           5
241           0          -4           5
257           0          -1           6
263           2           4           9
269          -1           0           6
271           2           8          -3
277           1          -7           5
281           0           2           7
283          -1           2           6
293          -1          -8           6
307           2          -1           6
311           0           5           6
337          -2           5           2
347           1           7           5
349           0          19         -10
359          -1           9          -3
373           2           5          10
397           1          -1           7
401           0         -68          37
409           3         -77          41
419           0          -7           6
421           0           7           2
431           1         -14           8
439           0           8          -1
457           0           1           8
461           0          -2           7
479           1          -8           6
491           0           7           4
499           0          13          -6
503          -1         -36          20
523           0           9          -2
541           2         -12           7
547           1         -11           7
557          -1          25         -13
563          -2         -11           8
569           0           8           1
571           1          -3           7
587           0         -29          16
593           3         -25          13
599          -1           0           8
601           0           7           6
607           0          11          -4
613           0           4           9
617           2          -1           8
659           0           8           3
673           0          -6           7
677           0         -17          10
683          -1           4           8
701           2          13          -6
733           1          10          -2
739          -1          14          -6
743          -2           1           8
757           0          81         -44
761          -1           8           2
769           0         -25          14
773          -1           7           5
787           2           5          12
809          -1         -10           8
811          -4           0           7
821          -1           3           9
827           2          10           7
853           0         -11           8
857          -2           3           8
863           0           9           2
877          -2         -15          10
883           0         -14           9
887           2          -3           8
907           0          -5           8
911           0           8           7
919           0          -2           9
929           1           7          11
937           3           8          14
941           3          -1           9
953          -1           6           8
967           1          13          -5
991           1         -35          19
997          -3           7           3


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