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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 $$ p = f(a,b,c)? $$

(A): 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 $p.$

(B): 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 $f(a,b,c) \equiv 0 \pmod {q^3}.$ Checked correct for $q=3,5.$ Maybe I will do a few more.

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 $x.$

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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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Primes represented by $x^2 + 11 y^2$ and then by $3 x^2 + 2 x y + 4 y^2,$ both up to $1000.$

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./primego
Input three coefficients a b c for positive f(x,y)= a x^2 + b x y + c y^2 
1  0  11
Discriminant  -44

Modulus for arithmetic progressions? 
11
Maximum number represented? 
1000
           p        p  mod 11 
          11           0
          47           3
          53           9
         103           4
         163           9
         199           1
         257           4
         269           5
         311           3
         397           1
         401           5
         419           1
         421           3
         499           4
         587           4
         599           5
         617           1
         683           1
         757           9
         773           3
         863           5
         883           3
         907           5
         911           9
         929           5
         991           1


    0    1    3    4    5    9

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./primego
Input three coefficients a b c for positive f(x,y)= a x^2 + b x y + c y^2 
3  2  4
Discriminant  -44

Modulus for arithmetic progressions? 
11
Maximum number represented? 
1000
           p        p mod 11
           3           3
           5           5
          23           1
          31           9
          37           4
          59           4
          67           1
          71           5
          89           1
          97           9
         113           3
         137           5
         157           3
         179           3
         181           5
         191           4
         223           3
         229           9
         251           9
         313           5
         317           9
         331           1
         353           1
         367           4
         379           5
         383           9
         389           4
         433           4
         443           3
         449           9
         463           1
         467           5
         487           3
         509           3
         521           4
         577           5
         619           3
         631           4
         641           3
         643           5
         647           9
         653           4
         661           1
         691           9
         709           5
         719           4
         727           1
         751           3
         797           5
         823           9
         829           4
         839           3
         859           1
         881           1
         947           1
         971           3
         977           9
         983           4


    1    3    4    5    9

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$

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joro asked about high powers being represented primitively. It turns out that the polynomial is not divisible by 8 unless $a,b,c$ are all even. This, despite the fact that 2 is represented. I believe this happens for all the (unrepresented) primes $q = 3 u^2 + 2 u v + 4 v^2$ as well, in the strongest manner: the polynomial is not divisible by $q$ itself unless $a,b,c$ are. I thought there might be trouble with the prime 11, but no. Anyway, here are some prime powers represented primitively, where $47 = 36 + 11$ and $53 = 9 + 44:$

           7           0           1           2
          49           1          -1           3
         343           6           4           5
        2401         -11          -3           9
       16807         -11          30          -8
      117649         -19          75         -29
      823543          -2        -117          82
     5764801         162          43          12
    40353607         205         -64         186



          11          -1           1           1
         121          10          15          16
        1331         -10          -2           7
       14641          12          28           9
      161051           1          25          59
     1771561          53         -78          70
    19487171          37          46         300
   214358881         171        -210         460




          13           1           3           3
         169          10          17          18
        2197          -4           3          10
       28561         -15          -8          24
      371293           8          71          34
     4826809         -54          98          77
    62748517        -257         125         167



          47           1           3           5
        2209          10          12           3
      103823         108         181         202
     4879681         104          32         153
   229345007        -128         319         432




          53          -1           1           3
        2809          10          23          24
      148877         100         163         170
     7890481         100          18         187
   418195493         342        -308         451

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Improve this question
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    $\begingroup$ Can $f$ represents large power with coprime $a,b,c$? In the bivariate case this is forbidden by an abc related conjecture. $\endgroup$
    – joro
    Commented Apr 11, 2013 at 13:02
  • $\begingroup$ @joro, I don't believe that will be a problem here, but I will do some checking. The main thing is that this behaves very much as the principal binary quadratic form of a discriminant, there is a rule for multiplication. $\endgroup$
    – Will Jagy
    Commented Apr 11, 2013 at 19:33
  • $\begingroup$ Sure, there are large powers with primitive representations. This follows from the multiplicativity that Will J. already noted. For instance, $f(3356898, 3732782, 5764967) = 7^{23}$ (I started from $f(2,1,0)=7$ and used $7^{23}=1^{10}7^{23}$ and $f(2,-1,0)=1$ to reduce the resulting $(a,b,c)=(472709258936428, 396738620092614, 257006830281609)$ ). $\endgroup$
    – Noam D. Elkies
    Commented Apr 11, 2013 at 20:32
  • $\begingroup$ @Noam, thanks. I've been checking, it appears the only primes to worry about are 2 and 11. I do think that the polynomial is not divisible by 8 unless all three variables are even ( a small finite check mod 8, not done yet). Perhaps something similar for 11, not sure yet. $\endgroup$
    – Will Jagy
    Commented Apr 11, 2013 at 21:12
  • $\begingroup$ $z^{k-1} x + y^k$ is surjective for all $k$ so large powers are represented primitively in arbitrary large degree with $z=1$. $\endgroup$
    – joro
    Commented Apr 12, 2013 at 8:16

1 Answer 1

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Your conjectures are correct. So was the "someone else at MSRI [who] muttered something about norm forms" (mentioned in earlier edits of the question), except for the part about laughing at you.

As you in effect note, $f(a,b,c)$ is the norm $N_{K/{\bf Q}}(a+bx+cx^2)$, where $x$ is one of the roots of $x^3-x^2-x-1 = 0$ and $K$ is the cubic number field ${\bf Q}(x)$. This field has discriminant $-44$, and ${\bf Z}[x]$ is the full ring of integers $O_K$ (equivalently, the field discriminant of $K/{\bf Q}$ equals the polynomial discriminant of $x^3-x^2-x-1$; to check this in gp, compute

poldisc(x^3-x^2-x-1)
nfdisc(x^3-x^2-x-1)

and observe that both return $-44$). Now for (A), you already know that $x^3-x^2-x-1$ has at least one root modulo any prime $q$ unless $q$ is represented by the nonprincipal quadratic form $3u^2+2uv+4v^2$ of discriminant $-44$. (For other $q$: there's a triple root for $q=2$, a double and a simple root for $q=11$, three distinct roots for $q=u^2+11v^2$, and one simple root for odd $q$ not congruent to a square $\bmod 11$.) Equivalently, $K$ has an ideal of norm $q$ unless $q = 3u^2+2uv+4v^2$. But $O_K$ is a principal ideal domain, so once there's an ideal of norm $q$ then it has a generator $a+bx+cx^2 \in O_K$, and then $q=f(a,b,c)$ (or $q=f(-a,-b,-c)$ if we chose $a+bx+cx^2$ of norm $-q$). The discriminant of $K$ is small enough that one can check unique factorization by hand using the Minkowski bound; nowadays this exercise can also be done routinely on the computer, e.g. in gp

K = bnfinit(x^3-x^2-x-1); K.cyc

(This functionality happens to be one of the "Usage examples" in the current Wikipedia page on gp.)

[EDIT In fact this $K$ happens to be one of the handful of number fields whose Minkowski bound is so tight that nothing needs to be checked! The discriminant $\Delta_K = -44$ is small enough in absolute value that the bound $$ \frac4\pi \frac{3!}{3^3} \left|\Delta_K\right|^{1/2} = 1.8768\ldots $$ is less than $2$, which means every ideal $I$ has a nonzero element of norm $\pm \left|I\right|$ and is thus automatically principal. TIDE]

(B) Translating the factorization of $x^3-x^2-x-1 \bmod q$ into the factorization of the ideal $(q)$ in $O_K$, we see that if $q = 3u^2+2uv+4v^2$ then $(q)$ remains prime in $O_K$, and thus that $q \mid N_{K/{\bf Q}}(a+bx+cx^2)$ iff $q \mid a+bx+cx^2$. For $q=2$ the ideal $(q)$ is the cube of $(1+x)$, so $8 \mid f(a,b,c)$ iff $a,b,c$ are all even. Any power of a prime $q$ other than those of the form $3u^2+2uv+4v^2$ can be represented primitively by $f$, even $q=11$ (for which $(q)$ factors as $(2+x)(3-2x)^2$). If we do not care about primitivity then we can also represent all powers of $2$, and all powers of $q^3$ for $q = 3u^2+2uv+4v^2$.

By multiplicativity this also proves the final conjecture: the nonzero $n \in {\bf Z}$ that are represented by $f$ are precisely those whose $q$-valuation is a multiple of $3$ for all primes $q = 3u^2+2uv+4v^2$.

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    $\begingroup$ Oh, EDIT spelled backwards is TIDE thus indicating the end of an edited section... $\endgroup$
    – Will Jagy
    Commented Apr 12, 2013 at 18:16

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