This is something I made up years ago when Kap got interested in the "tribonacci" numbers. A few weeks ago, I put this at [MSE][1] but no takers. Oh, years ago I asked some guy at MSRI about this, he muttered something about norm forms and laughed at me. 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 $f(a,b,c) \equiv 0 \pmod {q^3}.$ 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)][2], Theorem 1 and Table 1 on page 134. You get my version of the polynomial by negating their variable $x.$ =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ![enter image description here][3] =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= [1]: http://math.stackexchange.com/questions/336191/numbers-represented-by-a-cubic-form [2]: http://zakuski.utsa.edu/~jagy/Hudson_Williams_1991.pdf [3]: https://i.sstatic.net/xnDUC.jpg