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I made this up years ago when Kap got interested in the "tribonacci" numbers. Later I asked someone else at MSRI about this, he muttered something about norm forms and laughed at me.

(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 $f(a,b,c) \equiv 0 \pmod {q^3}.$

(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.

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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This is something I made up years ago when Kap got interested in the "tribonacci" numbers. Oh, years ago I asked some guy at MSRI about this, he muttered something about norm forms and laughed at me.

I made this up years ago when Kap got interested in the "tribonacci" numbers. Later I asked someone else at MSRI about this, he muttered something about norm forms and laughed at me.