Primes Produced by Quadratic Forms

[Max Alekseyev]

On Thu, Apr 24, 2008 at 5:36 PM, David Harden <oddleehr at alum.mit.edu> wrote:

>  >For degree(f_n(t)) > 2, there seems to be no direct relationship between
>  >the solveblity of f_n(t) modulo p and the determinant of f_n(t).
>  >However, I don't have a proof that the case h(-4n)>2 cannot >result in
>  >a simple description of the primes represented by x^2+n*y^2.
>
>  I have a counterexample: Q(sqrt(-30)) has class number 4.

[...]

>  The first form can, looking modulo 120, only produce primes p == 1, 31, 49 or 79 (mod 120).

Nice analysis.
Connecting it to the polynomial f_n(t), which in this case can be taken equal

f(t) = t^4 - 883067971104000*t^3 + 26329406807264910336000*t^2 -
2588458316335175909376000000*t + 4934510722321469030006784000000

we have that for a prime p>=7,
(-30/p)=1 and f(t)=0 modulo p has a root if and only if p == 1, 31, 49
or 79 (mod 120).
At first glance it still sounds quite unexpected. But perhaps, it can
be proved straightly from the value of f(t).

Regards,
Max