Primes Produced by Quadratic Forms
Max Alekseyev
maxale at gmail.com
Fri Apr 25 10:30:26 CEST 2008
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
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