Re: math -- values of f(x) (mod p)
- From: amy666 <tommy1729@xxxxxxxxxxx>
- Date: Tue, 15 Apr 2008 07:08:09 EDT
Two conjectures ...
Conjecture (1):
If n is an odd positive integer, then for all
sufficiently large
primes p (depending on n), there does not exist f in
Z_p[x], with
deg(f) = n, such that for all r in Z_p, f(r) is a
square in Z_p.
Conjecture (2):
If n is an even positive integer, then for all
sufficiently large
primes p (depending on n), if f in Z_p[x], with
deg(f) = n, is such
that for all r in Z_p, f(r) is a square in Z_p, then
f = g^2 for some
g in Z_p[x].
Remarks:
Conjecture (1) is trivially true for n = 1.
Conjecture (2) is true for n = 2, and can be proved
as a corollary to
the proposition I proved in the thread "quadratic
quadratic
non-residue".
For n = 3, 4, 5, 6, test data convincingly supports
the conjectures,
but of course, doesn't prove them.
Short of a general proof, a proof for n = 3 or n = 4
would be nice.
quasi
this is very close to a thread of myself.
good.
perhaps you can find inspiration in my posts.
regards
tommy1729
.
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- From: quasi
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