math -- values of f(x) (mod p)

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
.

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