Re: x^n + y^n (mod p)

On Sun, 21 Oct 2007 13:53:04 -0700, OwlHoot
<ravensdean@xxxxxxxxxxxxxx> wrote:

On Oct 21, 2:34 pm, quasi <qu...@xxxxxxxx> wrote:
On Sun, 21 Oct 2007 06:08:36 -0700, OwlHoot

<ravensd...@xxxxxxxxxxxxxx> wrote:
On Oct 21, 4:53 am, quasi <qu...@xxxxxxxx> wrote:

Given a positive integer n, and a prime p, define a map

f : Z_p x Z_p --> Z_p

by

f(x,y) = x^n + y^n

Two conjectures ...

Conjecture (1):

For each positive integer n>1, there is at least one prime p
such that f is not surjective.

You can narrow down the possible values of n
without such a prime p by noting that if q = (p - 1)/2 is a factor
of n for some odd prime p then z^n == 0, 1, or p - 1 mod p for any
integer z. So if p > 3 then f cannot be surjective for these values
of n.

Similarly, if phi(m) divides n for some integer m then z^n == 1 mod m,
and hence any prime factor of m gives a non-surjective f. Your result
would then follow if every integer > 0 is phi(m) (or phi(m)/2 if this
is an integer) for some m;

Nice observation.

but I'm not sure if this is true.

Me neither.

quasi
.

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