Re: x^n + y^n (mod p)
- From: Gerry Myerson <gerry@xxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Sun, 21 Oct 2007 23:02:57 GMT
In article <67glh3h7h39d6kd8e2svlh23ki3h3boukm@xxxxxxx>,
quasi <quasi@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.
Let p be a prime congruent to 1 (mod n), say, p = d n + 1.
Then there are only d (non-zero) n-th powers (mod p),
so only about d^2 values of f. Now it's surely true (though maybe
not yet proved) that d can be taken smaller than n, so d^2 < p,
so f is not surjective.
Conjecture (2):
For each positive integer n, and all sufficiently large primes p
(depending on n), f is surjective.
This is true, and follows from Weil's bounds on exponential
sums. I think it's known to be true for p > N_0(n), where
N_0(n) is something like n^4. I'll see if I can chase up some
references. I vaguely recall there was a paper by Chowla.
--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.
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