Proof

Theorem

Let p be an odd prime. The list of p^th powers mod θ does not contain two consecutive nonzero (mod θ) integers if and only if x^p + y^p + z^p = 0 mod θ implies x = 0 mod θ or y = 0 mod θ or z = 0 mod θ.

Proof

Suppose x, y, and z are all nonzero mod θ and x^p + y^p + z^p = 0 mod θ. Then there is a nonzero (mod θ) integer a such that xa = 1 mod θ, and nonzero integers b and c with yb = 1 mod θ and zc = 1 mod θ. Then

x^p + y^p + z^p = 0 mod θ
=> (ax)^p + (ay)^p + (az)^p = 0 mod θ
=> (ay)^p + (az)^p = -1 mod θ

Let k = (ay)^p mod θ. Note that k ≠ 0 mod θ since otherwise

(ay)^p = 0 mod θ
=> 0 = a^py^pb^p = a^p(yb)^p = a^p mod θ

which is impossible if a is nonzero mod θ.

Therefore (θ-az)^p = -(az)^p = k+1 mod θ. Thus (ay)^p and (θ-az)^p are consecutive nonzero p^th powers.

Conversely, if a^p = k and b^p = k+1 are two consecutive nonzero (mod θ) integers, then let x=θ-a, y=b, and z=θ-1. Then

x^p + y^p + z^p = -a^p + b^p - 1 = -k + k + 1 - 1 = 0 mod θ