I'm planning a challenge over on Code Golf.SE about integers $a, b, c \ge 0$ such that

$$a^n + b^n = c^n \pm 1$$

for a given integer $n > 2$. However, I'm interested in whether any non-trivial solutions to this exist for a given $n$. Here, I'm defining "non-trivial" solutions as triples $a, b, c$ such all three are unique and non-zero (i.e. to avoid $(a, 1, a)$ and $(a, 0, a)$, and related triples).

I've found this question which asks a related (and broader) question about the existence of such triples, and the accepted answer states

I think that if $n\ge5$ (and assuming the ABCD conjecture), then for any $k$, the equation $$ a^n + b^n - c^n = k $$ has only finitely many solutions $a,b,c\in\mathbb{Z}$ with $|a|,|b|,|c|$ distinct and non-zero.

However, this doesn't fully state whether there are a non-zero number of distinct, non-zero solutions.

This is a program which attempts to find such triples, with $0 \le a, b, c \le 100$, given an input $n$, but so far it hasn't found any for either $n = 4$ or $n = 5$, and it times out if you increase the upper limit by any significant amount.

Therefore, my question is:

• Can it be shown that, for all integers $n > 2$, the equation $a^n + b^n = c^n \pm 1$ has at least 1 non-trivial solution, for $a, b, c \ge 0$?
• If not, does expanding the range for $a, b, c$ to $\mathbb{Z}$ affect or change this?