Question 159
Find all quintuples of positive integers $(a, b, c, m, n)$ such that $m \le 2n - 2$ and
$$
a^n + b^n = (a, b)^m (a + b), \quad b^n + c^n = (b, c)^m (b + c), \quad c^n + a^n = (c, a)^m (c + a).
$$
Here, $(x, y)$ denotes the greatest common divisor of $x$ and $y$.
(Bilegdemberel Bat-Amgalan)
$$
a^n + b^n = (a, b)^m (a + b), \quad b^n + c^n = (b, c)^m (b + c), \quad c^n + a^n = (c, a)^m (c + a).
$$
Here, $(x, y)$ denotes the greatest common divisor of $x$ and $y$.
(Bilegdemberel Bat-Amgalan)