Question 145
Let us consider the equation
$$
(a^n - b^n)^2 = a^{n+m} - b^{n+m}
$$
for positive integers $n$ and $m$ such that $n \ge m \ge 1$. Show that this equation has no integer solution $(a, b)$ satisfying $\gcd(a, b) = 1$ and $|a| \gt |b| \gt 1$.
$$
(a^n - b^n)^2 = a^{n+m} - b^{n+m}
$$
for positive integers $n$ and $m$ such that $n \ge m \ge 1$. Show that this equation has no integer solution $(a, b)$ satisfying $\gcd(a, b) = 1$ and $|a| \gt |b| \gt 1$.