On the Diophantine equation $|ax^{n}-by^{n}|=1$

If $a, b$ and $n$ are positive integers with $b \geq a$ and $n \geq 3$, then the equation of the title possesses at most one solution in positive integers $x$ and $y$, with the possible exceptions of $( a, b, n )$ satisfying $b = a + 1$, $2 \leq a \leq \min \{ 0.3 n, 83 \}$ and $17 \leq n \leq 347$. The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometric functions, the theory of linear forms in logarithms and recent computational methods related to lattice-basis reduction. Additionally, we compare and contrast a number of these last mentioned techniques.