On the Diophantine equation |𝑎𝑥ⁿ-𝑏𝑦ⁿ|=1

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If a , b a, b and n n are positive integers with b ≥ a b \geq a and n ≥ 3 n \geq 3 , then the equation of the title possesses at most one solution in positive integers x x and y y , with the possible exceptions of ( a , b , n ) ( a, b, n ) satisfying b = a + 1 b = a + 1 , 2 ≤ a ≤ min { 0.3 n , 83 } 2 \leq a \leq \min \{ 0.3 n, 83 \} and 17 ≤ n ≤ 347 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.