On the Diophantine equation |axn - byn | = 1

If a, b and n are positive integers with b ≥ a and n ≥ 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 ≤ a ≤ min{0.3n, 83} and 17 ≤ n ≤ 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.

• Type: Journal
• Year: 1998
• Where: MOC
• Authors: Michael A. Bennett, Benjamin M. M. de Weger
• Keywords: Diophantine Approximation Techniques, MOC 1998, Positive Integers, Possible Exceptions