Class number bounds and Catalan's equation

We improve a criterion of Inkeri and show that if there is a solution to Catalan’s equation xp − yq = ±1,(1) with p and q prime numbers greater than 3 and both congruent to 3 (mod 4), then p and q form a double Wieferich pair. Further, we refine a result of Schwarz to obtain similar criteria when only one of the exponents is congruent to 3 (mod 4). Indeed, in light of the results proved here it is reasonable to suppose that if q ≡ 3 (mod 4), then p and q form a double Wieferich pair.

• Type: Journal
• Year: 1998
• Where: MOC
• Authors: Ray Steiner
• Keywords: Catalan’s Equation Xp, Double Wieferich Pair, MOC 1998, Prime Numbers