The theorem states that this is the only case of two consecutive powers. That is to say, that the only solution in the natural numbers of.
The history of the problem dates back at least to Gersonides , who proved a special case of the conjecture in where x , y was restricted to be 2, 3 or 3, 2. In , Robert Tijdeman applied Baker's method in transcendence theory to establish a bound on a,b and used existing results bounding x , y in terms of a , b to give an effective upper bound for x , y , a , b.
Catalan's Conjecture - Numberphile
Michel Langevin computed a value of exp exp exp exp for the bound. Nonetheless, the finite calculation required to complete the proof of the theorem was too time-consuming to perform.
It makes extensive use of the theory of cyclotomic fields and Galois modules. It is a conjecture that for every natural number n , there are only finitely many pairs of perfect powers with difference n.
Pillai's conjecture concerns a general difference of perfect powers sequence A in the OEIS : it is an open problem initially proposed by S. Pillai , who conjectured that the gaps in the sequence of perfect powers tend to infinity.
The general conjecture would follow from the ABC conjecture. From Wikipedia, the free encyclopedia.
Theorem about consecutive powers. Does each positive integer occur only finitely many times as a difference of perfect powers? Springer Monographs in Mathematics. Diophantine approximations and Diophantine equations.
Lecture Notes in Mathematics.
Categories : Conjectures Diophantine equations Theorems in number theory Conjectures that have been proved. Namespaces Article Talk.
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