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Perfect Powers: Pillai's works and their developments

Michel Waldschmidt (  Auteur correspondant ) 1

(2009)

A perfect power is a positive integer of the form $a^x$ where $a\ge 1$ and $x\ge 2$ are rational integers. Subbayya Sivasankaranarayana Pillai wrote several papers on these numbers. In 1936 and again in 1945 he suggested that for any given $k\ge 1$, the number of positive integer solutions $(a,\, b,\, x,\, y)$, with $x\ge 2$ and $y\ge 2$, to the Diophantine equation $a^x-b^y=k$ is finite. This conjecture amounts to saying that the distance between two consecutive elements in the sequence of perfect powers tends to infinity. After a short introduction to Pillai's work on Diophantine questions, we quote some later developments and we discuss related open problems.

1 :	Institut de Mathématiques de Jussieu (IMJ)
	CNRS : UMR7586 – Université Pierre et Marie Curie - Paris VI – Université Paris-Diderot - Paris VII

Domaine

:

Mathématiques/Théorie des nombres

Mots Clés : Diophantine Equations – Pillai's Conjecture – Catalan's Conjecture – perfect powers – abc Conjecture

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Contributeur : Michel Waldschmidt <>

Soumis le : Vendredi 24 Juillet 2009, 10:14:39

Dernière modification le : Vendredi 24 Juillet 2009, 14:31:30