2009, English, Article edition: Perfect Powers: Pillai's works and their developments Waldschmidt, Michel

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Physical Description
  • text
Published
  • HAL - CCSD
  • 2009
Language
  • English

Edition details

Title
  • Perfect Powers: Pillai's works and their developments
Author
  • Waldschmidt, Michel
Published
  • HAL - CCSD
  • 2009
Physical Description
  • text
Subjects
Notes
  • 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.
  • HAL:hal-00405119, version 1
  • HAL:http:/​/​hal.archives-ouvertes.fr/​hal-00405119/​en/​
Language
  • English
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