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A conjecture of Erdös on 3-powerful numbers


Author: J. H. E. Cohn
Journal: Math. Comp. 67 (1998), 439-440
MSC (1991): Primary 11P05
DOI: https://doi.org/10.1090/S0025-5718-98-00881-3
MathSciNet review: 1432127
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Abstract: Erdös conjectured that the Diophantine equation $x+y=z$ has infinitely many solutions in pairwise coprime 3-powerful integers, i.e., positive integers $n$ for which $p\mid n$ implies $p^3\mid n$. This was recently proved by Nitaj who, however, was unable to verify the further conjecture that this could be done infinitely often with integers $x$, $y$ and $z$ none of which is a perfect cube. This is now demonstrated.


References [Enhancements On Off] (What's this?)

  • [1] P. Erdös, Problems and results on consecutive integers, Eureka 38 (1975-76), 3-8.
  • [2] L. J. Mordell, Diophantine equations, Academic Press, London and New York, 1969, p. 78. MR 40:2600
  • [3] A. Nitaj, On a conjecture of Erdös on 3-powerful numbers, Bull. London Math. Soc. 27 (1995), 317-318. MR 96b:11045

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Additional Information

J. H. E. Cohn
Affiliation: Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom
Email: J.Cohn@rhbnc.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-98-00881-3
Received by editor(s): May 15, 1996
Received by editor(s) in revised form: September 13, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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