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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A conjecture of Erdös on 3-powerful numbers
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by J. H. E. Cohn PDF
Math. Comp. 67 (1998), 439-440 Request permission

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
  • P. Erdös, Problems and results on consecutive integers, Eureka 38 (1975–76), 3–8.
  • L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
  • Abderrahmane Nitaj, On a conjecture of Erdős on $3$-powerful numbers, Bull. London Math. Soc. 27 (1995), no. 4, 317–318. MR 1335280, DOI 10.1112/blms/27.4.317
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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
  • Received by editor(s): May 15, 1996
  • Received by editor(s) in revised form: September 13, 1996
  • © Copyright 1998 American Mathematical Society
  • 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