A conjecture of Erdös on 3-powerful numbers
Abstract: Erdös conjectured that the Diophantine equation has infinitely many solutions in pairwise coprime 3-powerful integers, i.e., positive integers for which implies . This was recently proved by Nitaj who, however, was unable to verify the further conjecture that this could be done infinitely often with integers , and none of which is a perfect cube. This is now demonstrated.
-  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, https://doi.org/10.1112/blms/27.4.317
- P. Erdös, Problems and results on consecutive integers, Eureka 38 (1975-76), 3-8.
- L. J. Mordell, Diophantine equations, Academic Press, London and New York, 1969, p. 78. MR 0249355
- A. Nitaj, On a conjecture of Erdös on 3-powerful numbers, Bull. London Math. Soc. 27 (1995), 317-318. MR 1335280
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J. H. E. Cohn
Affiliation: Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom
Received by editor(s): May 15, 1996
Received by editor(s) in revised form: September 13, 1996
Article copyright: © Copyright 1998 American Mathematical Society