On some Diophantine systems involving symmetric polynomials
Author:
Maciej Ulas
Journal:
Math. Comp. 83 (2014), 19151930
MSC (2010):
Primary 11D25, 11G05
Published electronically:
October 22, 2013
MathSciNet review:
3194135
Fulltext PDF
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Abstract: Let be the th elementary symmetric polynomial. In this note we generalize and extend the results obtained in a recent work of Zhang and Cai. More precisely, we prove that for each and rational numbers with , the system of diophantine equations has infinitely many solutions depending on free parameters. A similar result is proved for the system with and . Here, are rational numbers with . We also give some results concerning the general system of the form with suitably chosen rational values of and . Finally, we present some remarks on the systems involving three different symmetric polynomials.
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 I. Connell, Elliptic curve handbook, preprint, available at http://www.ucm.es/ BUCM/mat/doc8354.pdf
 [2]
 Richard K. Guy, Unsolved problems in number theory, 3rd ed., Problem Books in Mathematics, SpringerVerlag, New York, 2004. MR 2076335 (2005h:11003)
 [3]
 L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London, 1969. MR 0249355 (40 #2600)
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 A. Schinzel, Triples of positive integers with the same sum and the same product, Serdica Math. J. 22 (1996), no. 4, 587588. MR 1483607 (98g:11033)
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 Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, SpringerVerlag, New York, 1986. MR 817210 (87g:11070)
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 Th. Skolem, Diophantische Gleichungen, Chelsea, 1950.
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Additional Information
Maciej Ulas
Affiliation:
Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Mathematics, Łojasiewicza 6, 30  348 Kraków, Poland
Email:
maciej.ulas@uj.edu.pl
DOI:
http://dx.doi.org/10.1090/S002557182013027780
Keywords:
Symmetric polynomials,
elliptic curves
Received by editor(s):
August 27, 2012
Received by editor(s) in revised form:
November 12, 2012, and December 17, 2012
Published electronically:
October 22, 2013
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
