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Two more representation problems

Published online by Cambridge University Press:  20 January 2009

Andrew Bremner  and
Richard K. Guy
Andrew Bremner
Affiliation:
Department of Mathematics, Arizona State University, Tempe AZ 85287-1804, U.S.A. E-mail address: andrew@math.la.asu.edu
Richard K. Guy
Affiliation:
Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada T2N 1N4, E-mail address: rkg@cpsc.ucalgary.ca
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Abstract

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We discuss the problem of finding those integers which may be represented by (x + y + z)3/xyz, and also those which may be represented by x/y + y/z + z/x, where x, y, z are integers. For example,

satisfy (x + y + z)3/(xyz) = -47, and

satisfy x/y + y/z + z/x = -86.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 40 , Issue 1 , February 1997 , pp. 1 - 17
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

REFERENCES

1. Batut, C., Bernardi, D., Cohen, H. and Oliver, M., PARI/GP, Version 1.39.Google Scholar
2. Bremner, Andrew, Guy, Richard K. and Nowakowski, Richard J., Which integers are representable as the product of the sum of three integers with the sum of their reciprocals? Math. Comput. 61 (1993) (Lehmer Memorial issue), 117–130; MR 93k: 11021.Google Scholar
3. Cassels, J. W. S., On a Diophantine equation, Acta Arith. 6 (1960), 47–52.Google Scholar
4. Cassels, J. W. S. and Sansone, G., Sur le problème de M. Werner Mnich, Acta Arith. 7 (1962), 187–190.Google Scholar
5. Craig, Maurice, Integer Values of Σ(X2/yz), J. Number Theory 10 (1978), 62–63.CrossRefGoogle Scholar
6. Dofs, Erik, On some classes of homogeneous ternary cubic Diophantine equations, Ark. Mat. 13 (1975), 29–72.Google Scholar
7. Dofs, Erik, Solutions of x 3 + y 3 + z 3 = nxyz, Acta Arith., to appear.Google Scholar
8. Guy, Richard K., Problem 10382, Amer. Math. Monthly 101 (1994), 473; pre-publication solutions by Arthur Baragar and Brian Peterson.CrossRefGoogle Scholar
9. Guy, Richard K., Problem 93:31, Western Number Theory Problems, 93:12:21.Google Scholar
10. Kineke, Sharon A., Integers representable by (x + y + z)3/xyz, Internat. J. Math. Sci., submitted.Google Scholar
11. Konhauser, Joseph, Dan Velleman and Stan Wagon, Which Way Did the Bicycle Go?, Dolciani Series, Math. Assoc. Amer., 1996, Problem 83 (Problem 477 from the Joseph Konhauser collection), “A sum of fractions.” Some solutions by Underwood Dudley.Google Scholar
12. Mohanty, S. P., A system of cubic Diophantine equations, J. Number Theory 9 (1977), 153–159.CrossRefGoogle Scholar
13. Sierpiński, W., Remarques sur le travail de M. J. W. S. Cassels, “On a Diophantine Equation”, Acta Arith. 6 (1960), 469–471.Google Scholar
14. Sierpiński, W., 250 Problems in Number Theory (Elsevier, New York, 1970), 78–81.Google Scholar
15. Thomas, E. and Vasquez, A. T., Diophantine equations arising from cubic number fields, J. Number Theory 13 (1981), 398–414.Google Scholar
16. Wiles, Andrew, Modular elliptic curves and Fermat's Last Theorem, Ann. of Math. (2) 141 (1995), 443–551.Google Scholar