Abstract
In a conversation with Ramanujan, G.H. Hardy mentioned that 1729 seemed to be a dull number. Ramanujan answered, “No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.” He also stated that the answer to the corresponding problem for fourth powers “... must be very large.” In this paper, this problem and other generalizations to higher powers and larger sums is examined.
In particular, the computational results of Lander, Parkin and Selfridge (“... Equal Sums of Like Powers”) are extended.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E.T. Bell, Development of Mathematics, McGraw-Hill, New York, 1945.
S. Brudno, “Triples of Sixth Powers With Equal Sums,” Math. Comp. 30 (1976), 646–648.
L.E. Dickson, History of the Theory of Numbers, Carnegie Institute of Washington, Washington, D.C., 1920.
T. Estermann, Introduction to Modern Prime Number Theory, Cambridge Tract No. 41, Cambridge University Press, London, 1961.
G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers Oxford University Press, London, 1960.
L.J. Lander and T.R. Parkin, “A Counterexample to Euler's Sum of Powers Conjecture,” Math. Comp. 21 (1967), 101–103.
L.J. Lander, T.R. Parkin and J.L. Selfridge, “A Survey of Equal Sums of Like Powers,” Math. Comp. 21 (1967), 446–459.
J.R. Newman, World of Mathematics, Simon and Schuster, New York, 1956.
K.S. Rao, “On Sums of Sixth Powers,” J. London Math. Soc. 9 (1934), 172–173.
Author information
Authors and Affiliations
Editor information
Additional information
This paper is dedicated to my teacher, Professor Emil Grosswald, in honor of his sixty-eighth birthday.
Rights and permissions
Copyright information
© 1981 Springer-Verlag
About this paper
Cite this paper
Alter, R. (1981). Computations and generalizations on a remark of Ramanujan. In: Knopp, M.I. (eds) Analytic Number Theory. Lecture Notes in Mathematics, vol 899. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096461
Download citation
DOI: https://doi.org/10.1007/BFb0096461
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11173-3
Online ISBN: 978-3-540-38953-8
eBook Packages: Springer Book Archive