Sums of Squares, Cubes, and Higher Powers

William C. Jagy, Irving Kaplansky

Type: Article

Publication Date: 1995-01-01

Citations: 13

DOI: https://doi.org/10.1080/10586458.1995.10504318

Abstract

Any integer is expressible as a sum of two squares and a cube, mixed signs being allowed. We study the analogous question for a square and two cubes, and obtain an affirmative answer in the range from –4,000,000 to 2,000,000. For two squares and a cube with everything positive, computations support the possibility that there are only finitely many exceptions. However, x 2 + y 2 + z 9 admits infinitely many positive exceptions.

Locations

Experimental Mathematics - View
Project Euclid (Cornell University) - View - PDF

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Works Cited by This (10)

Action	Title	Year	Authors
+	Representation of Integers as Sums of a Square, a Positive Cube, and a Fourth Power of a Prime	1950	H. Halberstam
+	Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids	1990	William Duke Rainer Schulze‐Pillot
+	Proof that almost all Positive Integers are Sums of a Square, a Positive Cube and a Fourth Power	1949	K. F. Roth
+	Note on a Result in the Additive Theory of Numbers	1938	H. Davenport H. Heilbronn
+	On Waring's Problem : Two Cubes and One Square	1938	H. Davenport H. Heilbronn
+	On Sums of Four Cubes of Polynomials	1972	J. H. E. Cohn L. J. Mordell
+	The Hardy-Littlewood Method	1997	R. C. Vaughan
+	The Representation of Numbers as Sums of Unlike Powers	1995	Kevin Ford
+	Diophantine equations	1969	L. J. Mordell
+	The Hardy-Littlewood method	1996	M. N. Huxley