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The Prouhet-Tarry-Escott problem for Gaussian integers

Author: Timothy Caley
Journal: Math. Comp. 82 (2013), 1121-1137
MSC (2010): Primary 11D72, 11Y50; Secondary 11P05
Published electronically: October 22, 2012
MathSciNet review: 3008852
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Abstract: Given natural numbers $ n$ and $ k$, with $ n>k$, the Prouhet-Tarry-Escott (PTE) problem asks for distinct subsets of $ \mathbb{Z}$, say $ X=\{x_1,\ldots ,x_n\}$ and $ Y=\{y_1,\ldots ,y_n\}$, such that

$\displaystyle x_1^i+\ldots +x_n^i=y_1^i+\ldots +y_n^i$

for $ i=1,\ldots ,k$. Many partial solutions to this problem were found in the late 19th century and early 20th century.

When $ n=k-1$, we call a solution $ X=_{n-1}Y$ ideal. This is considered to be the most interesting case. Ideal solutions have been found using elementary methods, elliptic curves, and computational techniques. In 2007, Alpers and Tijdeman gave examples of solutions to the PTE problem over the Gaussian integers. This paper extends the framework of the problem to this setting. We prove generalizations of results from the literature, and use this information along with computational techniques to find ideal solutions to the PTE problem in the Gaussian integers.

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Additional Information

Timothy Caley
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

Received by editor(s): October 14, 2010
Received by editor(s) in revised form: February 10, 2011
Published electronically: October 22, 2012
Additional Notes: The author would like to thank NSERC and the University of Waterloo for funding.
Article copyright: © Copyright 2012 by the author

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