The Prouhet-Tarry-Escott problem for Gaussian integers
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- by Timothy Caley PDF
- Math. Comp. 82 (2013), 1121-1137
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 \[ x_1^i+\ldots +x_n^i=y_1^i+\ldots +y_n^i\] 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
- Email: tcaley@math.uwaterloo.ca
- 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.
- © Copyright 2012 by the author
- Journal: Math. Comp. 82 (2013), 1121-1137
- MSC (2010): Primary 11D72, 11Y50; Secondary 11P05
- DOI: https://doi.org/10.1090/S0025-5718-2012-02532-4
- MathSciNet review: 3008852