Ideal solutions in the Prouhet–Tarry–Escott problem
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- by Don Coppersmith, Michael J. Mossinghoff, Danny Scheinerman and Jeffrey M. VanderKam;
- Math. Comp. 93 (2024), 2473-2501
- DOI: https://doi.org/10.1090/mcom/3917
- Published electronically: November 6, 2023
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Abstract:
For given positive integers $m$ and $n$ with $m<n$, the Prouhet–Tarry–Escott problem asks if there exist two disjoint multisets of integers of size $n$ having identical $k$th moments for $1\leq k\leq m$; in the ideal case one requires $m=n-1$, which is maximal. We describe some searches for ideal solutions to the Prouhet–Tarry–Escott problem, especially solutions possessing a particular symmetry, both over $\mathbb {Z}$ and over the ring of integers of several imaginary quadratic number fields. Over $\mathbb {Z}$, we significantly extend searches for symmetric ideal solutions at sizes $9$, $10$, $11$, and $12$, and we conduct extensive searches for the first time at larger sizes up to $16$. For the quadratic number field case, we find new ideal solutions of sizes $10$ and $12$ in the Gaussian integers, of size $9$ in $\mathbb {Z}[i\sqrt {2}]$, and of sizes $9$ and $12$ in the Eisenstein integers.References
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Bibliographic Information
- Don Coppersmith
- Affiliation: Center for Communications Research, Princeton, New Jersey
- MR Author ID: 51455
- Email: dcopper@idaccr.org
- Michael J. Mossinghoff
- Affiliation: Center for Communications Research, Princeton, New Jersey
- MR Author ID: 630072
- ORCID: 0000-0002-7983-5427
- Email: m.mossinghoff@idaccr.org
- Danny Scheinerman
- Affiliation: Center for Communications Research, Princeton, New Jersey
- ORCID: 0000-0002-3985-325X
- Email: daniel.scheinerman@gmail.com
- Jeffrey M. VanderKam
- Affiliation: Center for Communications Research, Princeton, New Jersey
- MR Author ID: 606412
- Email: vanderkm@idaccr.org
- Received by editor(s): April 21, 2023
- Received by editor(s) in revised form: August 12, 2023, and September 20, 2023
- Published electronically: November 6, 2023
- Additional Notes: This work is dedicated to the memory of Peter Borwein.
The second author is the corresponding author - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 2473-2501
- MSC (2020): Primary 11D72, 11Y50; Secondary 11D79, 11P05, 11R11
- DOI: https://doi.org/10.1090/mcom/3917
- MathSciNet review: 4759381