Prouhet–Tarry–Escott problem

In mathematics, the Prouhet–Tarry–Escott problem asks for two disjoint multisets A and B of n integers each, whose first k power sum symmetric polynomials are all equal. That is, the two multisets should satisfy the equations

$\sum _{{a\in A}}a^{i}=\sum _{{b\in B}}b^{i}$

for each integer i from 1 to a given k. It has been shown that n must be strictly greater than k. Solutions with ${\displaystyle k=n-1}$ are called ideal solutions. Ideal solutions are known for ${\displaystyle 3\leq n\leq 10}$ and for ${\displaystyle n=12}$. No ideal solution is known for ${\displaystyle n=11}$ or for ${\displaystyle n\geq 13}$.^[1]

This problem was named after Eugène Prouhet, who studied it in the early 1850s, and Gaston Tarry and Escott, who studied it in the early 1910s. The problem originates from letters of Christian Goldbach and Leonhard Euler (1750/1751).

Examples

It has been shown that n must be strictly greater than k. The largest value of k for which a solution with n = k+1 is known is given by A = {±22, ±61, ±86, ±127, ±140, ±151}, B = {±35, ±47, ±94, ±121, ±146, ±148} for which k = 11.^[2]

An example for n = 6 and k = 5 is given by the two sets { 0, 5, 6, 16, 17, 22 } and { 1, 2, 10, 12, 20, 21 }, because:

0¹ + 5¹ + 6¹ + 16¹ + 17¹ + 22¹ = 1¹ + 2¹ + 10¹ + 12¹ + 20¹ + 21¹

0² + 5² + 6² + 16² + 17² + 22² = 1² + 2² + 10² + 12² + 20² + 21²

0³ + 5³ + 6³ + 16³ + 17³ + 22³ = 1³ + 2³ + 10³ + 12³ + 20³ + 21³

0⁴ + 5⁴ + 6⁴ + 16⁴ + 17⁴ + 22⁴ = 1⁴ + 2⁴ + 10⁴ + 12⁴ + 20⁴ + 21⁴

0⁵ + 5⁵ + 6⁵ + 16⁵ + 17⁵ + 22⁵ = 1⁵ + 2⁵ + 10⁵ + 12⁵ + 20⁵ + 21⁵.

Prouhet used the Thue–Morse sequence to construct a solution with $n=2^k$ for any $k$. Namely, partition the numbers from 0 to ${\displaystyle 2^{k+1}-1}$ into the evil numbers and the odious numbers; then the two sets of the partition give a solution to the problem.^[3] For instance, for $n=8$ and $k=3$, Prouhet's solution is:

0¹ + 3¹ + 5¹ + 6¹ + 9¹ + 10¹ + 12¹ + 15¹ = 1¹ + 2¹ + 4¹ + 7¹ + 8¹ + 11¹ + 13¹ + 14¹

0² + 3² + 5² + 6² + 9² + 10² + 12² + 15² = 1² + 2² + 4² + 7² + 8² + 11² + 13² + 14²

0³ + 3³ + 5³ + 6³ + 9³ + 10³ + 12³ + 15³ = 1³ + 2³ + 4³ + 7³ + 8³ + 11³ + 13³ + 14³.

Generalizations

A higher dimensional version of the Prouhet–Tarry–Escott problem has been introduced and studied by Andreas Alpers and Robert Tijdeman in 2007: Given parameters $n,k\in {\mathbb {N}}$, find two different multi-sets $\{(x_{1},y_{1}),\dots ,(x_{n},y_{n})\}$, $\{(x_{1}',y_{1}'),\dots ,(x_{n}',y_{n}')\}$ of points from $\mathbb {Z} ^{2}$ such that

$\sum _{{i=1}}^{n}x_{i}^{j}y_{i}^{{d-j}}=\sum _{{i=1}}^{n}{x'}_{i}^{j}{y'}_{i}^{{d-j}}$

for all $d,j\in \{0,\dots ,k\}$ with $j\leq d.$ This problem is related to discrete tomography and also leads to special Prouhet-Tarry-Escott solutions over the Gaussian integers (though solutions to the Alpers-Tijdeman problem do not exhaust the Gaussian integer solutions to Prouhet-Tarry-Escott).

A solution for $n=6$ and $k=5$ is given, for instance, by:

$\{(x_{1},y_{1}),\dots ,(x_{6},y_{6})\}=\{(2,1),(1,3),(3,6),(6,7),(7,5),(5,2)\}$ and

$\{(x'_{1},y'_{1}),\dots ,(x'_{6},y'_{6})\}=\{(1,2),(2,5),(5,7),(7,6),(6,3),(3,1)\}$.

No solutions for $n=k+1$ with $k\geq 6$ are known.

See also

• Euler's sum of powers conjecture
• Beal's conjecture
• Jacobi–Madden equation
• Taxicab number
• Pythagorean quadruple
• Sums of powers, a list of related conjectures and theorems
• Discrete tomography

Notes

1. Borwein, p. 85
2. Solution found by Nuutti Kuosa, Jean-Charles Meyrignac and Chen Shuwen, in 1999.
3. Wright, E. M. (1959), "Prouhet's 1851 solution of the Tarry–Escott problem of 1910", The American Mathematical Monthly, 66: 199–201, doi:10.2307/2309513, MR 0104622.

References

• Borwein, Peter B. (2002), "The Prouhet–Tarry–Escott problem", Computational Excursions in Analysis and Number Theory, CMS Books in Mathematics, Springer-Verlag, pp. 85–96, ISBN 0-387-95444-9, retrieved 2009-06-16 Chap.11.
• Alpers, Andreas; Rob Tijdeman (2007), "The Two-Dimensional Prouhet-Tarry-Escott Problem" (PDF), Journal of Number Theory, 123 (2): 403–412, doi:10.1016/j.jnt.2006.07.001, retrieved 2015-04-01.

External links

• Weisstein, Eric W. "Prouhet-Tarry-Escott problem". MathWorld.