Prouhet-Tarry-Escott problem

In mathematics, the Prouhet-Tarry-Escott problem asks for two distinct sets of "m" integers each, "A" and "B", such that:

:$sum\_\{ain\; A\}\; a^k\; =\; sum\_\{bin\; B\}\; b^k$

for each integer "k" from 1 to "n".

For example, for "m" = 6 and "n" = 5, a solution is 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

This problem was named after Eugène Prouhet, who studied it in early 1850s, and Gaston Tarry and Escott, who studied it in early 1910s.

The largest value of "n" known is given by "A" = {±22, ±61, ±86, ±127, ±140, ±151}, "B" = {±35, ±47, ±94, ±121, ±146, ±148} for which "n"=11.

ee also

* Thue-Morse sequence

References

*cite book | last=Borwein | first=Peter B | authorlink=Peter Borwein | title=Computational Excursions in Analysis and Number Theory | series=CMS Books in Mathematics | publisher=Springer-Verlag | year=2002 | isbn=0-387-95444-9 Chap.11.

External links

* [http://www.nabble.com/Prouhet-Tarry-Escott-problem-td10624352.html Prouhet-Tarry-Escott problem]
*mathworld | title = Prouhet-Tarry-Escott problem | urlname = Prouhet-Tarry-EscottProblem