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| In mathematics, the '''Prouhet–Tarry–Escott problem''' asks for two disjoint [[Set (mathematics)|sets]] ''A'' and ''B'' of ''n'' [[integer]]s each, such that:
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| :<math>\sum_{a\in A} a^i = \sum_{b\in B} b^i</math>
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| for each integer ''i'' from 1 to a given ''k''.<ref name="Borwein">{{harvnb|Borwein|p=85}}</ref>
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| This problem was named after {{link-interwiki|lang=fr|en=Eugène Prouhet}}, who studied it in the early 1850s, and [[Gaston Tarry]] and Escott, who studied it in the early 1910s.
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| 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.<ref>[http://euler.free.fr/eslp/TarryPrb.htm#Ideal%20symmetric Solution found by Nuutti Kuosa, Jean-Charles Meyrignac and Chen Shuwen, in 1999].</ref>
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| == Example ==
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| For example, a solution with ''n'' = 6 and ''k'' = 5 is the two sets { 0, 5, 6, 16, 17, 22 }
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| and { 1, 2, 10, 12, 20, 21 }, because:
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| : 0<sup>1</sup> + 5<sup>1</sup> + 6<sup>1</sup> + 16<sup>1</sup> + 17<sup>1</sup> + 22<sup>1</sup> = 1<sup>1</sup> + 2<sup>1</sup> + 10<sup>1</sup> + 12<sup>1</sup> + 20<sup>1</sup> + 21<sup>1</sup>
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| : 0<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup> + 16<sup>2</sup> + 17<sup>2</sup> + 22<sup>2</sup> = 1<sup>2</sup> + 2<sup>2</sup> + 10<sup>2</sup> + 12<sup>2</sup> + 20<sup>2</sup> + 21<sup>2</sup>
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| : 0<sup>3</sup> + 5<sup>3</sup> + 6<sup>3</sup> + 16<sup>3</sup> + 17<sup>3</sup> + 22<sup>3</sup> = 1<sup>3</sup> + 2<sup>3</sup> + 10<sup>3</sup> + 12<sup>3</sup> + 20<sup>3</sup> + 21<sup>3</sup>
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| : 0<sup>4</sup> + 5<sup>4</sup> + 6<sup>4</sup> + 16<sup>4</sup> + 17<sup>4</sup> + 22<sup>4</sup> = 1<sup>4</sup> + 2<sup>4</sup> + 10<sup>4</sup> + 12<sup>4</sup> + 20<sup>4</sup> + 21<sup>4</sup>
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| : 0<sup>5</sup> + 5<sup>5</sup> + 6<sup>5</sup> + 16<sup>5</sup> + 17<sup>5</sup> + 22<sup>5</sup> = 1<sup>5</sup> + 2<sup>5</sup> + 10<sup>5</sup> + 12<sup>5</sup> + 20<sup>5</sup> + 21<sup>5</sup>. | |
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| ==See also==
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| * [[Thue–Morse sequence#The_Prouhet–Tarry–Escott_problem|Thue–Morse sequence]]
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| * [[Euler's sum of powers conjecture]]
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| * [[Beal's conjecture]]
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| * [[Jacobi–Madden equation]]
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| * [[Taxicab number]]
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| * [[Pythagorean quadruple]]
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| * [[Sums of powers]], a list of related conjectures and theorems
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation | last=Borwein | first=Peter B. | authorlink=Peter Borwein | title=Computational Excursions in Analysis and Number Theory | chapter=The Prouhet–Tarry–Escott problem | pages=85–96 | series=CMS Books in Mathematics | publisher=[[Springer-Verlag]] | year=2002 | isbn=0-387-95444-9 | url=http://books.google.com/?id=A_ITwN13J6YC&pg=85#PPA85,M1 | accessdate=2009-06-16}} Chap.11.
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| ==External links==
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| *[http://www.nabble.com/Prouhet-Tarry-Escott-problem-td10624352.html Prouhet-Tarry-Escott problem]
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| *{{mathworld | title = Prouhet-Tarry-Escott problem | urlname = Prouhet-Tarry-EscottProblem }}
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| {{DEFAULTSORT:Prouhet-Tarry-Escott problem}}
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| [[Category:Number theory]]
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| [[Category:Mathematical problems]]
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Parole or Probation Officer Dorothy from Fort Saskatchewan, loves to spend some time house plants, como ganhar dinheiro na internet and badge collecting. During the previous year has completed a visit to Djoudj National Bird Sanctuary.
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