Combinatory categorial grammar

In mathematics, the Prouhet–Tarry–Escott problem asks for two disjoint sets A and B of n integers each, such that:

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

for each integer i from 1 to a given k.

This problem was named after Template:Link-interwiki, who studied it in the early 1850s, and Gaston Tarry and Escott, who studied it in the early 1910s.

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.

Example

For example, a solution with n = 6 and k = 5 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.

See also

• Thue–Morse sequence
• 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

Notes

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References

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External links

• Prouhet-Tarry-Escott problem
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