**∞**

**In 1769, Euler conjectured that at least n nth powers are required for n>2 to ****provide a sum that is itself an nth power. **

For example, for 5th powers, any solution would **require **5 (or more) TERMs on the left side of the equation.

Like so for n = 5 (x must be >= 5).

**T _{1}^{5} + T_{2}^{5} + .. T_{x}^{5} = Z^{5}**

**198 years later**, the conjecture was disproved by Lander and Parkin (1967 on a CDC 6600 computer) with the **counterexample shown below **(less than 5 terms on the left side of a 5th power equation):

**27 ^{5 }+ 84^{5 }+ 110^{5 }+ 133^{5 }= 144^{5}**

**Think about it… it took a lot of brilliant people almost 200 years to find the first counterexample… and as soon as it was found we knew that there were an infinite number of them! Or to put it another way, **

**if there were an infinite number of counterexamples then what took so long to find 1 of them!?**

**∞**

Also see Sum Of Powers – C#

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