For example: (n = 3): 13 + 123 = 93 + 103, made famous by Hardy's recollection of a conversation with Ramanujan about the number 1729 being the smallest number that can be expressed as a sum of two cubes in two distinct ways. There can also exist n − 1 positive integers whose nth powers sum to an nth power (though, by Fermat's last theorem, not for n = 3); these are counterexamples to Euler's sum of powers conjecture.

The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system.

It was discovered by Edgar James Banks shortly after 1900, and sold to George Arthur Plimpton in 1922, for $10. When searching for integer solutions, the equation is a Diophantine equation.

It was discovered by Edgar James Banks shortly after 1900, and sold to George Arthur Plimpton in 1922, for $10. When searching for integer solutions, the equation is a Diophantine equation.

The first proof was given by Andrew Wiles in 1994. ===n − 1 or n nth powers summing to an nth power=== Another generalization is searching for sequences of n + 1 positive integers for which the nth power of the last is the sum of the nth powers of the previous terms.

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