Euler's sum of powers conjecture

Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers $n$ and $k$ greater than 1, if the sum of $n$ $k$ th powers of non-zero integers is itself a $k$ th power, then $n$ is greater than or equal to $k$ .

In symbols, the conjecture falsely states that if

where $n > 1$ and $a 1, a 2, \dots, a n, b$ are non-zero integers, then $n \geq k$ .

The conjecture represents an attempt to generalize Fermat's last theorem, which is the special case $n = 2$ : if $a 1 k + a 2 k = b k$ , then $2 \geq k$ .

Although the conjecture holds for the case $k = 3$ (which follows from Fermat's last theorem for the third powers), it was disproved for $k = 4$ and $k = 5$ . It is unknown whether the conjecture fails or holds for any value $k \geq 6$ .

Euler had an equality for four fourth powers 59⁴ + 158⁴ = 133⁴ + 134⁴; this however is not a counterexample because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as in Plato's number 3³ + 4³ + 5³ = 6³ or the taxicab number 1729. The general solution for:

where $a$ and $b$ are any integers.

Euler's conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when, through a direct computer search on a CDC 6600, they found a counterexample for $k = 5$ . A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known:

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