Euler's sum of powers conjecture

The Euler's conjecture from 1769 is named after Leonhard Euler conjecture in number theory and generalized Fermat's Last Theorem. The Euler's conjecture is now disproved.

Presumption

There are no positive integer solutions of the equation for. Fermat proved the conjecture for. Euler could find for larger no proof nor a counterexample.

Refutations

N = 5

For the case n = 5 found 1966 LJ Lander and TR Parkin a counter-example:

N = 4

For n = 4 was 1988 Noam Elkies following counterexample:

Elkies proved also that there are infinitely many solutions for n = 4.

The smallest possible solution for n = 4 is

This minimal solution was found after the publication of the first solution by Elkies of Roger Frye.

Related question

Together with his presumption expressed Euler also that it should be possible to write the sum of four 4th powers given as the 4th power. This conjecture was answered positively in 1911 by R. Norrie:

For this general form

Was shown in 2008 by Lee W. Jacobi and Daniel J. Madden, that it has infinitely many integer solutions where all summands are non-zero. There was even a particularly aesthetic solution of the form

Found in whole numbers. The summands of this particular solution each have about 200 digits in the decimal system.