there are no numbers x, y, and z for which
is valid, when n>2.
Euler, in 1769 proposed that there are no sets of numbers such that
OR, in general terms, there are any sets of n-1 numbers such that the sum of their n-th power would also be an n-th power.
The conjecture was disproved in 1966 by Lander and Parkin who found counterexample for n=5:
Another counterexample was found by Noam Elkies in 1988:
Another counterexample was given by Roger Frye who found the smallest possible n=4 solution
Something for you to do:
First write down Euler's conjecture in algebraic notation.
Try combinations for n>5. No known solutions are found so far!
See more about Euler by cliking on his picture above.
See what the difference is between a conjecture and a theorem.
Fermat's Last Theorem - click on the equation.
You can download a presentation on Fermat's Last Theorem by clicking on the picture below.