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The assertion that this equation has no nontrivial solutions for 
has a long and fascinating history and is known as Fermat's last theorem.
Fermat Equation
See also
Fermat's Last TheoremExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Fermat Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FermatEquation.html