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for question related to conjectures.
10
votes
On Generalizations of Fermat's Conjecture
The smallest counterexample for $n = 4$ is
$95800^4 + 217519^4 + 414560^4 = 422481^4$.
This has been found out by Roger Frye in 1988, cf.
http://euler.free.fr/docs/euler88.ps.