Have anybody worked on FLT considering a fourth integer apart from triplets?

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Most of the people concerned always discuss about only the three integers to arrive a generalized form of the conjecture.; why so?

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It was only in 1987 that Noam Elkies solved this problem for the power $n=4$. I don't think any solutions have been found for $n>4$ with three on the left-hand side. This site has a lot of solutions with more than three numbers: http://euler.free.fr/