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95800^4 + 217519^4 + 414560^4 = 422481^4

Wow. How did you find that?

you said:

> which I have checked via exhaustive search

how far did you go checking?

Well, I was checking on a very old, underpowered desktop, so I tried to keep it under a billion combinations, so upper limit of 1000 on a, b, and c.