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Inspired by some popular book about Fermat's Last Theorem years ago, experimented a bit and found some interesting sequence:

$3 \neq 4$

$3^2 + 4^2 = 5^2$

$3^3 + 4^3 +5^3 = 6^3$

$3^4 + 4^4 + 5^4 + 6^4 \neq 7^4$

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    @anon - Well, in the question that I originally posted I wrongly set first line as 3 = 3. But, warned by lhf and corrected that. Before that correction it looked meaningful (first 3 valid, break at 4), but after that change it does not looked so nice to me any more. If I would have been aware of it on time, I would avoid posting this question.2011-07-12

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Looks like it was nothing to explain about. Since I originally failed to observe that first expression is inequalty, it looked interesting to me that sequence breaks at order of 4 (and is fulfilled for the first 3). After first expression correction, it was pointless.