This number is a prime.
4027^5 = 3301^5 + 3169^5 + 3037^5 + 2411^5 + 1481^5 + 859^5 + 569^5. Note that all base numbers and exponents are prime. Found by Takao Nakamura. [Meller]
(There are 2 curios for this number that have not yet been approved by an editor.)
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