The Third Not-so-easy Number Game

Number Theory Level pending

Now it's time for a little extreme... How many 5-digit numbers satisfy:

\(\overline { abcde } ={ a }^{ 5 }+{ b }^{ 5 }+{ c }^{ 5 }+{ d }^{ 5 }+{ e }^{ 5 }\)

This is one of my problems in Game Of Numbers.

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