Series, and Floors, and Numbers Oh My!!!!

Algebra Level 5

A number Z is equal to \( \frac { P }{ Q }\)

Where P is equal to the floor of 1000 times S, where S is:

\(\displaystyle\sum _{ n=1 }^{ 100 }{ \frac { 1 }{ n } } = S \)

\(\left\lfloor 1000S \right\rfloor = P\)

And Q is defined as the smallest positive integer solution to the equation:

\({ a }^{ 3 }+{ b }^{ 3 }={ c }^{ 3 }+{ d }^{ 3 } = Q\)

Where a,b,c, and d are positive distinct integers;

Then find Z, where:

\( \frac { P }{ Q } = Z\)

Details and Assumptions:

You may use a calculator if you wish, but you may not use Wolfram Alpha.

Good Luck!

If you have any questions or concerns, please notify me in the comments section.

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