Exponential product

Calculus Level 3

\[P=\prod _{ n=1 }^{ \infty }{ \left[ \prod _{ m=1 }^{ 4 }{ { \left( \frac { 2000 }{ { 2 }^{ m } } \right) }^{ \frac { 1 }{ { n }^{ 2 } } } } \right] }\]

If \(P\) can be expressed in the form \({ A }^{ { \pi }^{ B } }\), where \(A\) and \(B\) are positive integers, find the value of \(\frac { A }{ B }\).


This problem is original.

Picture credits: Sundog and Crepuscular Rays in 2340m Altitude by ThaliaTraianou, Wikipedia

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