Jake Lai's Problem

Number Theory Level 5

\[A=\left[ \prod _{ p=6k+1 \text{ is prime, } p>3 } (1+p^{ -1 }) \right] \left[ \prod _{ p=6k-1 \text{ is prime, }p>3 } (1-p^{ -1 }) \right] \]

Find the value of \(\left\lfloor 100000A \right\rfloor \).

Clue: Use the Legendre symbol and the fact that it is completely multiplicative.

My examinations are finally over! And I have time to re-look at one of Jake Lai's problems, which has unfortunately been deleted. So here I'm just going to re-post it.

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