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A puzzle related to mathematical facts and objects, whose solution needs mathematical arguments. General mathematics questions are off-topic but can be asked on Mathematics Stack Exchange.
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Four π's to make any integer
I also thought of Dirac's formula. Here's a formula for an arbitrary fraction:
$$\pm {m \over n} =
\pm { \ln {
\ln \sqrt { \sqrt { \cdots \sqrt {\pi}}}
\over \ln \pi
} \over \ln {
\ln \sqrt { …