# an infinite product identity

I encountered the following post on a website (www.quora.com) I quote the post verbatim-these are not my comments.

"Why is the number "42" so significant to mathematicians?

It is for the very simple reason that: $$42^{\frac{382835430481}{625000000000}}=6{\prod}_{p\ {\rm{Prime}}}{\frac{1}{1-{\frac{1}{p^2}}}}.$$ This is a very deep result, whose impact we do not yet fully understand."

I do not believe that this identity is true. Here is my reasoning. The right hand side is $$6{\zeta}(2)={\pi}^2$$. The left hand side on the other hand is of the form $$42^{\frac{m}{n}}$$ where $$m$$ and $$n$$ are both positive integers. If such an identity were true than we would be forced to conclude (raising both sides to the $$n$$-th power) that $$42^{m}={\pi}^{2n}.$$This would imply that $$\pi$$ is an algebraic integer, being a root of the polynomial $$f(x)= {x^{2n}}-42^{m}\in{{\bf Z}[x]}.$$ But this contradicts the fact that $$\pi$$ is a transcendental number. Am I missing something here? The reason I ask is that the author of the post seems to think that I am mistaken, but refuses to furnish a reason! Appreciate your thoughts!

• It is news to me that $42$ is specially significant to anyone outside of what amounts to dead memes these days. Sep 2, 2019 at 0:23
• This is because that identity is probably (definitely) not true. Troll? Sep 2, 2019 at 0:24
• @EeveeTrainer: Whats wrong with fans of Douglas Adams? Sep 2, 2019 at 0:25
• This seems like either a troll or the misguided victim of trolling. Close? Sep 2, 2019 at 0:25
• Also, for what it's worth, Wolfram gives the difference between that power of $42$ and $\pi^2$ to be about $-7.61 \times 10^{-12}$. My guess is it's just a close-miss troll attempt - not unlike the Fermat's Last Theorem near-miss contradictions seen often in The Simpson's - to get people antsy. He probably had some method of approximating it, and used this to play a joke on people. Sep 2, 2019 at 0:27

Just to continue the joke, whatever could be the rhs (say $$k$$) you always can write $$42^x=k \implies x=\frac{\log (k)}{\log (42)}$$ and find the closest rational number you want for a given accuracy.
For $$k=\pi^2$$ as in your example, take $$x=\frac{113324902276818252177817587796556698968660872730811}{185009166560153907338903 184470209689295067534049020}$$ and you will have $$100$$ correct significant figures.
For $$k=e$$, use $$x=\frac{14286931922365270336190072861145109626110493519818}{5339983138470746932953829 5633396756354619740280665}$$
For $$k=\zeta (3)$$, use $$x=\frac{4984293073556644693450328273976834743385651648058}{10122924587252622722204541 3352905624410070072168739}$$