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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!

Arturo Magidin
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student
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    It is news to me that $42$ is specially significant to anyone outside of what amounts to dead memes these days. – PrincessEev Sep 02 '19 at 00:23
  • This is because that identity is probably (definitely) not true. Troll? – Franklin Pezzuti Dyer Sep 02 '19 at 00:24
  • The person who posted it it was combative : I have a post-graduate degree in Pure Mathematics (sadly a lowly Masters), what is the likelihood that I know the difference between an algebraic and a transcendental number?

    In either case, I think a good rule is “think before you comment”. Of course it is true, are you suggesting that my decades of work on this equality are erroneous?

     – student Sep 02 '19 at 00:25
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    @EeveeTrainer: Whats wrong with fans of Douglas Adams? – Arturo Magidin Sep 02 '19 at 00:25
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    This seems like either a troll or the misguided victim of trolling. Close? – Franklin Pezzuti Dyer Sep 02 '19 at 00:25
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    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. – PrincessEev Sep 02 '19 at 00:27
  • "Whats wrong with fans of Douglas Adams?" -- Nothing at all xD – PrincessEev Sep 02 '19 at 00:27
  • @student It is proven that $\pi$ is transcendental, so until I see a proof of that unlikely identity of yours, I’m going to say it’s probably false. – Franklin Pezzuti Dyer Sep 02 '19 at 00:29
  • Sorry guys! I guess I deserve being trolled! I fell for it! – student Sep 02 '19 at 00:33
  • $\color{red}{42}$ is important to everyone (not only mathematicians) since it is "The "Answer to the Ultimate Question of Life, the Universe, and Everything". Just have a look at https://en.wikipedia.org/wiki/42_(number)#cite_note-18 – Claude Leibovici Sep 02 '19 at 03:30

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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 $$x=\frac{4984293073556644693450328273976834743385651648058}{10122924587252622722204541 3352905624410070072168739}$$

Claude Leibovici
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