# Is the equation $\phi(\pi(\phi^\pi)) = 1$ true? And if so, how?

$\phi(\pi(\phi^\pi)) = 1$

I saw it on an expired flier for a lecture at the university. I don't know what $\phi$ is, so I tried asking Wolfram Alpha to solve $x \pi x^\pi = 1$ and it gave me a bunch of results with $i$, and I don't know what that is either.

I apologize in advance if I'm using the wrong tag, I didn't know there was one more kind of algebra.

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Could you give us the title of the lecture? The $\ \pi \$ in that equation is likely not the number, but the name of a function or mapping... –  RecklessReckoner Jul 9 at 21:24
Something about teaching computers to resolve ambiguities. The flier was expired, so we had to throw it away. –  Mr. Brooks Jul 9 at 21:25
@Mr.Brooks That then makes perfect sense as the computer needs to distinguish between the number $\pi$ and the function $\pi$, and the same goes for $\phi$. –  Lord Soth Jul 9 at 21:26
@Mr.Brooks So I guess the equation does not mean anything by itself but it is an example of an ambiguous equation. –  Lord Soth Jul 9 at 21:26
@Mr.Brooks Not to mention whether these are multiplications (your interpretation) or functions, etc. –  Lord Soth Jul 9 at 21:28
It's a joke based on the use of the $\phi$ function (Euler's totient function), the $\pi$ function (the prime counting function), the constant $\phi$ (the golden ratio), and the constant $\pi$. Note $\phi^\pi\approx 4.5$, so there are two primes less than $\phi^\pi$ (they are $2$ and $3$), so $\pi(\phi^\pi)=2$. There is only one positive integer less than or equal to $2$ which is also relatively prime to $2$ (this number is $1$), so $\phi(2)=1$. Hence we have
$$\phi(\pi(\phi^\pi))=\phi(2)=1$$