What is wrong in this proof that $pi=2$ or $x=2$?

Let us consider the number

$$\Large\pi^{\pi^\pi}=\pi^{\pi\cdot\pi}=\pi^{\pi^2}$$

As the bases are equal, the exponents must be equal, So

$$\pi=2$$

You can take any $x$ instead of $\pi$.

What is wrong in this proof?

Answer

Lets write $a \uparrow b$ to mean $a^b$.

Then the following reasoning is correct:

$$(\pi \uparrow \pi)\uparrow \pi = \pi \uparrow (\pi \cdot \pi) = \pi \uparrow (\pi \uparrow 2)$$

However, we cannot necessarily deduce that the RHS equals

$$(\pi \uparrow \pi) \uparrow 2$$

because exponentiation isn't associative. Indeed, Google calculator tells me that:

• $\pi \uparrow (\pi \uparrow 2) \approx 80662.6659386$




• $(\pi \uparrow \pi) \uparrow 2 \approx 1329.48908322$

so if the calculator is correct to even the first decimal place, then

$$(\pi \uparrow \pi) \uparrow 2 \neq \pi \uparrow (\pi \uparrow 2).$$

Moral of the story: if in doubt, find better notation!