Does this equation yield only primes?

Question

Interested in solving this equation for $x$:

$\exp\Big(\frac{n}{\ln(\pi(x))}\Big)=\pi(x)$ for $n=1,2,3,...$

For $n=1$ up to $n=9,$ I got $x=5,11,13,19,29,37,47,59,73.$

$\pi(x)$ is the prime counting function.

I tried analytically solving the equation but could not isolate $x$.

This is the graph I'm getting when I plot it. The intersection occurs on the horizontal axis at $73.$

Question:

Does this equation yield only primes?

Answer 1

The equation $\exp\Big(\frac{n}{\ln(\pi(x))}\Big)=\pi(x)$ doesn't actually have any solutions for positive integers $n$, as pointed out by user. This is because the equation is equivalent to

$$\pi(x)=e^{\sqrt n},$$

and in the new equation, the left-hand side is always an integer, whereas the right-hand side is never an integer. (Actually, that might not be known; but it would certainly be extremely surprising if the right-hand side were ever an integer.)

But in any case, let's look at your original equation again:

$$\exp\Big(\frac{n}{\ln(\pi(x))}\Big)=\pi(x)$$

I think you're actually asking about the values of $x$ at which the left-hand side becomes smaller than the right-hand side.

The value of $\pi(x)$ only changes at prime numbers, which means that both sides of the equation only change at prime numbers. So, sure enough, $x$ is always a prime number at the crossing point.