Is there a known function $f(n) = P_n$, where $P_n$ denotes the $n$th prime number?

Question

Is there a known function $f:\mathbb{R}\to\mathbb{R}$, such that:

• The definition of $f$ does not contain the $!$ operator
• The definition of $f$ does not contain the $\sum$ operator
• The definition of $f$ does not contain the $\prod$ operator
• The definition of $f$ does not contain a continued fraction
• $\forall n\in\mathbb{N}:f(n) = P_n$, where $P_n$ denotes the $n$th prime number
• Needless to say, the definition of $f$ does not make an explicit use of $P_n$

If no such function is known, then is there any known proof that no such function exists?

Answer 1

Your question seems to reflect a misunderstanding of what a function is. Here are some examples of functions:

\begin{align*} f(x) &= \text{the number of primes less than or equal to } x \\ f(x) &= 1 \text{ if the third digit of } x \text{ is even, otherwise } 2 \\ f(x) &= 1 \text{ if } x \text{ is rational, otherwise } 2 \end{align*} A function simply means you take in an input and send out an output. A function does not have to have a simple "formula" like $x^2$ or $e^x$. In particular, it is extremely easy to make a function such that $f(n) = P_n$. Just define $f(x)$ to be the $x$th prime if $x$ is an integer, and then define it arbitrarily elsewhere.

• Needless to say, the definition of $f$ does not make an explicit use of $P_n$

This and all your other constraints don't really make mathematical sense. I can define $f(x)$ equals the product of all positive integers less than or equal to $x$. This results essentially in the factorial function, but I am not using the $!$ symbol. Or, I can define $f(x) = \int_0^{\infty} e^{-t} t^n \; dt$, and again I have defined factorials without really using factorials.

There are ways to make your idea of a "function" precise, but it's a lot harder than you think. For example, you can ask if there is an Elementary function satisfying $f(n) = P_n$. But it sounds like you want a broader scope than just exponentials, $n$th roots, and polynomials.

Some related references:

Can insight be derived from direct formulae for prime numbers?

Prime number formulas

What would be the immediate implications of a formula for prime numbers?