Suppose a property $\phi$ holds for a strictly positive proportion of the prime numbers, i.e. such that
$$\liminf_{x\to\infty}{\pi_\phi(x)\over \pi(x)}\gt 0,$$
where $\pi_\phi(x)$ counts the number of primes less than or equal to $x$ having the property $\phi$, and $\pi$ is the usual prime-counting function.
According to [here] and [here], cases of this include the following:
- $\phi_1(n)$="changing any single decimal digit of $n$ (not including leading zeros) produces a composite number"
- $\phi_2(n)$="changing any single decimal digit of $n$ (including any leading zero) produces a composite number"
E.g., $n=294001$ has property $\phi_1$, but not property $\phi_2$ (because $10294001$ is prime).
In fact:
Infinitely many primes (indeed a positive proportion) have property $\phi_2$, yet there is no known example.
Q1: What are some other number-theoretic properties proven to hold for a strictly positive proportion of the primes, yet there is no specific prime for which the property has been proven to hold?
Q2: Are there similar cases with respect to $\mathbb{N}?$ I.e., some property holds for a strictly positive proportion of $\mathbb{N},$ yet there is no specific $n\in\mathbb{N}$ for which the property has been proven to hold? In this case, I suppose the proportion of $\mathbb{N}$ satisfying property $\phi$ would be defined as
$$\liminf_{x\to\infty}{C_\phi(x)\over x},\ \text{ where }\ C_\phi(x)=\#\{n\in\mathbb{N}: n\le x,\ \phi(n) \}.$$
Best Answer
The current state of Artin’s conjecture can be put this way:
where $p$ is prime.
As of now, it is only known that $\phi(p)$ is true for all but at most two primes $p,$ but the proof gives no hint at what the two values are, and we don’t know $\phi(p)$ is true for any single prime $p.$
Still, the density of $\phi$ amongst primes is $1,$ and we could only do one better - if we have a theorem with one potential but unknown counterexample.