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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:

  1. $\phi_1(n)$="changing any single decimal digit of $n$ (not including leading zeros) produces a composite number"
  2. $\phi_2(n)$="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 $$\liminf_{x\to\infty}{C_\phi(x)\over x},\ \text{ where }\ C_\phi(x)=\#\{n\in\mathbb{N}: n\le x,\ \phi(n) \}.$$

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  • $\begingroup$ "n is prime" has density 0 so both conditions you list have density 0 as well. Am I missing something? $\endgroup$ May 21 at 21:25
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    $\begingroup$ @OlivierBégassat Thank you. I was wanting to generalize to $\mathbb{N}$ but made a misstep -- hopefully this makes sense now. $\endgroup$
    – r.e.s.
    May 21 at 21:44
  • $\begingroup$ A tempting answer would be the property "$n > 10^{10^{10^{10}}}$" - but I suppose that's not what you're looking for. $\endgroup$ May 21 at 21:57
  • $\begingroup$ @MiloBrandt I was going to say "positive and less than 1" to avoid cases like that, but it seems that would throw out other interesting cases too. $\endgroup$
    – r.e.s.
    May 21 at 22:14
  • $\begingroup$ The distinction between density of primes and density of $\mathbb N$ is kind of irrelevant, because you can take $\phi$ about primes and construct $\rho(n)=\phi_(p_n),$ as I did in my answer before you corrected the question to be about density in primes. $\endgroup$ May 21 at 22:38
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The current state of Artin’s conjecture can be put this way:

$\phi(p):$ $p$ is a multiplicative generator modulo $q$ for infinitely many primes $q.$

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.

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