# A positive proportion of the prime numbers is proven to have certain property, yet no specific prime is proven to have that property?

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 $$\liminf_{x\to\infty}{C_\phi(x)\over x},\ \text{ where }\ C_\phi(x)=\#\{n\in\mathbb{N}: n\le x,\ \phi(n) \}.$$

• "n is prime" has density 0 so both conditions you list have density 0 as well. Am I missing something? May 21 at 21:25
• @OlivierBégassat Thank you. I was wanting to generalize to $\mathbb{N}$ but made a misstep -- hopefully this makes sense now. May 21 at 21:44
• A tempting answer would be the property "$n > 10^{10^{10^{10}}}$" - but I suppose that's not what you're looking for. May 21 at 21:57
• @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. May 21 at 22:14
• 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. May 21 at 22:38

$$\phi(p):$$ $$p$$ 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.