Are there any interesting examples of subsets of $\mathbf{N}$ that are known to be nonempty, but of which no elements are known?

Question

There are many results in mathematics that establish the existence of some object without actually constructing said object. I am wondering if there are any interesting properties of the natural numbers such that it is known that there exists a natural number satisfying the property, but no such natural number has actually been found.

I guess such a number could be computed given sufficient time, so the question is really asking if there are any "interesting" natural numbers so huge that no one has had time to find them yet.

Of course, I suppose the solution to some NP-complete problem given some suitably large input would qualify, hence the qualification "interesting."

Edit: It seems that there are two basic categories of these examples so far: sets containing only numbers that are so large that it is so far computationally infeasible to find elements of them, and sets in which it is very difficult to determine membership.

Answer 1

While they aren't exactly defined by a property, the so-called Ramsey numbers contains good examples along the lines of your question. The number $R(6,6)$ mentioned below is the largest integer $n$ for which there exists a group of $n$ people within which there is no subset of 6 mutual acquaintances and also no subset of 6 mutual non-acquaintances. (An assumption is that acquaintances are only mutual. Two people are either both acquaintances of each other or neither is an acquaintance of the other.)

Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5,5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6,6). In that case, he believes, we should attempt to destroy the aliens. —Joel Spencer

Perhaps a logician or two will chime in, too. I have a hunch the situation gets much worse. It wouldn't surprise me if there are (under the assumption of some reasonable axioms about mathematics) properties for which the set of integers having that property is provably non-empty, but for which it is also provably the case that no specific integer in the set can ever be identified - because it's logically impossible, not because the calculation is intractable.

A logician might also clarify whether something like what’s described here answers your question affirmatively.

Answer 2

The Artin conjecture gives such a set. It has been shown that there are at most two prime numbers $q$ that don't generate infinitely many of the cyclic groups $(\Bbb Z/p\Bbb Z)^*$, $p$ prime. However, not a single prime that generates infinitely many of these groups is known. Thus, there is at least one prime with the property among $3,5,7$, yet no one knows them (or it).

Answer 3

The singleton consisting of the single number $T$, where $T$ is the truth value of the Riemann Hypothesis.

The set of truncated decimal expansions of Chaitin's constant.

(These two examples probably don't really fit your criteria but they illustrate a funny aspect of this problem.)

The set of counterexamples to Mertens' conjecture...