I'm looking for sets $S$ of natural numbers with the following properties:
- There is a known efficient algorithm for determining whether a given $n$ is in $S$ (let's say "efficient" means polynomial in the number of digits of $n,$ with reasonable degree/coefficients)
- It is known that $S$ has a small element (say below $10^{1,000}$)
- No element of $S$ has been identified (or is likely to be found with current technology)
- (Optional) $S$ is a singleton
One example would be the singleton $S$ containing the smaller prime factor of Reble's semiprime (a 1,084 digit number which was nonconstructively proven to be semiprime, see http://www.graysage.com/djr/isp.txt). This number has between 350 and 550 digits, and would be easy to verify if found.
What are some other examples of such sets?