At Factors of 1000 numbers up to googolplex, it's shown that for $G$=googolplex and $k \in (57, 101, 143, 167, 173, 219, 231, 257, 279, 303, 387, 587, 719, 741, 789, 813, 941, 971)$, the number $G-k$ has no prime factors under $3.5 \times 10^{14}$. I started wondering if any similarly gigantic numbers could be proven composite without the benefit of a known factor.
In 2005, Don Reble constructed an interesting semiprime with elliptic pseudo-curves and the Goldwasser-Kilian ECPP theorem. No factors are known, but we know the Reble number has exactly 2 prime factors.
Can a gigantic provably-composite number be constructed without the benefit of knowing any factors? Other than finding divisors, are there any primality tests that can quickly show that some carefully designed gigantic number isn't prime?
The number should have no "small" divisors.
Alternately, perhaps a provable-composite might be in a list of extremely rough numbers. A sample fake proof: "Assume for $k\in{1..1000}$ that all values $(10^{100}+k)! +1$ are prime. That would allow the construction of a Blahblah curve. By the Someguy theorem, such a construction is impossible. Therefore, at least one $k$ value yields a composite number". Are there any proofs similar to that useful for giving an example of a huge provable composite with no known divisors?