For demonstrating verification of a conjecture, surely you can do much simpler things than a zero-knowledge proof: Send one of the primes.

It would still take a nontrivial amount of computation to do all the verification afterwards. Back of the envelope calculations suggest it should less than 100x longer to find the two primes than to verify them.

It'd be neat to do the verification in the same manner by redistributing one client's results to another, therefore obtaining a proof modulo client collusion.

I am curious about alternatives or solutions in such a setting / context.

Say smaller prime is less then 10,000. Then this one or two Byte per Nummer. E.g. 100 Mio number is already 100mb or