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Take two relatively prime numbers, P and Q. P < Q, and P is not prime.

How many pairs (P,Q) are there so that ALL positive integers which are P modulo Q are composite? I couldn't think of any. For instance,

(6,11)

won't work because 17 is prime.

(8, 11) will choke on 19, and (8, 13) will choke on 47.

I know primes become much rarer when you get further up the number line. But do they get rare enough at some point for this to happen? I would suspect not, but I'm curious if there's a proof.

Thanks in advance,

ACG