- #1
ACG
- 46
- 0
Hi! I was thinking about primes and have a bit of a question. I apologize if this is too easy or obvious -- I haven't thought much about it.
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
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