Is there a finite set $P$ of primes such that for any integer $k\geq 1$, there exists an infinite list $a_1,a_2,\dots$ of positive integers such that $a_i$ and $a_i+k$ has only prime divisors in $P$ for all $i$?
If $|P|=1$ then this set cannot satisfy the condition because the difference between consecutive terms in the list $1,a,a^2,\dots$ is growing. But if $|P|\geq 2$ it is possible that this can work.
Answer
For $k=1$ you require infinite many pairs $(a,a+1)$ such that both $a$ and $a+1$ contain only prime factors in $P$.
It is well known that this is impossible for every finite set $P$ because of Stormer's theorem (See here : https://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem ) stating that only finite many pairs exist.
Maybe, for larger $k$ this is possible.
No comments:
Post a Comment