On primes less than a given number that do not divide it. Given an integer $n \geq3$ we consider $S(n)$ to be the set of all primes less than $n$ that do not divide $n$.
The question is: Are there two distincts numbers $n$ and $m$ such that $S(n)=S(m)$?
Another similar question is: Given a finite set of primes $P = \{p_1, ... , p_k\}$, does there exist a number $n > \underset{1 \leq i \leq k}{\max}p_i$ such that $p_i \nmid n$ for all $p_i \in P$, and that there are no primes between $\underset{1 \leq i \leq k}{\max}p_i$ and $n$?
For example is there a number $n$ between $53$ and $59$ such that none of the primes in $\lbrace 3, 17, 53\rbrace$ divide $n$? In this case the answer is yes because $n=55$ works.
 A: Assume n < n. If there is a prime n <= p < m then by definition m is a multiple of p.  If there is exactly one prime p then m >= 2p >= 2n, if there are two or more primes then m >= product of those primes. That should be enough to prove there cannot be such a prime between n and m which means n and m are rather close.
So the primes less than n and the primes less than m are the same, the ones not dividing n and m are the same, therefore the primes dividing n and m are the same. Let N be the product of the distinct prime factors, then n = kN, m = k’N, where k and k’ are different products formed from these primes. We have (m - n) = N (k’ - k) >= N.
As an example, with N = 6, k, k’ could be 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27 etc. Unfortunately there is always a prime in between, for example n = 86, m = 96, withd the prime p = 53 in between.
It is known that the gap between consecutive primes < 1.8 * 10^19 is less than 1,550. So wo only need the examine N <= 1550, we can exclude N primes, and for each N < 1550 which is the product of two or more distinct primes, we can quickly find all products k of those distinct primes. So we can verify this quickly for n, m < 1.8 * 10^19.
We could also show reasonably quickly that a counter example must be for example <= 10^50 with a prime gap >= 10,000.
