Take the first $n$ primes $p_1,…,p_n$ and the primorial $P_n$ .Denote by $p_i$ every prime bigger than $p_n$ and smaller than $P_n$.
1) Is that true that there always be a number in any interval of consecutive integers of length $P_n$ not divided by any $p_i$? (It's the same as taking a residue class $r_i\bmod p_i$ for every $p_i$ in every possible way and wondering if you can cover all the numbers in the interval $[0,P_n-1]$.)
ADDED:
2) Even if we do not know if we can cover this interval, can we have any good upper bound on the number of ways?
Best Answer
I did some computer programming to check plausibility. In future I request that you do this step yourself.
For $p_n = 3$ and $P_n = 6,$ the only prime in between is 5, and any interval of length 6 contains an integer not congruent to any prescribed value mod 5.
In C++ I was able to check up to 10,000,000. For definiteness I took the residue classes to all be 0, that is I checked multiples of the primes between $p_n$ and $P_n.$ For the $p_n$ I checked, I was able to find only relatively short intervals of consecutive numbers, each of which is divisible by at least one prime between $p_n$ and $P_n.$ That is, these intervals have lengths much shorter than $P_n$ itself. Thus in any interval of length $P_n,$ it should be quite easy to find numbers that are not divisible by any of those primes. Indeed, the probability of picking a success at random appears to increase with $p_n.$
For example, for $p_n = 5, P_n = 30,$ I tried to find long intervals where each number had at least one divisor in the set 7, 11, 13, 17, 19, 23, 29.
The bound of 10,000,000 is not on primes, it is on the output, such as 691565 < 10,000,000.