Given any finite positive integer, you can find that many consecutive numbers without a prime.
To be more precise, let n>1. Then the sequence (n+1)!+2 to (n+1)!+n+1 contains no primes.
For all k such that 1<k<=n+1, k divides (n+1)! and so it also divides (n+1)!+k. Thus, none of the n numbers from (n+1)!+2 to (n+1)!+k are prime.
For example, if n = 3, then the sequence this approach give is 26, 27, 28. 26 and 28 are both even and 27 is divisible by 3.
If n=1,000,000,000,000, then the 1,000,000,000,000 consecutive integers from 1,000,000,000,001!+2 to 1,000,000,000,001!+1,000,000,000,001 has no primes.
Note that (n+1)!+1 may or may not be prime. If it isn’t prime, then you have a sequence of n+2 consecutive numbers from (n+1)! to (n+1)!+n+1 with no prime.