Nope :) Consider the number 1000000! + 1 (one million factorial plus 1).

  1000000! + 2 is divisible by 2.  
  1000000! + 3 is divisible by 3.  
  ...  
  1000000! + 1000000 is divisible by 1000000.

Here we have a gap of 999999 numbers, none of which are prime.