(n+1)!+2,(n+1)!+3,(n+1)!+4,…,(n+1)!+n+1.
Each of these numbers is composite, as the first is divisible by 2(as each of the terms has 2 as a factor), the second is divisible by 3, the next by 4, and so on up until the final one in the list, which has n+1 as a factor. We therefore have, for any given n, a sequence ofn consecutive numbers, none ofwhich are prime.
Instead offocusing on numbers with the fewest possible factors(the primes), we shall in the next chapter turn to numbers with many factors, although we shall discover that here too there are surprising links to some very special prime numbers.