Ten in a Row, None Prime

Primes thin out as numbers grow, leaving longer and longer stretches with none at all. Find ten consecutive whole numbers, not one of them prime.

Solution

The first such run begins at $114$: $$114, 115, 116, 117, 118, 119, 120, 121, 122, 123,$$ none of which is prime. In fact the gap runs a little further, all the way to $126$, since $113$ and $127$ are the nearest primes on either side.

What is worth knowing is that runs of any length can be produced to order. For a chosen $n$, look at the $n - 1$ numbers $$n! + 2, \quad n! + 3, \quad \dots, \quad n! + n.$$ Each is composite, because $k$ divides $n!$ for every $k$ from $2$ to $n$, and so $k$ divides $n! + k$ as well. Taking $n = 11$ gives ten consecutive composites $11! + 2, \dots, 11! + 11$, though they are enormous. So prime-free stretches as long as you please always exist; it is only the smallest ten-in-a-row that happens to start at the modest $114$.