As it turns out, the answer to all these questions is “yes!” Even for a million! (You can find a million consecutive composite integers. That is, you can find a string of one million numbers without hitting a prime.) And – astonishingly – it turns out to be true for ANY number. Pick any number you like, no matter how big, and you can find that many consecutive numbers without a prime in them!! **
One of the ways this is phrased is that you “can find arbitrarily large gaps in the primes”.
This is particularly ironic – almost contradictory! – since the primes themselves are infinite! They NEVER run out – there is no largest one, they go on forever. And yet, we can find gaps in them as large as we’d like!!
** In each of these cases, one can even determine a number that starts the string. The reasoning is not too difficult (I used to show it to future elementary teachers all the time), but it involves learning a new symbol, and a few steps to think about. I’d be happy to elaborate for anyone who wants to know more. Feel free to contact me.