This is a fairly old one I first saw in a high school, at one of the Stellenbosch university olympiad camps. Enjoy.

It's possible to find a string of 1000 consecutive integers with no primes. For example

1001!+2,1001!+3,...,1001!+1001. Does there exist a string of consecutive integers with exactly 5 primes?

2-11?

ReplyDeletewell 2 through 11 is a string of length 10 (2,3,4,5,6,7,8,9,10,11) and the question is asking about a string of length 1000. So that doesn't work.

ReplyDelete"Does there exist a string of consecutive integers with exactly 5 primes?"

ReplyDeleteYes. But if you insist on 1000, then simply back up 5 primes before 1001!.