Consecutive primes are often far apart. In fact, show there are arbitrarily long sequences of consecutive non-prime integers.

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Not only are consecutive primes often far apart, but it's been conjectured that they are arbitrarily far apart arbitrarily often! A solution to the following conjecture will earn you $10,000; see A Tribute to Paul Erdos, Cambridge University Press, 1990, pp. 467-477

Conjecture: For every real number C, the difference between the n'th prime and n+1'st prime exceeds the quantity infinitely often.

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