Response from William Rapaport on November 19, 2009

It doesn't imply that, because that's false. If pi had an infinite number of the same digits repeated, it would be a rational number. But it's an irrational number. See Wikipedia's article on pi.

Response from Daniel J. Velleman on November 19, 2009

There are two questions here that need to be distinguished: (1) Does the decimal expansion of pi contain a large but finite string of consecutive 7's--say, 1000 consecutive 7's? (2) Does it contain an infinite string of consecutive 7's?

The second question is the easier one. The only way that pi could contain a string of infinitely many consecutive 7's is if all the digits are 7's from some point on. And, as William has pointed out, that can't happen because pi is irrational.

But the first question is harder. The digits of pi "look random." Imagine a number whose digits are generated by some random process--for example, we might roll a 10-sided die repeatedly to generate the digits. One could compute the probability that a string of 1000 consecutive 7's appears in the first n digits of this number. For n =1000, this number would be extremely small--you'd have to roll 1000 consecutive 7's on your first 1000 rolls, and that's very unlikely. But as n increases, the probability increases, and in fact as n approaches infinity the probability approaches 1. Thus, if you generate an infinite string of digits this way, then the probability that you will eventually get 1000 consecutive 7's is 1, although that doesn't mean that you're guaranteed to get 1000 consecutive 7's. (For more on the difference between "probability 1" and "guaranteed to happen," see this question. Well, actually the discussion there is about the difference between "probability 0" and "guaranteed not to happen," but the idea is the same.)

This heuristic argument makes it seem very likely that the decimal expansion of pi contains 1000 consecutive 7's. But it doesn't actually prove anything--after all, the digits of pi are not generated by a random process, they just "look random" to us. If pi is a normal number, then it must contain 1000 consecutive 7's. But proving that pi is normal is an unsolved problem.