TITLE: Prove there are no hidden messages in Pi
QUESTION [9 upvotes]: Assume that a proof that pi is normal existed.
Is it then possible that starting at some finite position x in pi, from there on every p(n)'th digit is 0, where p(n) is the n'th prime?
I know probability arguments say no, but do they also prove that it is impossible?
Is there a way to disprove the statement?
Also, the digits of Pi can tell us certain mathematical truths, for instance that the circumference of a circle is less then any other shape with same area.
The question is, is there a limit to the information one can extract from the digits of Pi?
Is it possible to reconstruct all theorems from the digits of Pi?
What sort of truths can be extracted from the digits of Pi?
Is entire math and all truths somehow encoded within the digits of Pi ?

REPLY [4 votes]: This is more of an informational comment than a direct answer to your question. (Later) Just before sending this answer in, I noticed the question and activity are from almost a year ago!
The property you're describing is shared by almost all real numbers in both the Baire category sense and the Lebesgue measure sense. In fact, its complement is even smaller than first-category-and-measure-zero, being a countable union of lower porous sets. Note there is a huge difference between the notion of a normal number (to base $10$) and the property you're talking about, since the set of normal numbers is large in one way (Lebesgue measure) but small in another way (Baire category), whereas the set you're talking about (sometimes called the "absolutely disjunctive" real numbers, for which you can google the phrase I put in quotes) is large in both the Lebesgue measure sense and in the Baire category sense (and, in fact, even larger than what the conjunction of these two notions could allow for). For more details, see my 19 February 2003 sci.math post at 
http://groups.google.com/group/sci.math/msg/4ec315328c1afdb8
An excerpt from the above post:
Disjunctive to base $b$ ($b$ being some integer greater than $1$) means that every finite $b$-word appears infinitely often in the $b$-ary expansion of the number (note this is equivalent to every finite $b$-word appearing at least once), and the adjective "absolutely" means that this property holds for each $b = 2,\; 3,\;...$ The result is virtually immediate [all but a $\sigma$-porous set of real numbers are absolutely disjunctive], since for each of the countably many ways of choosing a fixed $b$ and a fixed $b$-word, the collection of numbers whose $b$-ary expansions don't contain that $b$-word infinitely often is a uniform Cantor set, and hence is porous (even uniformly porous; in fact, even uniformly for the stronger lim-inf type of porosity that Julia set theorists use). Recall that the larger collection of numbers which fail to be absolutely normal (or which fail to be simply normal relative to a specific base) forms a measure zero (but with Hausdorff dimension $1$) co-meager set.