Do the digits of $\pi$ contain every possible finite-length digit sequence? [duplicate]

Possible Duplicate:
Prove there are no hidden messages in Pi

_{This is not a practical problem. I am asking out of curiosity. Any links/references are most welcome.}

Say, we write the digits of $\pi$ in base $10$. Does this sequence of digits contain every possible finite length digit sequence? What about $e$, $\sqrt{2}$ or some other commonly known irrational numbers?

Is this property of numbers independent of base? If a number has this property when written in base $10$, will it also have it in base $2$, $3$ and all other bases?

Solution 1:

According to the Wikipedia article on normal numbers, "It is not even known whether all digits occur infinitely often in the decimal expansions of those constants."

Solution 2:

It seems very little indeed is known (in terms of rigorous proofs) on this topic. But I thought I'd add some ideas for what I think seems true and logical and might hopefully eventually be proven.

I do think those numbers are normal in every base. But more than that as well. Normality is only one aspect of randomness and I think these numbers exhibit practically all of them.

I think the answer to your last question is no. I would hypothesise for example that you can "construct" a number which is normal in every base except for 2 where "11" never occurs (obviously not including bases of powers of 2). As restrictive as that is, there's still heaps of room for irrational, fancy numbers. In fact, I believe you could arbitrarily pick any "admissible" set of disallowed sequences in any bases and still make a number which is normal otherwise. I'd say uncountably many such numbers would exist. Randomness is the norm.

Basically, what I'm saying is that the bases are very independent of each-other and you can do whatever you like to each one without really affecting the others at all.

Another relevant example is this. Most would believe that numbers like $\pi$, $e$ and $\sqrt{2}$ probably don't have "patterns" in their base expansions and I'd agree. We know at least that irrational numbers never repeat indefinitely in any base. Now consider $0.1001000010000001000000001\dots$ in some chosen base with $1$'s in the positions $1,4,9,16,\dots$ It is clearly irrational (and probably transcendental). But the expansion has a simple pattern. Moreover I'm confident it's actually normal in every other base. Obviously, we could use any pattern we liked and the idea is the same.

SUMMARY: A pattern in any one base is absolute chaos in every other base with the one and only exception being rational numbers.

Now please shoot me down with some counter-examples!