21st Century Math Problem?

Question

(a) Let a sequence of a integer x have a length of n, n is less than infinity, and let this sequence be part of the infinite sequence of integers in the square root of 2. Is it true that for every sequence of length n there is a sequence of length n+1 for the same integer x?

(b) What about for an irrational number y = k sqrt(2)?

(c) Find a function f(x,n,j) that is equal to 1 when for a sequence of integers x of length n there is also a length n+1 contained in some real number j and zero when this is not true. Express this function in terms of an infinite series. (For example, f(9,2, pi)=1 because there is a sequence of 2 9s at about the hundreth decimal place.)

Answer 1

I am not completely certain that I understand your question, but here is a part of what is known in regard to this subject.

Almost all irrational numbers between 0 and 1, in the sense of Lebesgue measure, have the property that they are "normal", to all bases. In particular, that means that given any particular string of n digits whatsoever, it occurs with the right frequency, namely about 1/(10^n) of the time, in base 10 (and a comparable result is true for any base). It is much harder to answer questions of the type that you pose for specific constants like pi. It is not known at present whether pi has arbitrarily long strings of 9's, for example.

For (c), it follows that if we have almost any irrational number x in mind, then we would get f(x,n,j) = 1 for all n and j.

Answer 2

Are you insane?