I just watched the "Great Courses" series of lectures in number theory, in which Professor Burger stated that
using Kronecker's theorem
for any irrational number r, the sequence ({n * r}) where n >= 0 is dense in the interval [0,1)
we can prove there's some power of two yielding a number whose initial digits equal my social security number.
The sketch of the proof given in the course used the fact that log 2 is an irrational number. I tried to flesh out the whole proof to really understand it and I got stuck.
My attempt at a proof
First assume my social security number is 566...
From Kronecker:
there exists m | m * log 2 = X.566...
Since log 2^x = x * log 2, we have
m * log 2 = X.566...
log 2 ^ m = X.566...
What to do next? Or maybe I need to start on a different tack ? I'd be very grateful for any tips or guidance.
( kroenecker proof details here )
Let $N$ be the integer formed out of the required digits (for instance, your SSN). We want the first digits of $2^n$ to line up with those of $N$. Verify for yourself that it is equivalent to check that there exists a (non-negative) integer $k$ such that $$ 10^k \cdot N \leq 2^n \leq 10^k \cdot (N + 1) $$ Taking the log base $10$ of all of these, we see that $n$ is the exponent we want if there exists an integer $k$ such that we have the inequalities $$ k + \log_{10} N \leq n \log_{10}2 \leq k + \log_{10}(N + 1) $$ Perhaps now you can relate what you're looking for to Kronecker's theorem.