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 )