TITLE: Is 292229292292 the longest 29-smooth number made of 2's and 9's?
QUESTION [5 upvotes]: Is 292229292292 the longest 29-smooth number made of 2's and 9's? 
The factorization is $2^2 7^8*19*23*29$.  Is there a general way to find other numbers of this sort without resorting to brute force techniques?

REPLY [5 votes]: For for small fixed $B$, e.g. $B=29$, the number $\Psi(x,B)$ of $B$-smooth integers less than $x$ has an asymptotic estimate
$$ \Psi(x,B) \sim  \frac{1}{\pi(B)!} \prod_{p\le B}\frac{\log x}{\log p}. $$
So what were the odds of seeing such a 12-digit number that you found?
Roughly the probability would be $\Psi(10^{12},29) * (2/10)^{12} \approx 0.0002$.  So it seems lucky that such a number exists.  The probability will get exponentially smaller as the number of digits increases, so it seems rather unlikely that you'll find more of these.