Is 292229292292 the longest 29-smooth number made of 2's and 9's?

Question

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?

Answer 1

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.