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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?

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  • $\begingroup$ What does "29-smooth" mean? $\endgroup$ – Umberto P. Jun 2 '14 at 21:00
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    $\begingroup$ @UmbertoP. It means the number has no prime factor greater than 29. See en.wikipedia.org/wiki/Smooth_number $\endgroup$ – Matthew Conroy Jun 2 '14 at 21:11
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    $\begingroup$ $100\ldots 0$ is $10$-smooth and made up of $1$'s and $0$'s. :-) $\endgroup$ – vadim123 Jun 2 '14 at 21:25
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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.

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  • $\begingroup$ A short, elegant proof is likely impossible, so "highly unlikely, here's why" is good enough. $\endgroup$ – Ed Pegg Jun 3 '14 at 14:35

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