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?

  • $\begingroup$ What does "29-smooth" mean? $\endgroup$ – Umberto P. Jun 2 '14 at 21:00
  • 1
    $\begingroup$ @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
  • 2
    $\begingroup$ $100\ldots 0$ is $10$-smooth and made up of $1$'s and $0$'s. :-) $\endgroup$ – vadim123 Jun 2 '14 at 21:25

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.

| cite | improve this answer | |
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.