The number 3608528850368400786036725 | Hacker News Reader

The number 3608528850368400786036725

blog.republicofmath.com

59 pointswglb11 comments

I'm a little confused - why is this attributed to a recent blog post/tweet? The number was part of the original wikipedia page on "Polydivisible number", published in 2003[1]. Am I missing something?

P.S.: Just to clarify - it's an interesting problem. But the wording ("the Vitale property" / "Ben announced") seems to suggest there is something special about this not covered by polydivisible numbers.

[1] https://en.wikipedia.org/w/index.php?title=Polydivisible_num...

It's just a mistake. The author thought he found something new, not realizing it had been found before.

There is an addendum at the bottom of the article mentioning that.

Ben posted this on February 20, 2015 (and this link is up unlike the one in the original): https://benvitalenum3ers.wordpress.com/2015/02/20/num3er-360... Also, it's the longest such number and the only 25 digit one. This https://www.reddit.com/r/math/comments/2wl55r/fun_with_numbe... reddit thread gives you code to check.

Here's IPython equivalent.

http://nbviewer.ipython.org/gist/hirak99/e7902bd49454cbc23bc...

Thanks. Mathematica is indecipherable to me.

Interestingly, in hexadecimal there end up being three largest numbers of length 39:

  0x34e4a468166cd8604ec0f8106ab4326098286cf
  0xaa44ce207c78fc30003c3cc0d8382e2078d07ef
  0xfae06678c2e884607eb8b4e0b0a0f0603420342

You'd expect longer numbers in larger bases. The estimate at Wikipedia (https://en.wikipedia.org/wiki/Polydivisible_number#How_many_...) can be generalized: let F_k(n) be the number of n-digit polydivisible numbers in base k. Then F_k(n) ~ (k-1) * k^(n-1) / n!. This gets bigger as n increases up to n = k, then it gets smaller. If I'm doing the asymptotics right you have F_k(ek) approximately equal to 1 - so in base k the largest polydivisible number should have about ek digits.

The length of the longest polydivisible number in base k is in the OEIS (http://oeis.org/A109783) along with this conjecture.

For those sharing mathematical or other scientific facts it would be helpful to give the layman a short explanation of why the topic is important.

It isn't. But this is what makes mathematics so full of wonders, literally. You start out with the really simple Peano axioms so that you can do elementary counting and somehow it follows that the largest such number happens to be 25 digits long and this particular number. It's the kind of thing where you revert to a three year old and ask: Why? Why? Why?

This would make a good Project Euler problem.

this number looks pure beautiful XD I wonder about the value of polydivisble numbers written in digits different than 10 XD . and if there was any further algorithm to be found within it

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