## Posts Tagged ‘rounding’

### Rounding Up – Way Up

September 23, 2010

Ever heard of Dudeney numbers?  Neither had I, until yesterday, when I discovered them completely by accident while reading (Wikipedia, what else?) about narcissistic numbers.  A Dudeney number (named after famous English mathematician and puzzle author Henry Dudeney) is a number that is the cube of the sum of its digits.  For example,

$4913 = 17^3 = (4+9+1+3)^3$

There are only six Dudeney numbers.  Neat numbers, but I was a little disappointed by that.  What to do next?

Generalize, of course!  Generalized Dudeney numbers (discussed here, but the link appears to be dead, so I used Google’s cached version) are numbers that are some power of the sum of their digits:

$234256 = 22^4 = (2+3+4+2+5+6)^4$
$12157665459056928801 \times 10^{20} = 90^{20} = (1+2+\cdots+0+0)^{20}$

The largest number on the above site is $547210^{25662}$, which has 147253 digits.  The site links to Wolfram Alpha to confirm this.  Here’s where it gets weird:

How many digits is that?  About $10^6$?  About a million?  What kind of rounding is that?  It gets worse.  Try a number with just 100,002 digits (despite what Alpha says).  I think Alpha is a great tool, and I’ve had (far too much) fun playing with it, but I’m a tad disappointed (that’s twice in one post).  So, hey, get on that, Wolfram.