## Generalized Dudeney Numbers

In the following text I describe a way to generalize Dudeney Numbers and present the result of my search for big numbers. I also provide a Java applet which can be used by the reader to find large Generalized Dudeney Numbers himself.

### Dudeney Numbers

A Dudeney number is a positive integer that is a perfect cube such that the sum of its decimal digits is equal to the cube root of the number (Wikipedia). `So if ds(x) is the digit sum of x and ds(n3) = n, then n3 is a Dudeney Number.` There are exactly six Dudeney Numbers: 1, 512, 4913, 5832, 17576, and 19683 (here you will find the proof that no other Dudeney numbers exist):
``````    1 = 13  ;  1 = 1               512 = 83  ;  8 = 5+1+2
4913 = 173 ; 17 = 4+9+1+3        5832 = 183 ; 18 = 5+8+3+2
17576 = 263 ; 26 = 1+7+5+7+6     19683 = 273 ; 27 = 1+9+6+8+3``````

## Generalization

What happens if we generalize the condition and allow any exponents and not only 3? Here are the generalized Dudeney Numbers for ds(n4) = n:
``````      1 = 14  ;  1 = 1                   2401 = 74  ;  7 = 2+4+0+1
234256 = 224 ; 22 = 2+3+4+2+5+6       390625 = 254 ; 25 = 3+9+0+6+2+5
614656 = 284 ; 28 = 6+1+4+6+5+6      1679616 = 364 ; 36 = 1+6+7+9+6+1+6``````
Let's look for some bigger numbers with ds(n20) = n
``````1 = 120 ; ds(120) = 1
1215766545905692880100000000000000000000 = 9020 ; ds(9020) = 90
1424201691977055041360709423546231879609039601 = 18120 ; ds(18120) = 181
20864448472975628947226005981267194447042584001 = 20720 ; ds(20720) = 207``````

## Big Numbers

These numbers started to fascinate me. So I wrote a little Java program which helps me hunting for very big generalized Dudeney Numbers.

The biggest number which was found so far is 359000003122353. It is a number with 23589672 decimal digits. The digit sum is 35900000 again.

## Record List

Generalized Dudeney Number# Decimal DigitsDateFound by
359000003122353235896722012-07-21Resta (see here)
1001983370992226262012-07-16Steffen Jakob
653230301921755692012-07-15Steffen Jakob
547210256621472532010-01-12Steffen Jakob
458110218531237102010-01-11Steffen Jakob
35011017136950062010-01-07Steffen Jakob
20011010342548262010-01-06Steffen Jakob
522203103146402010-01-05Steffen Jakob

## Go Hunting!

Do you want to break the current record? You can play around with different value ranges for base and exponent with the following Java Applet. Enter some values and click on "Hunt!". The applet will then iterate over all base/exponent combinations. The largest Generalized Dudeney Number will be displayed next to the "Best" label. If you found a number which is bigger than the current record (have a look at the list above) then please write an email to . I will then add your number and name to the record list.

``` You need Java to run the applet. ```