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+3Generalization
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+61 = 120 ; ds(120) = 1
1215766545905692880100000000000000000000 = 9020 ; ds(9020) = 90
1424201691977055041360709423546231879609039601 = 18120 ; ds(18120) = 181
20864448472975628947226005981267194447042584001 = 20720 ; ds(20720) = 207Big 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 Digits | Date | Found by |
|---|---|---|---|
| 359000003122353 | 23589672 | 2012-07-21 | Resta (see here) |
| 100198337099 | 222626 | 2012-07-16 | Steffen Jakob |
| 65323030192 | 175569 | 2012-07-15 | Steffen Jakob |
| 54721025662 | 147253 | 2010-01-12 | Steffen Jakob |
| 45811021853 | 123710 | 2010-01-11 | Steffen Jakob |
| 35011017136 | 95006 | 2010-01-07 | Steffen Jakob |
| 20011010342 | 54826 | 2010-01-06 | Steffen Jakob |
| 522203103 | 14640 | 2010-01-05 | Steffen 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.