Generalized Dudeney Numbers
A Dudeney number is a perfect cube whose decimal digits sum to its cube root. Generalize the exponent and you get a far richer hunt - this page lets you search for big ones right in your browser.
Dudeney numbers
Take the number 8 and cube it: 8 × 8 × 8 = 512. Now add up the digits of
512: 5 + 1 + 2 = 8 - right back where you started. A
Dudeney number is a perfect cube that does exactly that: the digits of the cube
add up to its cube root. Writing ds(x) for that
digit sum (so
ds(4913) = 4+9+1+3 = 17), a cube n3 qualifies exactly when
ds(n3) = n. There are only six - past a point a cube has too few
digits for them to ever reach its root, so the hunt is finite:
Generalization
What if we allow any exponent, not just 3? We look for integers with
ds(nk) = n. For the fourth power (k = 4):
The numbers grow fast. A few solutions for k = 20:
Go hunting!
Pick ranges for the base n and the exponent k and press
Hunt!. The hunter iterates over every base/exponent combination, and the
largest number it finds with ds(nk) = n is shown next to
Biggest. The computation runs in a background thread, so the page stays
responsive and you can stop it any time.
Searches every nk whose digits add up to n.
Hits found
The current record is 2000000000014764030693 (a number with 152 084 723 026 digits). Raise the ranges to hunt for bigger numbers - very large values take a long time to compute.
Record history
This is the record as it grew over time - each entry was the largest known generalized Dudeney number when it was found, each one beating the previous. The top row is the current record. Click any entry to load it into the hunter and recompute it. Be warned: the current record has 4.4 billion significant digits - rebuilding it in-browser took about 3 hours and needs tens of gigabytes of memory (measured on a 34 GB desktop), and it will crash phones or low-memory machines. The rows below it are far quicker. The record is also verified offline with the tools in the repo:
| Generalized Dudeney number | # digits | Date | Found by |
|---|---|---|---|
| 152084723026 | 2026-06-08 | Steffen Jakob | |
| 23589672 | 2012-07-21 | Resta (link) | |
| 222626 | 2012-07-16 | Steffen Jakob | |
| 175569 | 2012-07-15 | Steffen Jakob | |
| 147253 | 2010-01-12 | Steffen Jakob | |
| 123710 | 2010-01-11 | Steffen Jakob | |
| 95006 | 2010-01-07 | Steffen Jakob | |
| 54826 | 2010-01-06 | Steffen Jakob | |
| 14640 | 2010-01-05 | Steffen Jakob |
Beat the record? The hunter shows the digit count of every number it finds. If you turn up something bigger than the top row, it is worth reporting.