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 I found so far is 547210²⁵⁶⁶². It is a number with 147253 decimal digits. The digit sum is 547210 again.

Record List

Generalized Dudeney Number	# Decimal Digits	Date	Found by
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

547210²⁵⁶⁶² =
3225502971203402335424358792796366704623490239249837760734138923506524772733429374719803440895273606166655526677431369272211666971011339066766776275363818819772742124073983437203598527805676574316913886551263028819535403294767650631107816812701710965042559766511320813016983120294761283193648478247748302373186883163209130991705178465984844288313563073217304732630104305668912699572209858687773090209446713385162834441746742604185961025727186444257823099983228907981694271741276337592194235504500354878832533778528816974118924342288609329109555831172831883755068759799614184757405100972854566242422805104146792702061481100497355001348838214760288209771458239309446418199836722159731753778678478572605208940928392075966073529994443456301678277196661239786837253076660929710683574523991018333984670805903514396224114157661768863045119930661914904004736178091336614956623176615798453688372124384090157749887742187445817954115164672681235970935508436679826357263675077543339251115873956996250296077500788
9406955765551776918192715661256156791379733339793222076040283985953064009813231631001070992205509116538048715447979344005540679190931564797975565471566404083846589193398601847428864146163728992862088656226110004941641568362428547424279152968990432412156750802893451729106787218510464771189877945205949602705301667623455448287247309716825020551471424704126298808094035797079695354627049596906705303688988841313119769605247486094911258635466994132654670440779370284842562796506280268826053940805515145899864430316942743613554938229716020464255059531334071494204842865435933366244080298344199876655336294481506143355044726829721858862204706462145974620553008318030441437468744571122143660266590868856633675807929912290363505069004779555796354865038345657590232677698407522575120446969620728728761562161273741838617283803807517754477527161963079301703177201725288237090092322515077817821551966591522608914681504778654570642989592352538587837119514180329405863105109958452917308542098014935409291986678251
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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 email . I will then add your number and name to the record list.


	
			
			
			
			
			You need Java to run the applet.