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 2016-07-12, 18:40 #1 firejuggler     Apr 2010 Over the rainbow 46538 Posts a^x+b^x+c^x="ABC" such as 166³ + 500³ + 333³ = 166,500,333 296³ + 584³ + 415³ = 296,584,415 710³ + 656³ + 413³ = 710,656,413 828³ + 538³ + 472³ = 828,538,472 I know this is useless, and I picked those from twitter. is there an easy way to find some more, or even larger one?
2016-07-12, 20:18   #2
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

2·709 Posts

Quote:
 Originally Posted by firejuggler I know this is useless, and I picked those from twitter. is there an easy way to find some more, or even larger one?
(For x=3 and one digits numbers these are known as Armstrong numbers, and seeing your example I've allowed leading zero only if a,b,c has one digit). For x=3 all solutions up to a,b,c<10^5, so up to abc<10^15, repeating the above known solutions:

Code:
1^3+5^3+3^3=153
3^3+7^3+0^3=370
3^3+7^3+1^3=371
4^3+0^3+7^3=407
16^3+50^3+33^3=165033
22^3+18^3+59^3=221859
34^3+10^3+67^3=341067
44^3+46^3+64^3=444664
48^3+72^3+15^3=487215
98^3+28^3+27^3=982827
98^3+32^3+21^3=983221
166^3+500^3+333^3=166500333
296^3+584^3+415^3=296584415
710^3+656^3+413^3=710656413
828^3+538^3+472^3=828538472
1420^3+5170^3+1000^3=142051701000
1666^3+5000^3+3333^3=166650003333
2626^3+6214^3+1664^3=262662141664
3423^3+5887^3+4614^3=342358874614
4126^3+6984^3+1211^3=412669841211
7548^3+3884^3+6433^3=754838846433
9984^3+1126^3+1211^3=998411261211
11762^3+44982^3+29233^3=117624498229233
12768^3+41454^3+37883^3=127684145437883
16666^3+50000^3+33333^3=166665000033333
36770^3+65970^3+31376^3=367706597031376
74530^3+27300^3+67749^3=745302730067749
81918^3+41244^3+58413^3=819184124458413
88086^3+22064^3+57149^3=880862206457149
From this you can also easily spot a general solution.

 2016-07-12, 23:19 #3 R. Gerbicz     "Robert Gerbicz" Oct 2005 Hungary 2·709 Posts After almost two hours of wall-clock time on my core-i3 got all 18 digits solutions for x=3 (and here stopped): Code: 157233^3+469369^3+368258^3=157233469369368258 166666^3+500000^3+333333^3=166666500000333333 194132^3+209572^3+562113^3=194132209572562113 211464^3+569598^3+258168^3=211464569598258168 245980^3+610270^3+156251^3=245980610270156251 272168^3+408414^3+568653^3=272168408414568653 339294^3+534660^3+528237^3=339294534660528237 616881^3+707455^3+303863^3=616881707455303863 771111^3+670497^3+223517^3=771111670497223517 906988^3+422208^3+440737^3=906988422208440737 ps. note that in these searches assumed that a,b,c has the same number of digits (otherwise there could be more solutions).
 2016-07-12, 23:59 #4 firejuggler     Apr 2010 Over the rainbow 32×52×11 Posts Thanks a lot.
2016-07-13, 06:42   #5
Nick

Dec 2012
The Netherlands

5E016 Posts

Quote:
 Originally Posted by firejuggler 166³ + 500³ + 333³ = 166,500,333 296³ + 584³ + 415³ = 296,584,415 710³ + 656³ + 413³ = 710,656,413 828³ + 538³ + 472³ = 828,538,472 I know this is useless, and I picked those from twitter. is there an easy way to find some more, or even larger one?
As the calculations of R. Gerbicz suggest, your first example can be extended to any length.

For squares: take a positive integer k, and suppose we want positive integers a and b each of at most k decimal digits such that
$a^2+b^2=10^ka+b$
These exist if and only if $$10^{2k}+1$$ is not prime.

Example (with k=6): $$123288^2+328768^2=123288328768$$.

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