20160712, 18:40  #1 
Apr 2010
Over the rainbow
4653_{8} 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? 
20160712, 20:18  #2  
"Robert Gerbicz"
Oct 2005
Hungary
2·709 Posts 
Quote:
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 

20160712, 23:19  #3 
"Robert Gerbicz"
Oct 2005
Hungary
2·709 Posts 
After almost two hours of wallclock time on my corei3 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 
20160712, 23:59  #4 
Apr 2010
Over the rainbow
3^{2}×5^{2}×11 Posts 
Thanks a lot.

20160713, 06:42  #5  
Dec 2012
The Netherlands
5E0_{16} Posts 
Quote:
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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