mersenneforum.org  

Go Back   mersenneforum.org > Extra Stuff > Miscellaneous Math

Reply
 
Thread Tools
Old 2016-07-12, 18:40   #1
firejuggler
 
firejuggler's Avatar
 
Apr 2010
Over the rainbow

46538 Posts
Default 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?
firejuggler is online now   Reply With Quote
Old 2016-07-12, 20:18   #2
R. Gerbicz
 
R. Gerbicz's Avatar
 
"Robert Gerbicz"
Oct 2005
Hungary

2·709 Posts
Default

Quote:
Originally Posted by firejuggler View Post
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.
R. Gerbicz is offline   Reply With Quote
Old 2016-07-12, 23:19   #3
R. Gerbicz
 
R. Gerbicz's Avatar
 
"Robert Gerbicz"
Oct 2005
Hungary

2·709 Posts
Default

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).
R. Gerbicz is offline   Reply With Quote
Old 2016-07-12, 23:59   #4
firejuggler
 
firejuggler's Avatar
 
Apr 2010
Over the rainbow

32×52×11 Posts
Default

Thanks a lot.
firejuggler is online now   Reply With Quote
Old 2016-07-13, 06:42   #5
Nick
 
Nick's Avatar
 
Dec 2012
The Netherlands

5E016 Posts
Default

Quote:
Originally Posted by firejuggler View Post
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): \(123288^2+328768^2=123288328768\).
Nick is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Stockfish game: "Move 8 poll", not "move 3.14159 discussion" MooMoo2 Other Chess Games 5 2016-10-22 01:55
Aouessare-El Haddouchi-Essaaidi "test": "if Mp has no factor, it is prime!" wildrabbitt Miscellaneous Math 11 2015-03-06 08:17
P-1 B1/B2 selection with "Test=" vs "Pfactor=" James Heinrich Software 2 2005-03-19 21:58
Would Minimizing "iterations between results file" may reveal "is not prime" earlier? nitai1999 Software 7 2004-08-26 18:12

All times are UTC. The time now is 20:40.

Sun Nov 29 20:40:16 UTC 2020 up 80 days, 17:51, 4 users, load averages: 0.76, 1.04, 1.13

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.