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Powers of 3 (Posted on 2016-08-25) Difficulty: 3 of 5
Find powers of 3 which can be written as the sum of the kth powers (k > 1) of two coprime integers.

No Solution Yet Submitted by Ady TZIDON    
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Hints/Tips re(2): Computer exploration | Comment 3 of 5 |
(In reply to re: Computer exploration by Jer)

please edit

(3^n)^3 + (2*3^n)^3 = (1+2^3)*(3^n) = 9*(3^n)= 3^(n+2)

So if the coprime requirement is lifted -  your answer would become

The (3m+2) powers of 3 can be written as the sum of two 3rd powers of integers (with a common factor of 3^3m):
3^(3m+2)=(3^(3m))(1+8)= 3^(3m)   + (2*3^m)^3

e.g.  for m=1   3^5= 3^3+6^3= 27+216=243

Edited on August 26, 2016, 8:38 am
  Posted by Ady TZIDON on 2016-08-26 08:34:06

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