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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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re: Computer exploration | Comment 2 of 5 |
(In reply to Computer exploration by Charlie)

Of course the numbers in your table are all due to the fact that

1 + 2^3 = 3^2 so for all n,

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

So if the coprime requirement is lifted your answer would become

The (3n+2) powers of 3 can be written as the sum of two k=3rd powers of integers (with a common factor of 3^n)




Edited on August 26, 2016, 10:15 am
  Posted by Jer on 2016-08-25 21:27:27

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