Powers of 2 and 3 as a sum of two powers

 

3^{3 \, n+2} \; = \; ( \,3^{n} \,)^3 \; + \; ( \,2 \, \times \, 3^{n} \,)^3
 
  3^2 \; = \; 1^3 \; + \; 2^3
  3^5 \; = \; 3^3 \; + \; 6^3
  3^8 \; = \; 9^3 \; + \; 18^3
3^{11} \; = \; 27^3 \; + \; 54^3
3^{14} \; = \; 81^3 \; + \; 162^3
3^{17} \; = \; 243^3 \; + \; 486^3
3^{20} \; = \; 729^3 \; + \; 1458^3
3^{23} \; = \; 2187^3 \; + \; 4374^3
3^{26} \; = \; 6561^3 \; + \; 13122^3
3^{29} \; = \; 19683^3 \; + \; 39366^3
3^{32} \; = \; 59049^3 \; + \; 118098^3
3^{35} \; = \; 177147^3 \; + \; 354294^3
3^{38} \; = \; 531441^3 \; + \; 1062882^3
3^{41} \; = \; 1594323^3 \; + \; 3188646^3
3^{44} \; = \; 4782969^3 \; + \; 9565938^3
3^{47} \; = \; 14348907^3 \; + \; 28697814^3

 

( \,2^{3 \, n+2} \,)^2 \; + \; ( \,2^{6 \, n+3} - 1 \,)^2 \; = \; ( \,2^{4 \, n+2} \,)^3 \; + \; 1

  ( \,2^5 \,)^2 \; + \; ( \,2^9 - 1 \,)^2 \; = \; ( \,2^6 \,)^3 \; + \; 1
  ( \,2^8 \,)^2 \; + \; ( \,2^{15} - 1 \,)^2 \; = \; ( \,2^{10} \,)^3 \; + \; 1
( \,2^{11} \,)^2 \; + \; ( \,2^{21} - 1 \,)^2 \; = \; ( \,2^{14} \,)^3 \; + \; 1
( \,2^{14} \,)^2 \; + \; ( \,2^{27}- 1 \,)^2 \; = \; ( \,2^{18} \,)^3 \; + \; 1
( \,2^{17} \,)^2 \; + \; ( \,2^{33} - 1 \,)^2 \; = \; ( \,2^{22} \,)^3 \; + \; 1
( \,2^{20} \,)^2 \; + \; ( \,2^{39} - 1 \,)^2 \; = \; ( \,2^{26} \,)^3 \; + \; 1
( \,2^{23} \,)^2 \; + \; ( \,2^{45} - 1 \,)^2 \; = \; ( \,2^{30} \,)^3 \; + \; 1
( \,2^{26} \,)^2 \; + \; ( \,2^{51} - 1 \,)^2 \; = \; ( \,2^{34} \,)^3 \; + \; 1
( \,2^{29} \,)^2 \; + \; ( \,2^{57} - 1 \,)^2 \; = \; ( \,2^{38} \,)^3 \; + \; 1
( \,2^{32} \,)^2 \; + \; ( \,2^{63} - 1 \,)^2 \; = \; ( \,2^{42} \,)^3 \; + \; 1
( \,2^{35} \,)^2 \; + \; ( \,2^{69} - 1 \,)^2 \; = \; ( \,2^{46} \,)^3 \; + \; 1
( \,2^{38} \,)^2 \; + \; ( \,2^{75} - 1 \,)^2 \; = \; ( \,2^{50} \,)^3 \; + \; 1
( \,2^{41} \,)^2 \; + \; ( \,2^{81} - 1 \,)^2 \; = \; ( \,2^{54} \,)^3 \; + \; 1
( \,2^{44} \,)^2 \; + \; ( \,2^{87} - 1 \,)^2 \; = \; ( \,2^{58} \,)^3 \; + \; 1
( \,2^{47} \,)^2 \; + \; ( \,2^{93} - 1 \,)^2 \; = \; ( \,2^{62} \,)^3 \; + \; 1

 

( \,2^{n+1} \,)^2 \; + \; ( \,2^{2 \, n+1} - 1 \,)^2 \; = \; ( \,2^{4 \, n+2} \,)^1 \; + \; 1

( \,2^{3 \, n+2} \,)^2 \; + \; ( \,2^{6 \, n+3} - 1 \,)^2 \; = \; ( \,2^{4 \, n+2} \,)^3 \; + \; 1

( \,2^{5 \, n+3} \,)^2 \; + \; ( \,2^{10 \, n+5} - 1 \,)^2 \; = \; ( \,2^{4 \, n+2} \,)^5 \; + \; 1

( \,2^{7 \, n+4} \,)^2 \; + \; ( \,2^{14 \, n+7} - 1 \,)^2 \; = \; ( \,2^{4 \, n+2} \,)^7 \; + \; 1

( \,2^{9 \, n+5} \,)^2 \; + \; ( \,2^{18 \, n+9} - 1 \,)^2 \; = \; ( \,2^{4 \, n+2} \,)^9 \; + \; 1

( \,2^{11 \, n+6} \,)^2 \; + \; ( \,2^{22 \, n+11} - 1 \,)^2 \; = \; ( \,2^{4 \, n+2} \,)^{11} \; + \; 1

and so on.

 

( \,n \,(n^2 - 2) \,)^2 \; + \; ( \,2 \, n^2 - 1 \,)^2 \; = \; ( \,n^3 \,)^2 \; + \; 1

( \,n^2 \: (n^4-2) \,)^2 \; + \; ( \,2 \, n^4 - 1 \,)^2 \; = \; ( \,n^3 \,)^4 \; + \; 1

( \,n^3 \: (n^6-2) \,)^2 \; + \; ( \,2 \, n^6 -1 \,)^2 \; = \; ( \,n^3 \,)^6 \; + \; 1

( \,n^4 \: (n^8-2) \,)^2 \; + \; ( \,2 \, n^8 - 1 \,)^2 \; = \; ( \,n^3 \,)^8 \; + \; 1

( \,n^5 \: (n^{10} - 2) \,)^2 \; + \; ( \,2 \, n^{10} - 1 \,)^2 \; = \; ( \,n^3 \,)^{10} \; + \; 1

and so on.

 

(n^3 - 2 \, n)^2 + (2 \, n^2 - 1)^2 \; = \; n^6 \; + \; 1

(n^5 - 2 \, n^3 + 2 \, n)^2 + (2 \, n^4 - 2 \, n^2 + 1)^2 \; = \; n^{10} \; + \; 1

(n^7 - 2 \, n^5 + 2 \, n^3 - 2 \, n)^2 + (2 \, n^6 - 2 \, n^4 + 2 \, n^2 - 1)^2 \; = \; n^{14} \; + \; 1

(n^9 - 2 \, n^7 + 2 \, n^5 - 2 \, n^3 + 2 \, n)^2 + (2 \, n^8 - 2 \, n^6 + 2 \, n^4 - 2 \, n^2 + 1)^2 \; = \; n^{18} \; + \; 1

and so on.

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , . Bookmark the permalink.

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