All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
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    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts Computer exploration | Comment 1 of 5
DefDbl A-Z
Dim crlf$

Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 p3 = 3
 For base3 = 2 To 31
   p3 = p3 * 3
   k = 1
   Do
     k = k + 1
     n1 = 1
     t1 = n1
     didOne = 0
     Do
       n1 = n1 + 1
       t1 = Int(n1 ^ k + 0.5)
       If t1 > p3 Then Exit Do
       didOne = 1
       t2 = p3 - t1
       n2 = Int(t2 ^ (1 / k) + 0.5)
       If t2 = Int(n2 ^ k + 0.5) And t2 > 1 Then
      '  If gcd(n1, n2) = 1 Then
         Text1.Text = Text1.Text & base3 & Str(p3) & " = " & n1 & "^" & k & "+" & n2 & "^" & k
         Text1.Text = Text1.Text & "   " & gcd(n1, n2) & crlf
         ct = ct + 1
      '  End If
       End If
       DoEvents
     Loop
   Loop Until didOne = 0
 Next
 
 Text1.Text = Text1.Text & crlf & ct & " done"
  
End Sub

Function gcd(a, b)
  x = a: y = b
  Do
   q = Int(x / y)
   z = x - q * y
   x = y: y = z
  Loop Until z = 0
  gcd = x
End Function

was originally written to check that the two integers were coprime, but no solutions were found, so that condition was commented out.  The results now show that all the solutions have two numbers that are multiples of 3, and the powers (k) are all 3. The gcd's of the two numbers are all powers of 3.

     power
     of 3                                                       gcds

   3^5  = 243 = 3^3+6^3   3
   3^5  = 243 = 6^3+3^3   3
   3^8  = 6561 = 9^3+18^3   9
   3^8  = 6561 = 18^3+9^3   9
   3^11 = 177147 = 27^3+54^3   27
   3^11 = 177147 = 54^3+27^3   27
   3^14 = 4782969 = 81^3+162^3   81
   3^14 = 4782969 = 162^3+81^3   81
   3^17 = 129140163 = 243^3+486^3   243
   3^17 = 129140163 = 486^3+243^3   243
   3^20 = 3486784401 = 729^3+1458^3   729
   3^20 = 3486784401 = 1458^3+729^3   729
   3^23 = 94143178827 = 2187^3+4374^3   2187
   3^23 = 94143178827 = 4374^3+2187^3   2187
   3^26 = 2541865828329 = 6561^3+13122^3   6561
   3^26 = 2541865828329 = 13122^3+6561^3   6561
   3^29 = 68630377364883 = 19683^3+39366^3   19683
   3^29 = 68630377364883 = 39366^3+19683^3   19683

Edited on August 25, 2016, 2:12 pm
  Posted by Charlie on 2016-08-25 14:11:11

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information