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(x−y)^ n = xy (Posted on 2017-01-07) Difficulty: 3 of 5
For what triplets (x, y, n) of positive integers is the equation (x−y)^ n = xy ?

No Solution Yet Submitted by Ady TZIDON    
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Some Thoughts computer exploration | Comment 1 of 2
DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For tot = 1 To 3000
 
 For n = 1 To 50
   For y = 1 To (tot - n) / 2
     DoEvents
     x = tot - n - y
     pwr = (x - y) ^ n
     If pwr = x * y Then
       Text1.Text = Text1.Text & x & Str(y) & Str(n) & crlf
     End If
   Next
 Next
 
 Next tot
 
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

finds
                                            
4 2 3       
18 12 3     
48 36 3     
100 80 3    
180 150 3   
294 252 3   
448 392 3   
648 576 3   
900 810 3   
1210 1100 3 

annotated, this is:

              ratio
 x,y,n         x:y                          x            y
4 2 3          2:1      2
18 12 3        3:2      6     4        (k+1)*(k^2+k)  k*(k^2+k)
48 36 3        4:3     12     6
100 80 3       5:4     20     8
180 150 3      6:5     30    10
294 252 3      7:6     42    12
448 392 3      8:7     56    14
648 576 3      9:8     72    16
900 810 3     10:9     90    18
1210 1100 3   11:10   110    20

Apparently you can set x = (k+1)*(k^2+k) and y = k*(k^2+k) for any k. And n=3 only.

  Posted by Charlie on 2017-01-07 14:46:42
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