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

 No Solution Yet Submitted by Ady TZIDON No Rating

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 computer exploration | Comment 1 of 2
DefDbl A-Z
Dim crlf\$

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            y4 2 3          2:1      218 12 3        3:2      6     4        (k+1)*(k^2+k)  k*(k^2+k)48 36 3        4:3     12     6100 80 3       5:4     20     8180 150 3      6:5     30    10294 252 3      7:6     42    12448 392 3      8:7     56    14648 576 3      9:8     72    16900 810 3     10:9     90    181210 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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