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Repdigit Rigor (Posted on 2014-05-11) Difficulty: 3 of 5
Find all positive integers x, y, z so that:
  • x2 + y = z and:
  • Base ten expansion of each of x and y has precisely n digits and that of z has precisely 2n digits, where n > 1.
  • All the digits of x are the same, all the digits of y are the same and all the digits of z are the same.
Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts computer exploration | Comment 1 of 2


After some single-digit z's, the program produced

x y and z

 2 7 11
 3 2 11
 4 6 22
 5 8 33
 6 8 44
 7 6 55
 8 2 66
 9 7 88
 33 22 1111
 66 88 4444
 88 33 7777
 333 222 111111
 666 888 444444
 3333 2222 11111111
 6666 8888 44444444
 33333 22222 1111111111
 66666 88888 4444444444
 333333 222222 111111111111
 666666 888888 444444444444
 3333333 2222222 11111111111111
 6666666 8888888 44444444444444

It looks like all 3's for x and all 2's for y produce all 1's for z, and all 6's for x and all 8's for y produce all 4's for z, for as large values of n as you like.


DefDbl A-Z
Dim crlf$

Private Sub Form_Load()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 DoEvents

  xt = 0
  For xl = 1 To 7
   xt = 10 * xt + 1
  For xd = 1 To 9
   x = xt * xd
   yt = 0
   For yl = 1 To 15
    yt = 10 * yt + 1
   For yd = 1 To 9
     y = yt * yd
     z = x * x + y
     zs$ = LTrim$(Str(z))
     good = 1
     For i = 2 To Len(zs$)
       If Mid(zs$, i, 1) <> Left(zs$, 1) Then good = 0: Exit For
     Next
     If good Then Text1.Text = Text1.Text & Str(x) & Str(y) & Str(z) & crlf$
   Next
   Next
  Next
  Next

End Sub


  Posted by Charlie on 2014-05-11 16:07:53
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