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A special short-list (Posted on 2015-12-08) Difficulty: 3 of 5
Some integers are one more than the sum of the squares of their digits in base 10.

Prove that the set of such integers is finite and list all of them.

No Solution Yet Submitted by Ady TZIDON    
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Solution solution with computer-generated list | Comment 2 of 3 |
Finiteness:

9^2 + 1 = 82
2(9^2) + 1 =  163
3(9^2) + 1 =  244
4(9^2) + 1 =  325

so no more than 3 digits is possible, as even if they're all 9's, the sum is only 3 digits until you get even more digits needed, and in fact, no higher than 244; in fact much less as it certainly can't start with 9, but we can check up to that value.


Evaluate the list:

DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For n = 1 To 244
  ns$ = LTrim(Str(n))
  t = 0
  For i = 1 To Len(ns)
   t = t + Val(Mid(ns, i, 1)) * Val(Mid(ns, i, 1))
  Next
  If t + 1 = n Then
    Text1.Text = Text1.Text & n & crlf
  End If
 Next n
 

 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

finds 35 and 75.

  Posted by Charlie on 2015-12-08 10:45:22
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