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Square Sum in Pairs (Posted on 2016-06-27) Difficulty: 3 of 5
(i) X and Y are two positive integers such that:
Each of 2016+X, 2016+Y and X+Y is a perfect square.

Find the two smallest values of X+Y

(ii) If in addition to given conditions in (i), it is known that X is itself a perfect square and Y is two times a perfect square- then, what is the smallest value of X+Y?

No Solution Yet Submitted by K Sengupta    
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Solution computer solution | Comment 1 of 2
The first two values for x+y are 576 and 1521:

 x   y       2016+    x+y   square roots
288 288    2304 2304 576     48 48 24
288 1233   2304 3249 1521    48 57 39
288 1828   2304 3844 2116    48 62 46
1120 1584  3136 3600 2704    56 60 52
288 4068   2304 6084 4356    48 78 66
585 5040   2601 7056 5625    51 84 75
2608 3168  4624 5184 5776    68 72 76
688 5553   2704 7569 6241    52 87 79
2340 4384  4356 6400 6724    66 80 82
900 7200   2916 9216 8100    54 96 90  **
3609 5040  5625 7056 8649    75 84 93
2209 7200  4225 9216 9409    65 96 97  **
3168 6633  5184 8649 9801    72 93 99

Marked **, x and y values 900 and 7200, respectively are squares and total 8100, as are 2209 and 7200 later in the list, totaling 9409.

Likewise 2209 and 7200.

DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For tot = 1 To 10000
   For x = 1 To tot / 2
    DoEvents
     y = tot - x
     sq1 = x + 2016
     sr1 = Int(Sqr(sq1) + 0.5)
     If sr1 * sr1 = sq1 Then
        sq2 = y + 2016
        sr2 = Int(Sqr(sq2) + 0.5)
        If sr2 * sr2 = sq2 Then
          sr3 = Int(Sqr(tot) + 0.5)
          If sr3 * sr3 = tot Then
            Text1.Text = Text1.Text & x & Str(y) & "   "
            Text1.Text = Text1.Text & sq1 & Str(sq2) & Str(tot) & "    "
            Text1.Text = Text1.Text & sr1 & Str(sr2) & Str(sr3)
            sr = Int(Sqr(x) + 0.5)
            If sr * sr = x And y Mod 2 = 0 Then
                sr = Int(Sqr(y / 2) + 0.5)
                If sr * sr = y / 2 Then
                  Text1.Text = Text1.Text & "  **"
                End If
            End If
            Text1.Text = Text1.Text & crlf
          End If
        End If
     End If
   Next
 Next
 
 
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

The program chose x <= y, as the first part was symmetrical with regard to x and y.

A version checking part b for the reversed values for part ii did not find any, so that version is not included here.

  Posted by Charlie on 2016-06-27 13:57:18
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