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Positively Positive (Posted on 2016-10-18) Difficulty: 3 of 5
Find all pairs (A, B) of positive integers such that each of the equations
x2 - A*x + B = 0 and x2 - B*x + A=0 has positive integer roots.

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Some Thoughts findings | Comment 1 of 7
First equation roots:

x = (A +/- sqrt(A^2 - 4*B) / 2

Second equation roots:

x = (B +/- sqrt(B^2 - 4*A) / 2

4*B <= A^2

4*A <= B^2

4*sqrt(A) <= A^2, remembering A is positive
4*sqrt(A) <= (sqrt(A))^4

(sqrt(A))^3 >= 4
A >= 2.5198... and since integer, >= 3

Same for B

sqrt(A^2 - 4*B) must be of same parity as A, and must be smaller than A so as to make both roots positive. (Each equation must have positive roots -- singular equation, plural roots).

The same with sqrt(B^2 - 4*A) vs B.

Found that satisfy:

A B     roots
4 4    2 2
5 6    4.5 4.5

Also B and A can be interchanged due to the symmetry of the equations. The roots would be reversed as well, but the roots are the same in each case.

However, 4.5 is not an integer so A=4; B=4 would seem the only pair.

DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For tot = 6 To 10000
   For a = 3 To tot / 2
     b = tot - a
     If 4 * b <= a * a Then
      If 4 * a <= b * b Then
        tst = Sqr(a * a - 4 * b)
        If tst = Int(tst) And (tst - a) Mod 2 = 0 And tst < a Then
          tst = Sqr(b * b - 4 * a)
          If tst = Int(tst) And (tst - b) Mod 2 = 0 And tst < b Then
            
            Text1.Text = Text1.Text & a & Str(b) & "    "
            Text1.Text = Text1.Text & (a + tst) / 2 & Str((a + tst) / 2) & crlf
            
          End If
        End If
      End If
     End If
     DoEvents
   Next a
 Next tot

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


  Posted by Charlie on 2016-10-18 16:18:25
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