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 Positively Positive (Posted on 2016-10-18)
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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 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\$

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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