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 List of pairs (Posted on 2015-12-04)
There are infinitely many ordered pairs (m,n) of positive integers for which
m + (m+1) + (m+2) + ... (n-1) + n = mn.

List the first five pairs ordered by values of m.

 No Solution Yet Submitted by Ady TZIDON No Rating

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 computer solution -- first eight pairs | Comment 1 of 3
(2m+(n-m))/2 * (n-m+1) = nm

(m+n)(n-m+1) = 2nm

mn - m^2 + m + n^2 - nm + n = 2nm

n^2 - m^2 + m + n - 2nm = 0

n^2 + (1-2m)n + m - m^2 = 0

so

DefDbl A-Z
Dim crlf\$

Form1.Visible = True

Text1.Text = ""
crlf = Chr\$(13) + Chr\$(10)

For m = 1 To 100000
b = 1 - 2 * m
c = m - m * m
det = b * b - 4 * c
If det >= 0 Then
s1 = (-b - Sqr(det)) / 2
s2 = (-b + Sqr(det)) / 2
If s1 > 0 And Abs(s1 - Int(s1)) < 0.000001 Then
Text1.Text = Text1.Text & m & Str(s1) & crlf
End If
If s2 > 0 And Abs(s2 - Int(s2)) < 0.000001 Then
Text1.Text = Text1.Text & m & Str(s2) & crlf
End If
End If
Next

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

End Sub

producing these first eight pairs:

1 1
3 6
15 35
85 204
493 1189
2871 6930
16731 40391
97513 235416

 Posted by Charlie on 2015-12-04 11:27:37

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