The sum of the first m integers is

m(m+1)/2

DefDbl A-Z

Dim crlf$

Private Sub Form_Load()

 Form1.Visible = True

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 

 sqs = 0

 For n = 1 To 9999999

   sqs = sqs + n * n

   mp = Sqr(sqs * 2)

   m = Int(mp)

   If m * (m + 1) / 2 = sqs Then

      Text1.Text = Text1.Text & n & Str(m) & "    " & Str(sqs) & crlf

   End If

  

 Next

 

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

  

End Sub

Sums the first n squares and attempts to find an m that will work.

The results are

n m    sums

1 1     1

5 10     55

6 13     91

85 645     208335

2696556 3615502423     6.53592888716719E+18

4432596 7619766609     2.90304215916256E+19

4767069 8498290955     3.61104745821666E+19

6506645 13551584866     9.18227261969759E+19

8888711 21637774295     2.340966382315E+20

9936291 25573540422     3.27002984870621E+20

In the cases above n=85, m=645, the results are probably spurious, as the sum of the squares exceeds the precision available with double-precision floating point.

This points to 85 being the largest n for which this is possible.