Fundamental Solution to Pell's Equation

From Wikipedia,

Pell's equation

is any Diophantine equation of the form

where n is a nonsquare integer and x and y are integers.

 

We can calculate the x,y for certain n with the following codes:

 

Function Pellsolution(ByVal d As Long) As String If InStr(Sqr(d), ".") = 0 Then Pellsolution = "No Fundamental Solution was found to Pell's Equation X^2-" & d & "Y^2=1.": Exit Function Dim s() As String, x, y With CreateObject("Microsoft.XMLHTTP") .Open "get", "http://www.bioinfo.rpi.edu/zukerm/cgi-bin/pell.cgi?D=" & d & "&V=NO&Q=Q", False .send Pellsolution = .responsetext End With s = Split(Pellsolution, "</span></td>") x = Mid(s(1), InStrRev(s(1), ">") + 1) y = Mid(s(4), InStrRev(s(4), ">") + 1) Pellsolution = "Fundamental Solution to Pell's Equation X^2-" & d & "Y^2=1 was:" & vbCrLf & String(55, "-") & vbCrLf & "X=" & x & vbCrLf & "Y=" & y & vbCrLf End Function Sub Main() Dim d As Long For d = 1001 To 1024 Debug.Print Pellsolution(d) Next End Sub

 It returns:

 

 

Fundamental Solution to Pell's Equation X^2-1001Y^2=1 was:
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X=1060905
Y=33532

Fundamental Solution to Pell's Equation X^2-1002Y^2=1 was:
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X=206869247
Y=6535248

Fundamental Solution to Pell's Equation X^2-1003Y^2=1 was:
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X=9026
Y=285

Fundamental Solution to Pell's Equation X^2-1004Y^2=1 was:
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X=27009633024199
Y=852416459730

Fundamental Solution to Pell's Equation X^2-1005Y^2=1 was:
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X=2950149761
Y=93059568

Fundamental Solution to Pell's Equation X^2-1006Y^2=1 was:
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X=14125267563780214605455
Y=445346140255574921748

Fundamental Solution to Pell's Equation X^2-1007Y^2=1 was:
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X=476
Y=15

Fundamental Solution to Pell's Equation X^2-1008Y^2=1 was:
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X=127
Y=4

Fundamental Solution to Pell's Equation X^2-1009Y^2=1 was:
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X=583201
Y=18360

Fundamental Solution to Pell's Equation X^2-1010Y^2=1 was:
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X=3395619
Y=106846

Fundamental Solution to Pell's Equation X^2-1011Y^2=1 was:
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X=8426
Y=265

Fundamental Solution to Pell's Equation X^2-1012Y^2=1 was:
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X=3222617399
Y=101302110

Fundamental Solution to Pell's Equation X^2-1013Y^2=1 was:
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X=309159979019115849
Y=9713562669309460

Fundamental Solution to Pell's Equation X^2-1014Y^2=1 was:
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X=4656965
Y=146246

Fundamental Solution to Pell's Equation X^2-1015Y^2=1 was:
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X=352871
Y=11076

Fundamental Solution to Pell's Equation X^2-1016Y^2=1 was:
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X=255
Y=8

Fundamental Solution to Pell's Equation X^2-1017Y^2=1 was:
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X=2900932297217
Y=90965584992

Fundamental Solution to Pell's Equation X^2-1018Y^2=1 was:
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X=15155578752298003
Y=475005498176334

Fundamental Solution to Pell's Equation X^2-1019Y^2=1 was:
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X=6089923321730
Y=190776436539

Fundamental Solution to Pell's Equation X^2-1020Y^2=1 was:
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X=511
Y=16

Fundamental Solution to Pell's Equation X^2-1021Y^2=1 was:
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X=198723867690977573219668252231077415636351801801
Y=6219237759214762827187409503019432615976684540

Fundamental Solution to Pell's Equation X^2-1022Y^2=1 was:
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X=1023
Y=32

Fundamental Solution to Pell's Equation X^2-1023Y^2=1 was:
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X=32
Y=1

No Fundamental Solution was found to Pell's Equation X^2-1024Y^2=1.