(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 24898, 708] NotebookOptionsPosition[ 22329, 624] NotebookOutlinePosition[ 22666, 639] CellTagsIndexPosition[ 22623, 636] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"L", ",", "L1", ",", "de", ",", "quo"}], "]"}]], "Input", CellChangeTimes->{{3.40019794852553*^9, 3.400197954492199*^9}, { 3.40023921452179*^9, 3.40023921513883*^9}, {3.400239888255788*^9, 3.400239889379815*^9}}], Cell[BoxData[ RowBox[{"(*", " ", RowBox[{ "These", " ", "are", " ", "the", " ", "notes", " ", "on", " ", "the", " ", "quadratic", " ", "integral", " ", RowBox[{"L", "[", RowBox[{"a", ",", "m"}], "]"}], " ", "discussed", " ", "in", " ", "class"}], " ", "*)"}]], "Input", CellChangeTimes->{{3.400197545512731*^9, 3.400197577116798*^9}}], Cell[BoxData[ RowBox[{"(*", " ", RowBox[{ "We", " ", "now", " ", "implement", " ", "the", " ", "recurrence", " ", "proved", " ", "in", " ", "class"}], " ", "*)"}]], "Input", CellChangeTimes->{{3.400197702378953*^9, 3.4001977192445374`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"L", "[", RowBox[{"a_", ",", "m_"}], "]"}], ":=", RowBox[{ RowBox[{"L", "[", RowBox[{"a", ",", "m"}], "]"}], "=", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{"m", "\[Equal]", "0"}], ",", "b", ",", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"2", "*", "m"}], "-", "1"}], ")"}], "/", RowBox[{"(", RowBox[{"2", "*", "m"}], ")"}]}], ")"}], "*", RowBox[{ RowBox[{"L", "[", RowBox[{"a", ",", RowBox[{"m", "-", "1"}]}], "]"}], "/", RowBox[{"(", RowBox[{"1", "-", RowBox[{"a", "^", "2"}]}], ")"}]}]}], "-", RowBox[{"a", "/", RowBox[{"(", RowBox[{"2", "*", "m", "*", RowBox[{"(", RowBox[{"1", "-", RowBox[{"a", "^", "2"}]}], ")"}]}], ")"}]}]}]}], "]"}], "//", "Together"}]}]}], ";"}]], "Input", CellChangeTimes->{{3.3992178559919577`*^9, 3.3992179555706873`*^9}, { 3.3992184587086563`*^9, 3.399218471738408*^9}, {3.400197723567539*^9, 3.400197726279825*^9}, {3.400197770151866*^9, 3.400197784456512*^9}, 3.4001979753295803`*^9, {3.400239599839147*^9, 3.400239604238883*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"L", "[", RowBox[{"a", ",", "0"}], "]"}]], "Input", CellChangeTimes->{{3.400197730884535*^9, 3.400197732444171*^9}, { 3.400197790453437*^9, 3.400197790519146*^9}, 3.400239614733758*^9}], Cell[BoxData["b"], "Output", CellChangeTimes->{3.400197732848316*^9, 3.400197790944026*^9, 3.4001979809614573`*^9, 3.400238989839241*^9, 3.400239110056814*^9, 3.400239250029896*^9, 3.400239517967882*^9, 3.400239615040019*^9, 3.400239902311998*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"L", "[", RowBox[{"a", ",", "1"}], "]"}]], "Input", CellChangeTimes->{{3.400197755363736*^9, 3.400197757763571*^9}, { 3.4001977936448402`*^9, 3.40019779628438*^9}, 3.400239617238769*^9}], Cell[BoxData[ FractionBox[ RowBox[{"a", "-", "b"}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["a", "2"]}], ")"}]}]]], "Output", CellChangeTimes->{3.400197796808131*^9, 3.4001979830154753`*^9, 3.400238991726383*^9, 3.400239110123684*^9, 3.4002392528344927`*^9, 3.400239519821734*^9, 3.400239617613118*^9, 3.400239902351039*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"L", "[", RowBox[{"a", ",", "2"}], "]"}]], "Input", CellChangeTimes->{{3.400239641701981*^9, 3.40023964394762*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{ RowBox[{"-", "5"}], " ", "a"}], "+", RowBox[{"2", " ", SuperscriptBox["a", "3"]}], "+", RowBox[{"3", " ", "b"}]}], RowBox[{"8", " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["a", "2"]}], ")"}], "2"]}]]], "Output", CellChangeTimes->{3.4002396445480633`*^9, 3.4002399024027853`*^9}] }, Open ]], Cell[BoxData[ RowBox[{"(*", " ", RowBox[{ "The", " ", "new", " ", "function", " ", "L1", " ", "will", " ", "eliminate", " ", "the", " ", "power", " ", "of", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"a", "^", "2"}]}], ")"}], " ", "in", " ", "the", " ", "denominator"}], " ", "*)"}]], "Input", CellChangeTimes->{{3.4002395548834457`*^9, 3.400239593060277*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"L1", "[", RowBox[{"a_", ",", "m_"}], "]"}], ":=", RowBox[{ RowBox[{"L1", "[", RowBox[{"a", ",", "m"}], "]"}], "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"a", "^", "2"}]}], ")"}], "^", "m"}], "*", RowBox[{"L", "[", RowBox[{"a", ",", "m"}], "]"}]}], "//", "Together"}]}]}], ";"}]], "Input", CellChangeTimes->{{3.399217967007819*^9, 3.399217985057005*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"L1", "[", RowBox[{"a", ",", "0"}], "]"}]], "Input", CellChangeTimes->{{3.400239632985544*^9, 3.400239634171687*^9}}], Cell[BoxData["b"], "Output", CellChangeTimes->{3.4002396347283497`*^9, 3.400239902512879*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"L1", "[", RowBox[{"a", ",", "1"}], "]"}]], "Input", CellChangeTimes->{{3.400239636425001*^9, 3.400239637555599*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"a", "-", "b"}], "2"]], "Output", CellChangeTimes->{3.40023963798522*^9, 3.400239902550824*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"L1", "[", RowBox[{"a", ",", "2"}], "]"}]], "Input", CellChangeTimes->{{3.4002396469292088`*^9, 3.400239648594832*^9}}], Cell[BoxData[ RowBox[{ FractionBox["1", "8"], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "5"}], " ", "a"}], "+", RowBox[{"2", " ", SuperscriptBox["a", "3"]}], "+", RowBox[{"3", " ", "b"}]}], ")"}]}]], "Output", CellChangeTimes->{3.400239649239984*^9, 3.400239902600947*^9}] }, Open ]], Cell[BoxData[ RowBox[{"(*", " ", RowBox[{ "We", " ", "now", " ", "study", " ", "the", " ", "denominator", " ", "of", " ", "L1"}], " ", "*)"}]], "Input", CellChangeTimes->{{3.400239654260401*^9, 3.400239664978448*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "de", "]"}]], "Input", CellChangeTimes->{{3.400239738525757*^9, 3.400239740025796*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"de", "[", "m_", "]"}], ":=", RowBox[{"Denominator", "[", RowBox[{"L1", "[", RowBox[{"a", ",", "m"}], "]"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.399218022144261*^9, 3.39921804088242*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"de", "[", "m", "]"}], ",", RowBox[{"{", RowBox[{"m", ",", "0", ",", "10"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.399218041764165*^9, 3.399218059864201*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "1", ",", "2", ",", "8", ",", "48", ",", "384", ",", "1280", ",", "15360", ",", "215040", ",", "229376", ",", "4128768", ",", "82575360"}], "}"}]], "Output", CellChangeTimes->{{3.399218048362012*^9, 3.399218060270986*^9}, 3.399220253778632*^9, 3.399278073527272*^9, 3.3992795691431932`*^9, 3.399279615727339*^9, 3.399285175333289*^9, 3.3993367921504927`*^9, 3.400239129871381*^9, 3.4002396778047943`*^9, 3.400239745060131*^9, 3.400239902745365*^9}] }, Open ]], Cell[BoxData[ RowBox[{"(*", " ", RowBox[{ "After", " ", "some", " ", "experimenting", " ", "we", " ", "decided", " ", "that", " ", "this", " ", "quotient", " ", "was", " ", "interesting"}], " ", "*)"}]], "Input", CellChangeTimes->{{3.400239681429738*^9, 3.400239705020196*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "quo", "]"}]], "Input", CellChangeTimes->{{3.400239752367693*^9, 3.400239755283599*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"quo", "[", "m_", "]"}], ":=", RowBox[{ RowBox[{"quo", "[", "m", "]"}], "=", RowBox[{ RowBox[{"de", "[", "m", "]"}], "/", RowBox[{"(", RowBox[{"2", "*", "m", "*", RowBox[{"de", "[", RowBox[{"m", "-", "1"}], "]"}]}], ")"}]}]}]}], ";"}]], "Input", CellChangeTimes->{{3.3992181239634542`*^9, 3.3992181364076357`*^9}, { 3.399218232675403*^9, 3.399218234511866*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"quo", "[", "m", "]"}], ",", RowBox[{"{", RowBox[{"m", ",", "2", ",", "20"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.399278084446053*^9, 3.399278095374322*^9}, { 3.4002397121901093`*^9, 3.40023971422663*^9}, {3.4002397627211742`*^9, 3.400239762787527*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", "1", ",", "1", ",", FractionBox["1", "3"], ",", "1", ",", "1", ",", FractionBox["1", "15"], ",", "1", ",", "1", ",", FractionBox["1", "21"], ",", "1", ",", FractionBox["1", "5"], ",", FractionBox["1", "9"], ",", "1", ",", "1", ",", FractionBox["1", "33"], ",", FractionBox["1", "35"], ",", "1", ",", FractionBox["1", "39"]}], "}"}]], "Output", CellChangeTimes->{{3.399278092776825*^9, 3.39927809599162*^9}, 3.399279569214299*^9, 3.39927961581066*^9, 3.399285175411922*^9, 3.39933679224093*^9, 3.400239139878861*^9, 3.400239714735429*^9, { 3.400239759388227*^9, 3.4002397632015343`*^9}, 3.400239902836928*^9}] }, Open ]], Cell[BoxData[ RowBox[{"(*", " ", RowBox[{"Now", " ", "we", " ", "state", " ", "our", " ", "conjecture"}], " ", "*)"}]], "Input", CellChangeTimes->{{3.4002397664944887`*^9, 3.400239776347941*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"m", ":=", "1"}], ")"}], "&&", RowBox[{"While", "[", RowBox[{ RowBox[{"m", "\[LessEqual]", "20"}], ",", RowBox[{"(", RowBox[{ RowBox[{"Print", "[", RowBox[{"m", ",", "\"\< \>\"", ",", RowBox[{ RowBox[{"2", "*", "m"}], "-", "1"}], ",", "\"\< \>\"", ",", RowBox[{"PrimeQ", "[", RowBox[{ RowBox[{"2", "*", "m"}], "-", "1"}], "]"}], ",", "\"\< \>\"", ",", RowBox[{"quo", "[", "m", "]"}]}], "]"}], ";", RowBox[{"m", "++"}]}], ")"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.399292072968398*^9, 3.399292096690455*^9}, { 3.3992922889364347`*^9, 3.399292297754683*^9}, {3.3992923379693737`*^9, 3.399292338035972*^9}, {3.4002397983413563`*^9, 3.400239799141309*^9}}], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "1", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "False", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "1"}], SequenceForm[1, " ", 1, " ", False, " ", 1], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.400239902891292*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "2", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "3", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "True", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "1"}], SequenceForm[2, " ", 3, " ", True, " ", 1], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.400239902893093*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "3", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "5", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "True", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "1"}], SequenceForm[3, " ", 5, " ", True, " ", 1], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.4002399028969803`*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "4", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "7", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "True", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "1"}], SequenceForm[4, " ", 7, " ", True, " ", 1], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.4002399029004793`*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "5", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "9", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "False", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", FractionBox["1", "3"]}], SequenceForm[5, " ", 9, " ", False, " ", Rational[1, 3]], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.4002399029036407`*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "6", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "11", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "True", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "1"}], SequenceForm[6, " ", 11, " ", True, " ", 1], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.4002399029068003`*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "7", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "13", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "True", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "1"}], SequenceForm[7, " ", 13, " ", True, " ", 1], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.400239902909995*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "8", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "15", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "False", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", FractionBox["1", "15"]}], SequenceForm[8, " ", 15, " ", False, " ", Rational[1, 15]], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.400239902913159*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "9", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "17", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "True", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "1"}], SequenceForm[9, " ", 17, " ", True, " ", 1], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.400239902916366*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "10", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "19", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "True", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "1"}], SequenceForm[10, " ", 19, " ", True, " ", 1], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.400239902919531*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "11", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "21", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "False", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", FractionBox["1", "21"]}], SequenceForm[11, " ", 21, " ", False, " ", Rational[1, 21]], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.400239902922695*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "12", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "23", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "True", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "1"}], SequenceForm[12, " ", 23, " ", True, " ", 1], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.400239902925878*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "13", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "25", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "False", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", FractionBox["1", "5"]}], SequenceForm[13, " ", 25, " ", False, " ", Rational[1, 5]], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.400239902929113*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "14", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "27", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "False", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", FractionBox["1", "9"]}], SequenceForm[14, " ", 27, " ", False, " ", Rational[1, 9]], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.400239902932355*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "15", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "29", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "True", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "1"}], SequenceForm[15, " ", 29, " ", True, " ", 1], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.400239902935605*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "16", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "31", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "True", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "1"}], SequenceForm[16, " ", 31, " ", True, " ", 1], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.400239902938848*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "17", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "33", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "False", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", FractionBox["1", "33"]}], SequenceForm[17, " ", 33, " ", False, " ", Rational[1, 33]], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.40023990294205*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "18", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "35", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "False", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", FractionBox["1", "35"]}], SequenceForm[18, " ", 35, " ", False, " ", Rational[1, 35]], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.400239902945256*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "19", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "37", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "True", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "1"}], SequenceForm[19, " ", 37, " ", True, " ", 1], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.4002399029484777`*^9}], Cell[BoxData[ InterpretationBox[ RowBox[{ "20", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "39", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "False", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", FractionBox["1", "39"]}], SequenceForm[20, " ", 39, " ", False, " ", Rational[1, 39]], Editable->False]], "Print", CellChangeTimes->{3.3992920971528*^9, 3.399292298127898*^9, 3.399292338443379*^9, 3.399336792292798*^9, 3.400239800501446*^9, 3.4002399029517307`*^9}] }, Open ]], Cell[BoxData[ RowBox[{"Null", "&&", "Null"}]], "Output", CellChangeTimes->{3.399292097392147*^9, 3.3992922981619473`*^9, 3.399292338805468*^9, 3.399336792976659*^9, 3.4002398007302227`*^9, 3.400239902954883*^9}] }, Open ]], Cell[BoxData[ RowBox[{"(*", " ", RowBox[{ "This", " ", "is", " ", "stated", " ", "on", " ", "page", " ", "35", " ", "of", " ", "the", " ", "book"}], " ", "*)"}]], "Input", CellChangeTimes->{{3.4002398046025143`*^9, 3.4002398236764517`*^9}}] }, WindowSize->{1440, 744}, WindowMargins->{{-4, Automatic}, {Automatic, 0}}, FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (April 20, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[568, 21, 266, 5, 33, "Input"], Cell[837, 28, 358, 8, 33, "Input"], Cell[1198, 38, 251, 5, 33, "Input"], Cell[1452, 45, 1347, 38, 33, "Input"], Cell[CellGroupData[{ Cell[2824, 87, 215, 4, 33, "Input"], Cell[3042, 93, 257, 4, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3336, 102, 216, 4, 33, "Input"], Cell[3555, 108, 397, 10, 61, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3989, 123, 142, 3, 33, "Input"], Cell[4134, 128, 416, 14, 66, "Output"] }, Open ]], Cell[4565, 145, 412, 10, 33, "Input"], Cell[4980, 157, 533, 18, 33, "Input"], Cell[CellGroupData[{ Cell[5538, 179, 144, 3, 33, "Input"], Cell[5685, 184, 94, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5816, 190, 144, 3, 33, "Input"], Cell[5963, 195, 134, 3, 53, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6134, 203, 146, 3, 33, "Input"], Cell[6283, 208, 315, 10, 54, "Output"] }, Open ]], Cell[6613, 221, 228, 5, 33, "Input"], Cell[6844, 228, 125, 2, 33, "Input"], Cell[6972, 232, 260, 7, 33, "Input"], Cell[CellGroupData[{ Cell[7257, 243, 237, 6, 33, "Input"], Cell[7497, 251, 514, 10, 33, "Output"] }, Open ]], Cell[8026, 264, 291, 6, 33, "Input"], Cell[8320, 272, 126, 2, 33, "Input"], Cell[8449, 276, 458, 13, 33, "Input"], Cell[CellGroupData[{ Cell[8932, 293, 339, 8, 33, "Input"], Cell[9274, 303, 701, 14, 54, "Output"] }, Open ]], Cell[9990, 320, 203, 4, 33, "Input"], Cell[CellGroupData[{ Cell[10218, 328, 842, 21, 33, "Input"], Cell[CellGroupData[{ Cell[11085, 353, 516, 10, 25, "Print"], Cell[11604, 365, 514, 10, 25, "Print"], Cell[12121, 377, 516, 10, 25, "Print"], Cell[12640, 389, 516, 10, 25, "Print"], Cell[13159, 401, 557, 12, 46, "Print"], Cell[13719, 415, 518, 10, 25, "Print"], Cell[14240, 427, 516, 10, 25, "Print"], Cell[14759, 439, 559, 12, 46, "Print"], Cell[15321, 453, 516, 10, 25, "Print"], Cell[15840, 465, 518, 10, 25, "Print"], Cell[16361, 477, 561, 12, 45, "Print"], Cell[16925, 491, 518, 10, 25, "Print"], Cell[17446, 503, 559, 12, 46, "Print"], Cell[18008, 517, 559, 12, 46, "Print"], Cell[18570, 531, 518, 10, 25, "Print"], Cell[19091, 543, 518, 10, 25, "Print"], Cell[19612, 555, 560, 12, 46, "Print"], Cell[20175, 569, 561, 12, 46, "Print"], Cell[20739, 583, 520, 10, 25, "Print"], Cell[21262, 595, 563, 12, 46, "Print"] }, Open ]], Cell[21840, 610, 219, 4, 33, "Output"] }, Open ]], Cell[22074, 617, 251, 5, 33, "Input"] } ] *) (* End of internal cache information *)