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FontSize->24,ExpressionUUID->"c245b79a-23f4-4727-b458-a5a26220374f"], Cell[TextData[{ "Consider the integers mod p. These form a field, so \ \[OpenCurlyDoubleQuote]+\[OpenCurlyDoubleQuote] and \[OpenCurlyDoubleQuote]x\ \[CloseCurlyDoubleQuote] are defined and so are their inverses.\nFor example \ the integers mod 7 are ", Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox["\[DoubleStruckCapitalF]", "7"]}], TraditionalForm]], ExpressionUUID->"80dac367-e9e5-4546-823a-34b4e4f09329"], " = {0, 1, 2, 3, 4, 5, 6, 7} . Apart from 0, each has an inverse\n\n1 x 1 = \ 1, 4 x 2 = 1, 5 x 3 = 1 , 2 x 4 = 1 , 3 x 5 = 1 , 6 x 6 = 1\n\nSo ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"1", "/", "2"}], " ", "=", " ", "4"}], TraditionalForm]], ExpressionUUID->"36130d01-6e2d-46a7-9a17-bceb2994b126"], " , 1/3 = 5 , 1/4 = 2 etc. The lesson is that in ", Cell[BoxData[ FormBox[ SubscriptBox["\[DoubleStruckCapitalF]", "7"], TraditionalForm]], ExpressionUUID->"4cc9ee75-40bb-476b-bcf5-0fee0d7bff23"], " rational numbers m/n (n not divisible by 7) are integers. So it makes \ sense to talk of 1/2 mod 7, for example.\n", Cell[BoxData[ FormBox[ SubscriptBox["\[DoubleStruckCapitalF]", "p"], TraditionalForm]], ExpressionUUID->"7bd0f71b-5c12-4a7d-8069-7339684e7081"], " is the set of integers mod p. This is a field for every prime p.\nIt is \ important that p is prime. The set of integers mod 6 , say, is not a field: \ 2 x 3 = 0, so neither 2 nor 3 have multiplicative inverses.\n\nWe have seen \ that we can use information about residues mod p to find an integer, can we \ use information mod p to find a rational number?" }], "Text", CellChangeTimes->{{3.821294259858843*^9, 3.821294309049903*^9}, { 3.821294357229473*^9, 3.82129445330792*^9}, {3.8212944872640142`*^9, 3.821294588619349*^9}, {3.82129481164497*^9, 3.8212954196805573`*^9}, { 3.821295476024448*^9, 3.821295758770697*^9}}, FontSize->24,ExpressionUUID->"9bdfff51-413c-4fd6-95ef-0158a6f49cee"], Cell[CellGroupData[{ Cell["\<\ A fraction modulo a collection of primes. 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Perhaps there is a rational number of low height that \ reproduces the data?\n\nConsider the problem ", Cell[BoxData[ FormBox[ FractionBox["a", "b"], TraditionalForm]],ExpressionUUID-> "501e1ac4-aae7-4ae1-aada-0d643b372d6d"], " = z mod m and take a and b small compared to m, in the sense that |a|, \ |b| < ", Cell[BoxData[ FormBox[ SqrtBox["m"], TraditionalForm]],ExpressionUUID-> "d9dd1ab9-55d1-42cc-a6d4-7e0af78ce9e2"], ".\nWe have b z = a + c m, for some integer c and \n\n", Cell[BoxData[ FormBox[ RowBox[{"|", " ", RowBox[{ RowBox[{"(", RowBox[{"z", "/", "m"}], ")"}], "-", RowBox[{"(", RowBox[{"c", "/", "b"}], ")"}]}], "|"}], TraditionalForm]], ExpressionUUID->"4865351d-22db-470a-bc71-54d3edfa94e6"], " = |a/(m b)| < ", Cell[BoxData[ FormBox[ RowBox[{"1", "/", SqrtBox["m"]}], TraditionalForm]],ExpressionUUID-> "20e27369-3437-45e4-9581-57b030b719cd"], " , since b \[GreaterEqual] 1 and a \[LessEqual] ", Cell[BoxData[ FormBox[ SqrtBox["m"], TraditionalForm]],ExpressionUUID-> 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FrameMargins->{{0, 4}, {0, 2}}], BaseStyle->"InformationTitleBackground", StripOnInput->False]}, { ItemBox[ PaneBox[ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"PowerMod\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"a\\\", \\\"TI\\\"], \\\",\\\", StyleBox[\\\"b\\\", \ \\\"TI\\\"], \\\",\\\", StyleBox[\\\"m\\\", \\\"TI\\\"]}], \\\"]\\\"}]\\) \ gives \\!\\(\\*RowBox[{SuperscriptBox[StyleBox[\\\"a\\\", \\\"TI\\\"], \ StyleBox[\\\"b\\\", \\\"TI\\\"]], \\\" \\\", \\\"mod\\\", \\\" \\\", \ StyleBox[\\\"m\\\", \\\"TI\\\"]}]\\). \\n\\!\\(\\*RowBox[{\\\"PowerMod\\\", \ \\\"[\\\", RowBox[{StyleBox[\\\"a\\\", \\\"TI\\\"], \\\",\\\", \ RowBox[{\\\"-\\\", \\\"1\\\"}], \\\",\\\", StyleBox[\\\"m\\\", \\\"TI\\\"]}], \ \\\"]\\\"}]\\) finds the modular inverse of \\!\\(\\*StyleBox[\\\"a\\\", \ \\\"TI\\\"]\\) modulo \\!\\(\\*StyleBox[\\\"m\\\", \ \\\"TI\\\"]\\).\\n\\!\\(\\*RowBox[{\\\"PowerMod\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"a\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"1\\\", \ \\\"/\\\", StyleBox[\\\"r\\\", \\\"TI\\\"]}], \\\",\\\", StyleBox[\\\"m\\\", \ \\\"TI\\\"]}], \\\"]\\\"}]\\) finds a modular \\!\\(\\*StyleBox[\\\"r\\\", \\\ \"TI\\\"]\\)\\!\\(\\*SuperscriptBox[\\\"\[Null]\\\", \\\"th\\\"]\\) root of \ \\!\\(\\*StyleBox[\\\"a\\\", \\\"TI\\\"]\\).\"\>", "InformationUsageText", StripOnInput->False, LineSpacing->{1.5, 1.5, 3.}], FrameMargins->{{10, 10}, {8, 10}}], BaseStyle->"InformationUsageSubtitleBackground", StripOnInput->False], ItemBox["\<\"\"\>", BaseStyle->"InformationUsageSubtitleBackground", StripOnInput->False]}, { PaneBox[GridBox[{ { DynamicModuleBox[{System`InformationDump`open$$ = { False, False, False, False, False, False}}, StyleBox[GridBox[{ { TagBox[ TooltipBox[ StyleBox["\<\" Documentation\"\>", "InformationRowLabel", StripOnInput->False], "\"Documentation\"", TooltipStyle->"TextStyling"], Annotation[#, "Documentation", "Tooltip"]& ], TemplateBox[{ TemplateBox[{ "\"Local \[RightGuillemet]\"", "paclet:ref/PowerMod", "paclet:ref/PowerMod", "Link", { RGBColor[0.9686274509803922, 0.4666666666666667, 0.]}, BaseStyle -> { RGBColor[0.0784313725490196, 0.1568627450980392, 0.6]}}, "HyperlinkTemplate"], "\" \"", StyleBox[ "\"|\"", "InformationRowLabel", StripOnInput -> False], "\" \"", TemplateBox[{"\"Web \[RightGuillemet]\"", { URL[ "http://reference.wolfram.com/language/ref/PowerMod.html"]\ , None}, "http://reference.wolfram.com/language/ref/PowerMod.html", "Hyperlink", { RGBColor[0.9686274509803922, 0.4666666666666667, 0.]}, BaseStyle -> { RGBColor[0.0784313725490196, 0.1568627450980392, 0.6]}}, "HyperlinkTemplate"]}, "RowDefault"]}, { TagBox[ TooltipBox[ StyleBox["\<\" Attributes\"\>", "InformationRowLabel", StripOnInput->False], "\"Attributes\"", TooltipStyle->"TextStyling"], Annotation[#, "Attributes", "Tooltip"]& ], RowBox[{"{", RowBox[{"Listable", ",", "Protected"}], "}"}]}, { TagBox[ TooltipBox[ StyleBox["\<\" Full Name\"\>", "InformationRowLabel", StripOnInput->False], "\"FullName\"", TooltipStyle->"TextStyling"], Annotation[#, "FullName", "Tooltip"]& ], "\<\"System`PowerMod\"\>"} }, AutoDelete->False, GridBoxAlignment->{"Columns" -> {Right, Left}}, GridBoxDividers->None, GridBoxItemSize->{"Columns" -> {Automatic, Automatic}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.8]}, Offset[0.2]}}], "DialogStyle", StripOnInput->False], DynamicModuleValues:>{}]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxDividers->{"Columns" -> {{False}}, "Rows" -> {{False}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[3.6]}, Offset[0.2]}}], FrameMargins->{{6, 6}, {6, 3}}], ""}, { ItemBox[ TagBox[ ButtonBox[ PaneSelectorBox[{False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "UpPointerOpener"]], True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "UpPointerOpenerHot"]]}, Dynamic[ System`InformationDump`mouseOver$$]], Alignment->Left, Appearance->{"Default" -> None}, ButtonFunction:>FEPrivate`Set[ System`InformationDump`open$$, False], Evaluator->Automatic, FrameMargins->{{9, 0}, {0, 0}}, ImageMargins->0, ImageSize->Full, Method->"Preemptive"], EventHandlerTag[{ "MouseEntered" :> FEPrivate`Set[System`InformationDump`mouseOver$$, True], "MouseExited" :> FEPrivate`Set[System`InformationDump`mouseOver$$, False], Method -> "Preemptive", PassEventsDown -> Automatic, PassEventsUp -> True}]], BaseStyle->"InformationTitleBackground", StripOnInput->False], "\[SpanFromLeft]"} }, AutoDelete->False, FrameStyle->Directive[ GrayLevel[0.8], Thickness[Tiny]], GridBoxAlignment->{"Columns" -> {Left, Right}, "Rows" -> {{Center}}}, GridBoxDividers->{ "Columns" -> {{None}}, "Rows" -> {False, {True}, False}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Grid"], False-> TagBox[GridBox[{ { ItemBox[ PaneBox[ StyleBox["\<\" Symbol\"\>", "InformationTitleText", StripOnInput->False], FrameMargins->{{4, 0}, {-1, 1}}], BaseStyle->"InformationTitleBackground", StripOnInput->False], ItemBox[ PaneBox[ TooltipBox[ ButtonBox[ PaneSelectorBox[{False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "InformationHelpIcon"], ImageSizeCache->{20., {6., 14.}}], True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "InformationHelpIconHot"], ImageSizeCache->{20., {6., 14.}}]}, Dynamic[ CurrentValue["MouseOver"]]], Appearance->None, BaseStyle->"Link", ButtonData->"paclet:ref/PowerMod", ButtonNote->"paclet:ref/PowerMod"], "\"paclet:ref/PowerMod\""], FrameMargins->{{0, 4}, {0, 2}}], BaseStyle->"InformationTitleBackground", StripOnInput->False]}, { ItemBox[ PaneBox[ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"PowerMod\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"a\\\", \\\"TI\\\"], \\\",\\\", StyleBox[\\\"b\\\", \ \\\"TI\\\"], \\\",\\\", StyleBox[\\\"m\\\", \\\"TI\\\"]}], \\\"]\\\"}]\\) \ gives \\!\\(\\*RowBox[{SuperscriptBox[StyleBox[\\\"a\\\", \\\"TI\\\"], \ StyleBox[\\\"b\\\", \\\"TI\\\"]], \\\" \\\", \\\"mod\\\", \\\" \\\", \ StyleBox[\\\"m\\\", \\\"TI\\\"]}]\\). \\n\\!\\(\\*RowBox[{\\\"PowerMod\\\", \ \\\"[\\\", RowBox[{StyleBox[\\\"a\\\", \\\"TI\\\"], \\\",\\\", \ RowBox[{\\\"-\\\", \\\"1\\\"}], \\\",\\\", StyleBox[\\\"m\\\", \\\"TI\\\"]}], \ \\\"]\\\"}]\\) finds the modular inverse of \\!\\(\\*StyleBox[\\\"a\\\", \ \\\"TI\\\"]\\) modulo \\!\\(\\*StyleBox[\\\"m\\\", \ \\\"TI\\\"]\\).\\n\\!\\(\\*RowBox[{\\\"PowerMod\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"a\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"1\\\", \ \\\"/\\\", StyleBox[\\\"r\\\", \\\"TI\\\"]}], \\\",\\\", StyleBox[\\\"m\\\", \ \\\"TI\\\"]}], \\\"]\\\"}]\\) finds a modular \\!\\(\\*StyleBox[\\\"r\\\", \\\ \"TI\\\"]\\)\\!\\(\\*SuperscriptBox[\\\"\[Null]\\\", \\\"th\\\"]\\) root of \ \\!\\(\\*StyleBox[\\\"a\\\", \\\"TI\\\"]\\).\"\>", "InformationUsageText", StripOnInput->False, LineSpacing->{1.5, 1.5, 3.}], FrameMargins->{{10, 10}, {8, 10}}], BaseStyle->"InformationUsageSubtitleBackground", StripOnInput->False], ItemBox["\<\"\"\>", BaseStyle->"InformationUsageSubtitleBackground", StripOnInput->False]}, { ItemBox[ TagBox[ ButtonBox[ PaneSelectorBox[{False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "DownPointerOpener"], ImageSizeCache->{15., {2., 13.}}], True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "DownPointerOpenerHot"], ImageSizeCache->{15., {2., 13.}}]}, Dynamic[ System`InformationDump`mouseOver$$]], Alignment->Left, Appearance->{"Default" -> None}, ButtonFunction:>FEPrivate`Set[ System`InformationDump`open$$, True], Evaluator->Automatic, FrameMargins->{{9, 0}, {0, 0}}, ImageMargins->0, ImageSize->Full, Method->"Preemptive"], EventHandlerTag[{ "MouseEntered" :> FEPrivate`Set[System`InformationDump`mouseOver$$, True], "MouseExited" :> FEPrivate`Set[System`InformationDump`mouseOver$$, False], Method -> "Preemptive", PassEventsDown -> Automatic, PassEventsUp -> True}]], BaseStyle->"InformationTitleBackground", StripOnInput->False], "\[SpanFromLeft]"} }, AutoDelete->False, FrameStyle->Directive[ GrayLevel[0.8], Thickness[Tiny]], GridBoxAlignment->{"Columns" -> {Left, Right}, "Rows" -> {{Center}}}, GridBoxDividers->{ "Columns" -> {{None}}, "Rows" -> {False, {True}, False}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Grid"]}, Dynamic[System`InformationDump`open$$], BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], DynamicModuleValues:>{}], BaseStyle->"InformationGridFrame", StripOnInput->False], "InformationGridPlain", StripOnInput->False], InformationData[<| "ObjectType" -> "Symbol", "Usage" -> "\!\(\*RowBox[{\"PowerMod\", \"[\", RowBox[{StyleBox[\"a\", \"TI\"], \ \",\", StyleBox[\"b\", \"TI\"], \",\", StyleBox[\"m\", \"TI\"]}], \"]\"}]\) \ gives \!\(\*RowBox[{SuperscriptBox[StyleBox[\"a\", \"TI\"], StyleBox[\"b\", \ \"TI\"]], \" \", \"mod\", \" \", StyleBox[\"m\", \"TI\"]}]\). \n\ \!\(\*RowBox[{\"PowerMod\", \"[\", RowBox[{StyleBox[\"a\", \"TI\"], \",\", \ RowBox[{\"-\", \"1\"}], \",\", StyleBox[\"m\", \"TI\"]}], \"]\"}]\) finds the \ modular inverse of \!\(\*StyleBox[\"a\", \"TI\"]\) modulo \ \!\(\*StyleBox[\"m\", \"TI\"]\).\n\!\(\*RowBox[{\"PowerMod\", \"[\", \ RowBox[{StyleBox[\"a\", \"TI\"], \",\", RowBox[{\"1\", \"/\", StyleBox[\"r\", \ \"TI\"]}], \",\", StyleBox[\"m\", \"TI\"]}], \"]\"}]\) finds a modular \ \!\(\*StyleBox[\"r\", \"TI\"]\)\!\(\*SuperscriptBox[\"\[Null]\", \"th\"]\) \ root of \!\(\*StyleBox[\"a\", \"TI\"]\).", "Documentation" -> <| "Local" -> "paclet:ref/PowerMod", "Web" -> "http://reference.wolfram.com/language/ref/PowerMod.html"|>, "OwnValues" -> None, "UpValues" -> None, "DownValues" -> None, "SubValues" -> None, "DefaultValues" -> None, "NValues" -> None, "FormatValues" -> None, "Options" -> None, "Attributes" -> {Listable, Protected}, "FullName" -> "System`PowerMod"|>, False]]], "Output", 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