(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 1810609, 34270] NotebookOptionsPosition[ 1780335, 33334] NotebookOutlinePosition[ 1781203, 33362] CellTagsIndexPosition[ 1781160, 33359] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Outstanding Question(s)", "Subtitle", CellChangeTimes->{{3.481656147338875*^9, 3.481656182917*^9}, { 3.561917487784*^9, 3.561917489552*^9}, {3.561927852948*^9, 3.5619278533*^9}} ], Cell["By Dustin Darcy, Aug. 15th 2010, updated Dec. 28th 2024", "Text", CellChangeTimes->{{3.5619276945550003`*^9, 3.561927701134*^9}, { 3.561927813744*^9, 3.561927842389*^9}, {3.6541289668019457`*^9, 3.654128978401419*^9}, {3.657138367324494*^9, 3.6571383703206654`*^9}, { 3.7805871542365623`*^9, 3.7805871626847696`*^9}, {3.944402106301813*^9, 3.944402112723547*^9}}], Cell[TextData[{ StyleBox["The big question this notebook seeks to answer is whether there is \ a recurrence relation between the summed rows of triangular numbers of the \ form ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"tri", "(", "n", ")"}], "=", RowBox[{"(", FractionBox[ RowBox[{ SuperscriptBox["n", "2"], "+", "n"}], "2"], ")"}]}], TraditionalForm]], FontWeight->"Plain"], StyleBox[" where", FontWeight->"Plain"], " ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["n", "2"], "=", "36"}], TraditionalForm]], FontWeight->"Plain"], StyleBox[", ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["n", "1"], "=", "4"}], TraditionalForm]], FontWeight->"Plain"], StyleBox[", and primes to the ", FontWeight->"Plain"], StyleBox["p", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["-th power ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["p", "2"], "=", " ", "2"}]}], TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "1"], "=", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], ")", StyleBox[". \n\nThe initial observation that kicked off the investigation is \ a curious link between ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["n", FontWeight->"Plain"], "1"], StyleBox["=", FontWeight->"Plain"], StyleBox["4", FontWeight->"Plain"]}], TraditionalForm]]], StyleBox[" ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"meaning", " ", "triangular", " ", "number", " ", RowBox[{"tri", "(", "4", ")"}]}], "=", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["4", "2"], "+", "4"}], "2"], "=", "10"}]}], ")"}], TraditionalForm]], FontWeight->"Plain"], " ", StyleBox["and ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["n", "2"], "=", "36"}], TraditionalForm]], FormatType->"TraditionalForm"], StyleBox[" ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"or", " ", RowBox[{"tri", "(", "36", ")"}]}], "=", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["36", "2"], "+", "36"}], "2"], "=", "666"}]}], ")"}], TraditionalForm]], FontWeight->"Plain"], ".", StyleBox["\n\nThe link is a bit easier to notice if we think of ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"tri", "(", "4", ")"}], FontWeight->"Plain"], TraditionalForm]]], " ", StyleBox["as a summation. It\[CloseCurlyQuote]s trivial to see that ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"tri", "(", "4", ")"}], FontWeight->"Plain"], TraditionalForm]]], " ", StyleBox["is the same as", FontWeight->"Plain"], " ", Cell[BoxData[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"x", "=", "1"}], "4"], "x"}], "=", "10"}]], FontWeight->"Plain"], StyleBox[". Geometrically this creates a 4 row triangular number sequence \ composed of 10 digits (see the diagram below) whose base-10 rows sum to: ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"0123", "+", "456", "+", "78", "+", "9"}], "=", "666"}], TraditionalForm]], FontWeight->"Plain"], StyleBox[". Since ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["n", "2"], "=", "36"}], TraditionalForm]], FontWeight->"Plain"], StyleBox[", ", FontWeight->"Plain"], Cell[BoxData[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"x", "=", "1"}], "36"], "x"}], "=", "666"}]], FontWeight->"Plain"], StyleBox[". This suggested that ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["n", "2"], "=", "36"}], TraditionalForm]]], StyleBox[" might be a part of another \[OpenCurlyDoubleQuote]tri-row-sum \ sequence\[CloseCurlyDoubleQuote] as derived from the footprint of ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["n", "1"], "=", "4"}], TraditionalForm]], FontWeight->"Plain"], " through some as of yet unknown recursive series (presumably in base-666).\n\ \nThe second observation which made this connection curious enough to warrant \ deeper investigation is that 10 is the sum of the first 3 primes to the first \ power (i.e. ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["2", "1"], "+", SuperscriptBox["3", "1"], "+", SuperscriptBox["5", "1"]}], "=", "10"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "1"], "=", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], ") and 666 is the sum of the first seven squared 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", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"tri", "(", "3", ")"}], "=", "6"}], TraditionalForm]]], ") is added to the sum of the ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " of the next (ie. ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"n", "=", "4"}], ";", RowBox[{"tri", "(", "4", ")"}]}], TraditionalForm]]], ") to arrive at the larger ", Cell[BoxData[ FormBox[ RowBox[{"tri", "(", "n", ")"}], TraditionalForm]]], " value (ex. ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"tri", "(", "3", ")"}], "+", "4"}], "=", RowBox[{ RowBox[{"tri", "(", "4", ")"}], "=", "10"}]}], TraditionalForm]]], "). Proof:" }], "Text", CellChangeTimes->{{3.6541337117939835`*^9, 3.654133722483841*^9}, { 3.6541337757085996`*^9, 3.6541337943179626`*^9}, {3.6541338461100397`*^9, 3.6541338943056593`*^9}, {3.654134188894068*^9, 3.6541342148918686`*^9}, { 3.6541369818442273`*^9, 3.654137037881843*^9}, {3.6541371733945513`*^9, 3.6541371811455355`*^9}, {3.654147853024192*^9, 3.6541479220579576`*^9}, { 3.6541479571939197`*^9, 3.6541479824481263`*^9}, {3.6541480327500143`*^9, 3.6541480485585213`*^9}, {3.654148096884158*^9, 3.6541481233125143`*^9}, { 3.6541487340820723`*^9, 3.6541487466481676`*^9}, {3.6571861211842594`*^9, 3.657186145866671*^9}, {3.657192288313035*^9, 3.657192289479183*^9}, { 3.657420972236487*^9, 3.657420977786192*^9}, {3.6574804222851844`*^9, 3.657480442560259*^9}, {3.657480476736599*^9, 3.657480477750228*^9}, { 3.6574943907184505`*^9, 3.657494403590085*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"tri", "[", "x", "]"}], "+", RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}]}], "\[Equal]", RowBox[{"tri", "[", RowBox[{"x", "+", "1"}], "]"}]}], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.6541340322246733`*^9, 3.65413415828168*^9}, { 3.6574202425033226`*^9, 3.657420243687973*^9}}], Cell[BoxData["True"], "Output", CellChangeTimes->{3.658015287132229*^9, 3.749088480885354*^9, 3.7817825809325695`*^9, 3.7832858282763195`*^9, 3.7950382492987127`*^9, 3.7950387203114166`*^9, 3.795045281906385*^9, 3.7950475385330267`*^9, 3.795047629851453*^9, 3.795047707317074*^9}] }, Open ]], Cell[TextData[{ "This behavior may factor into the special relationship between ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", "4"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", "36"}], TraditionalForm]]], " in some way. " }], "Text", CellChangeTimes->{{3.6541337117939835`*^9, 3.654133722483841*^9}, { 3.6541337757085996`*^9, 3.6541337943179626`*^9}, {3.6541338461100397`*^9, 3.6541338943056593`*^9}, {3.654134188894068*^9, 3.6541342148918686`*^9}, { 3.6541369818442273`*^9, 3.654137037881843*^9}, {3.6541371733945513`*^9, 3.6541371811455355`*^9}, {3.654147853024192*^9, 3.6541479220579576`*^9}, { 3.6541479571939197`*^9, 3.6541479824481263`*^9}, {3.6541480327500143`*^9, 3.6541480485585213`*^9}, {3.654148096884158*^9, 3.654148130803965*^9}}], Cell[TextData[{ "To calculate the sum of the rows of a triangular number (ex. ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"tri", "(", "3", ")"}], " ", "=", "6"}], ";", " ", RowBox[{ RowBox[{"012", " ", "+", " ", "34", " ", "+", " ", "5"}], " ", "=", " ", "51"}]}], TraditionalForm]]], ") we\[CloseCurlyQuote]ll need to know how many rows to count. The ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " in ", Cell[BoxData[ FormBox[ RowBox[{"tri", "(", "n", ")"}], TraditionalForm]]], " is equal to the total number of rows. \n\nSince ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " could be a bit more descriptive. We\[CloseCurlyQuote]ll encapsulate the ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " in a function called max increment (", Cell[BoxData[ FormBox[ RowBox[{"mi", "(", ")"}], TraditionalForm]]], " for short) to indicate the maximum upper bound of the summation (", Cell[BoxData[ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "0"}], RowBox[{"mi", "(", ")"}]], "..."}], TraditionalForm]]], "). This will make it easier later to see why and how we are using ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " and to explicitly show the total number of times the summation will have \ to increment. \n\nThis convention will be used commonly throughout the \ notebook even though the function itself is very simple (", StyleBox["mi()", FontSlant->"Italic"], " is a glorified subtraction: ", Cell[BoxData[ FormBox[ RowBox[{"n", "-", "1"}], TraditionalForm]]], "). The algorithm starts at 0 so we need to subtract 1. \n\nWe define \ function ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"mi", "(", ")"}], FontSlant->"Italic"], TraditionalForm]]], " to take the ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " of ", Cell[BoxData[ FormBox[ RowBox[{"tri", "(", "n", ")"}], TraditionalForm]]], " and second parameter to offset the value to remove 1. 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In base-666 we need 666 unique numerals to \ represent the values for each location in the triangle. The bottom row will \ have 36 unique characters in it. So the bottom row might look like this:\n\n\ 0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ \n\nHow to sum this then with the row \ above it? \n\nThinking about how triangular number ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"tri", "(", "4", ")"}], "=", "10"}], TraditionalForm]]], " is structured in base-10. 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Or simply 0+", StyleBox["4", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " = ", StyleBox["4", FontColor->RGBColor[1, 0.5, 0]], "; ", StyleBox["4", FontColor->RGBColor[1, 0.5, 0]], "+", StyleBox["3", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " = ", StyleBox["7", FontColor->RGBColor[1, 0.5, 0]], "; ", StyleBox["7", FontColor->RGBColor[1, 0.5, 0]], "+", StyleBox["2", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " = ", StyleBox["9", FontColor->RGBColor[1, 0.5, 0]], "; last ", StyleBox["9", FontColor->RGBColor[1, 0.5, 0]], "+", StyleBox["1", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " = 10 (done). Since we know the number-base (", Cell[BoxData[ FormBox[ RowBox[{"b", "=", "10"}], TraditionalForm]]], ") we know the 4 in advance because we can solve for ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " in the triangular equation (", Cell[BoxData[ FormBox[ RowBox[{"tri", "[", "n", "]"}], TraditionalForm]]], ") using the quadratic equation (i.e. ", Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["n", "2"], "+", "n"}], "2"], "=", RowBox[{ RowBox[{"b", "\[Rule]", RowBox[{ FractionBox[ RowBox[{ RowBox[{"-", RowBox[{"(", FractionBox["1", "2"], ")"}]}], "+", SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", FractionBox["1", "2"], ")"}], "2"], "-", " ", RowBox[{"4", RowBox[{"(", FractionBox["1", "2"], ")"}], RowBox[{"(", RowBox[{"-", "b"}], ")"}]}]}]]}], RowBox[{"2", RowBox[{"(", FractionBox["1", "2"], ")"}]}]], "\[Equal]", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"-", "1"}], "2"], "+", SqrtBox[ RowBox[{ FractionBox["1", "4"], "+", RowBox[{"2", "b"}]}]]}], ")"}], "\[Equal]", RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], "+", SqrtBox[ RowBox[{ FractionBox["1", "4"], "+", RowBox[{ FractionBox["8", "4"], "b"}]}]]}], ")"}], "==", RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], "+", RowBox[{ FractionBox["1", "2"], SqrtBox[ RowBox[{"1", "+", RowBox[{"8", "b"}]}]]}]}], ")"}]}]}], "="}]}]], "Input", CellChangeTimes->{{3.780590019893527*^9, 3.780590020562498*^9}}], Cell[BoxData[ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SqrtBox[ RowBox[{"1", "+", RowBox[{"8", "b"}]}]]}], ")"}]}]], CellChangeTimes->{{3.588172617802457*^9, 3.588172632784314*^9}, { 3.588172666214226*^9, 3.5881726970499897`*^9}}], ". 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Once we know the second row from the bottom of a 10 element \ triangle starts at number 4 it becomes very easy to count forward to get 4,5, \ and 6. The second row only has three elements in it because ", Cell[BoxData[ FormBox[ RowBox[{"tri", "(", "4", ")"}], TraditionalForm]]], " implies the base of the triangle is 4 numbers wide. Thus the second to \ bottom row has to be 3 numbers wide.\n\nThe easiest way to generate the ", StyleBox["4", FontWeight->"Bold"], " (3rd row 1st number), ", StyleBox["7", FontWeight->"Bold"], " (2nd row 1st number), ", StyleBox["9", FontWeight->"Bold"], " (1st row & only number) sequence is to figure out the length or \ cardinality of the set of numbers of each of the preceding rows and then sum \ them up to calculate the first number of the desired row. \n\nFor example, if \ we wanted the starting number for the 2nd row (", Cell[BoxData[ FormBox[ RowBox[{"r", "=", "2"}], TraditionalForm]]], ") in a 10 element triangular number (or to think of it another way the \ total number of digits we have to iterate over to get to the first number of \ the 2nd row) represented by tri[n] where n=4 , then counting from the bottom \ of the triangle to the top we have: (4th row) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1", ",", "2", ",", "3"}], "}"}], "=", "4"}], TraditionalForm]]], " elements; (3rd row) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"{", RowBox[{"4", ",", "5", ",", "6"}], "}"}], " ", "=", " ", RowBox[{"3", " ", "elements"}]}], TraditionalForm]]], "; gives ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"4", "+", "3"}], "=", "7"}], TraditionalForm]]], ". \n\nLets imagine one triangle is full of As and the bigger triangle is \ full of As and Bs.\n\n B A row #1\n BB \ AA row #2\n ABB BBB row #3\nAABB \ or BBBB row #4\n\nSince all triangular numbers where ", Cell[BoxData[ FormBox[ RowBox[{"tri", "(", "n", ")"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{"n", ">", "1"}], TraditionalForm]]], " can be broken into smaller triangular numbers, then one way to do this \ would be to subtract the top of the triangle (represented by the As) which \ includes the numbers we haven\[CloseCurlyQuote]t counted to yet ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"tri", "[", RowBox[{ RowBox[{"(", RowBox[{"n", "-", "x"}], ")"}], "-", "1"}], 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We just reverse the order.\ \>", "Text", CellChangeTimes->{{3.657156328235664*^9, 3.657156351609632*^9}, { 3.657157078103385*^9, 3.657157111913678*^9}, {3.657161171401668*^9, 3.657161190957651*^9}, {3.657169247379687*^9, 3.6571692734179935`*^9}, 3.657184497026363*^9, {3.65741979927804*^9, 3.65741982321408*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"tri", "[", "4", "]"}], "-", RowBox[{"tri", "[", RowBox[{"4", "-", "x", "+", "1"}], "]"}]}], ",", " ", RowBox[{"{", RowBox[{"x", ",", "1", ",", "4"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.657156377122872*^9, 3.657156426109092*^9}, { 3.6571565575707855`*^9, 3.6571565761386433`*^9}, {3.6571571469551277`*^9, 3.6571571517827415`*^9}, {3.6574202860993586`*^9, 3.657420286827451*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"0", ",", "4", ",", "7", ",", "9"}], "}"}]], "Output", CellChangeTimes->{{3.6571563779644785`*^9, 3.6571564263606243`*^9}, { 3.657156558770938*^9, 3.657156576708716*^9}, {3.657157147492196*^9, 3.657157152019271*^9}, 3.65716396248359*^9, 3.657164168609265*^9, 3.657420779895563*^9, 3.6574214277313275`*^9, 3.657422626121504*^9, 3.6574227726356087`*^9, 3.6574235425423746`*^9, 3.657424355771141*^9, 3.657494273974126*^9, 3.657496590572296*^9, 3.6574969554311275`*^9, 3.657497005181945*^9, 3.6574972297829657`*^9, 3.6575210204254937`*^9, 3.6575532392052593`*^9, 3.6575533011171207`*^9, 3.6575812707953186`*^9, 3.6575851313610487`*^9, 3.657590465681921*^9, 3.65759291252063*^9, 3.65801438252343*^9, 3.749088481467353*^9, 3.7817825811948442`*^9, 3.7832858288066015`*^9, 3.7950382497759843`*^9, 3.7950387207681932`*^9, 3.795045282381115*^9, 3.795047539014738*^9, 3.7950476303062353`*^9, 3.795047707731962*^9}] }, Open ]], Cell["\<\ To get an arbitrary row length and sum them manually we would do: \ \>", "Text", CellChangeTimes->{{3.657156328235664*^9, 3.657156351609632*^9}, { 3.657156629439912*^9, 3.6571566464855766`*^9}, {3.6571579804419675`*^9, 3.65715799274753*^9}, {3.6571581104814806`*^9, 3.657158144789837*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"x", "=", "0"}], RowBox[{"n", "-", "z"}]], RowBox[{"(", RowBox[{ RowBox[{"tri", "[", RowBox[{"n", "-", "x"}], "]"}], "-", RowBox[{"tri", "[", RowBox[{ RowBox[{"(", RowBox[{"n", "-", "x"}], ")"}], "-", "1"}], "]"}]}], ")"}]}]], "Input",\ CellChangeTimes->{{3.490220125285*^9, 3.490220362085*^9}, { 3.4902204097869997`*^9, 3.4902204190179996`*^9}, {3.490220505586*^9, 3.490220523308*^9}, {3.589410476077479*^9, 3.5894104768425226`*^9}, { 3.657420287785573*^9, 3.65742028855317*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"n", "+", SuperscriptBox["n", "2"]}], ")"}]}], "+", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"1", "-", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "z"}], ")"}], "2"], "-", "z"}], ")"}]}]}]], "Output", CellChangeTimes->{3.658015385349367*^9, 3.749088481517351*^9, 3.781782581374362*^9, 3.783285828849681*^9, 3.7950382498104076`*^9, 3.7950387207981453`*^9, 3.7950452824120317`*^9, 3.795047539046653*^9, 3.795047630336156*^9, 3.795047707763875*^9}] }, Open ]], Cell[TextData[{ "As established earlier this can be simplified because the total number of \ elements in the immediate preceding smaller triangle (or ", Cell[BoxData[ FormBox[ RowBox[{"tri", "(", RowBox[{"n", "-", "1"}], ")"}], TraditionalForm]]], ") is equal to ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"tri", "(", "n", ")"}], "-", "n"}], TraditionalForm]]], ". So we can just cancel out the n\[CloseCurlyQuote]s (or more correctly the \ ", Cell[BoxData[ FormBox[ RowBox[{"n", "-", "x"}], TraditionalForm]]], "\[CloseCurlyQuote]s) of the two triangles.\n\nTo make \ what\[CloseCurlyQuote]s happening easier to see, just think of a ", Cell[BoxData[ FormBox[ SubscriptBox["base", "infinity"], TraditionalForm]]], " number line going on indefinitely. If we write out the counting numbers \ and put a pipe (|) before each triangular number we can see the pattern more \ clearly:\n\n|0|12|345|6789|ABCDE|FGHIJK|...\n |\n \:2514 Starts at 1 \ \[LongDash] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"tri", "[", "1", "]"}], "=", "1"}], ",", RowBox[{ RowBox[{"tri", "[", "2", "]"}], "=", "3"}], ",", RowBox[{ RowBox[{"tri", "[", "3", "]"}], "=", "6"}], ",", RowBox[{ RowBox[{"tri", "[", "4", "]"}], "=", RowBox[{"10", " ", "or", " ", SubscriptBox[ StyleBox["A", FontSlant->"Plain"], "\[Infinity]"]}]}], ",", RowBox[{ RowBox[{"tri", "[", "5", "]"}], "=", RowBox[{"15", " ", "or", " ", SubscriptBox[ StyleBox["F", FontSlant->"Plain"], "\[Infinity]"]}]}]}], TraditionalForm]]], ",...\n \nThe whole number line is a triangular number with an infinite \ sized base. \n\nWorking with triangular number ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"tri", "(", "4", ")"}], "=", "10"}], TraditionalForm]]], " causes us to have four breaks.\n\n0123|456|78|9|\n\nAnother way to do this \ rather than subtracting {tri(1), tri(2), tri(3), tri(4)} from ", Cell[BoxData[ FormBox[ RowBox[{"tri", "(", "n", ")"}], TraditionalForm]]], " is to use the summation form ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"tri", "(", "n", ")"}], "=", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"x", "=", "1"}], "n"], "x"}]}], TraditionalForm]]], ". So instead of subtracting the top of the triangle from the larger \ triangle (", Cell[BoxData[ FormBox["n", TraditionalForm]]], ") we just count and sum all the triangular numbers up to the row (", Cell[BoxData[ FormBox["z", TraditionalForm]]], ") that we are currently at. 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This allows us to specify the correct row without having to manually \ offset by adding one to the z parameter. If we don\[CloseCurlyQuote]t add ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"z", "+", "1"}], ")"}], TraditionalForm]]], " to ", Cell[BoxData[ FormBox[ RowBox[{"leftedge", "(", ")"}], TraditionalForm]]], " then we would need to do ", Cell[BoxData[ FormBox[ RowBox[{"leftedge", "(", RowBox[{"n", ",", RowBox[{"z", "+", "1"}]}], ")"}], TraditionalForm]]], " on each call.\n\n", Cell[BoxData[ FormBox["n", TraditionalForm]]], " = quadratic equation of the total number of elements in the triangle (ex. \ tri(4) = 10, so n = 4)\n", Cell[BoxData[ FormBox["z", TraditionalForm]]], " = specific arbitrary row number\n", Cell[BoxData[ FormBox["f", TraditionalForm]]], " = optional base form" }], "Text", CellChangeTimes->{{3.654146661137841*^9, 3.6541466931959124`*^9}, { 3.6541470756299753`*^9, 3.654147144564229*^9}, {3.6541498293426523`*^9, 3.6541498474459515`*^9}, {3.6571845569227886`*^9, 3.6571845918727875`*^9}, {3.657187812426993*^9, 3.6571878151771507`*^9}, { 3.657188689777175*^9, 3.6571887084652433`*^9}, {3.657228738085513*^9, 3.6572287386135435`*^9}, {3.6574213730143795`*^9, 3.657421375791732*^9}, { 3.6574257030002174`*^9, 3.657425704629925*^9}, {3.657494899526561*^9, 3.657494949562414*^9}, {3.657495077442153*^9, 3.657495133836314*^9}, 3.7807333840429783`*^9}], Cell[BoxData[ RowBox[{ RowBox[{"leftedge", "[", RowBox[{"n_", ",", "z_"}], "]"}], ":=", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"1", "+", "n", "-", RowBox[{"(", RowBox[{"z", "+", "1"}], ")"}]}], ")"}], " ", RowBox[{"(", RowBox[{"n", "+", RowBox[{"(", RowBox[{"z", "+", "1"}], ")"}]}], ")"}]}]}]], "Input", InitializationCell->True, CellChangeTimes->{{3.490220816399*^9, 3.490220843585*^9}, 3.5894099857684345`*^9, 3.589410078210722*^9, {3.589410204042919*^9, 3.589410222344966*^9}, {3.589410284096498*^9, 3.5894102923189683`*^9}, { 3.6541466401596775`*^9, 3.654146649171322*^9}, {3.6541467455015545`*^9, 3.654146776301465*^9}, 3.657421337414859*^9, {3.6574883248881874`*^9, 3.657488336687686*^9}, {3.657494957144377*^9, 3.6574949618299723`*^9}, { 3.657495141876335*^9, 3.657495145875843*^9}, {3.657496853350665*^9, 3.6574968609896345`*^9}, {3.657496978330535*^9, 3.6574969860880203`*^9}, 3.6575209100474772`*^9, {3.6575860648510866`*^9, 3.657586073118636*^9}}], Cell[TextData[{ "Starting from ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"z", "=", "1"}], FontWeight->"Plain", FontSlant->"Plain"], TraditionalForm]], FontWeight->"Bold"], " specifies the row at the top of the triangle. ", "A good description for this function would be \ \[OpenCurlyDoubleQuote]outputs numbers on the left edge of the triangle.\ \[CloseCurlyDoubleQuote]" }], "Text", CellChangeTimes->{{3.6541455797000165`*^9, 3.654145689449953*^9}, { 3.6541463454762573`*^9, 3.6541463571017337`*^9}, 3.6541466540539417`*^9, { 3.654147163033074*^9, 3.6541471790236044`*^9}, {3.654149856469597*^9, 3.6541498644221067`*^9}, {3.6571878415226574`*^9, 3.657187844802845*^9}, { 3.6574867792524166`*^9, 3.65748679424082*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"leftedge", "[", RowBox[{"4", ",", "x"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "1", ",", " ", "4"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.4902208296029997`*^9, 3.4902208785039997`*^9}, 3.490221040314*^9, {3.5894100241796317`*^9, 3.589410026052739*^9}, { 3.5894100842970705`*^9, 3.5894100876092596`*^9}, {3.5894102168886538`*^9, 3.589410225753161*^9}, {3.654146807217391*^9, 3.6541468084590487`*^9}, 3.6571630002374*^9, 3.6574213411638346`*^9, {3.657496993127414*^9, 3.6574970007413807`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"9", ",", "7", ",", "4", ",", "0"}], "}"}]], "Output", CellChangeTimes->{{3.490220830809*^9, 3.490220879091*^9}, 3.490221040939*^9, 3.490540749949*^9, 3.4905408654110003`*^9, 3.490540978961*^9, 3.5619281782539997`*^9, 3.5787104950635066`*^9, 3.5881732670905943`*^9, 3.5881743948240967`*^9, 3.5881745190001993`*^9, 3.588175745014323*^9, 3.5881766726823826`*^9, 3.5881806383932085`*^9, 3.589075723760638*^9, 3.589078317569996*^9, 3.589409732636956*^9, 3.5894099893516397`*^9, 3.5894100264707627`*^9, {3.589410081973937*^9, 3.58941008796228*^9}, { 3.5894102069480853`*^9, 3.589410226108181*^9}, {3.5894102858605986`*^9, 3.5894102941940756`*^9}, 3.5894103559926105`*^9, 3.5894104838179216`*^9, 3.5894105385110493`*^9, 3.65413190279377*^9, 3.654135366962663*^9, 3.654140096177697*^9, {3.6541468126365795`*^9, 3.6541468143412957`*^9}, 3.6571406037654114`*^9, 3.657163001314537*^9, 3.657163962586603*^9, 3.6571641686552706`*^9, 3.6574207800890875`*^9, 3.6574214277923355`*^9, 3.657422626179511*^9, 3.6574227726946163`*^9, 3.6574235426068826`*^9, 3.65742435584315*^9, 3.6574883393480234`*^9, 3.657494274030133*^9, 3.6574950752868795`*^9, 3.6574965906383047`*^9, 3.6574969555116377`*^9, { 3.6574969892169175`*^9, 3.6574970052489533`*^9}, 3.657497229883478*^9, 3.657521020484001*^9, 3.6575532392882695`*^9, 3.6575533015456753`*^9, 3.6575812708338237`*^9, 3.6575851314090548`*^9, 3.6575904657379284`*^9, 3.6575929125616355`*^9, 3.6580143825984297`*^9, 3.7490884816433525`*^9, 3.781782581617753*^9, 3.7832858289782133`*^9, 3.79503824989917*^9, 3.7950387208879128`*^9, 3.795045282500825*^9, 3.7950475391314263`*^9, 3.79504763042093*^9, 3.795047707850645*^9}] }, Open ]], Cell[TextData[{ "This is correct. The base of the ", Cell[BoxData[ FormBox[ RowBox[{"tri", "(", "4", ")"}], TraditionalForm]]], " triangular number starts at 0 to form {0,1,2,3}. The second row starts at \ 4 to form {4,5,6} and so on. 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" }], "Text", CellChangeTimes->{{3.6574881491403704`*^9, 3.6574881546600714`*^9}, { 3.657488209578045*^9, 3.6574882944583235`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{ FractionBox["1", "2"], RowBox[{"(", RowBox[{"n", "-", "z"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "n", "+", "z"}], ")"}]}], "+", RowBox[{"(", RowBox[{"z", "-", "1"}], ")"}]}], "]"}]], "Input", CellChangeTimes->{{3.65748602146369*^9, 3.6574860265803394`*^9}, { 3.6574881205782433`*^9, 3.6574881330203238`*^9}}], Cell[BoxData[ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", "n", "+", SuperscriptBox["n", "2"], "+", "z", "-", SuperscriptBox["z", "2"]}], ")"}]}]], "Output", CellChangeTimes->{3.6574858313970547`*^9, 3.657486028757616*^9, 3.6574942740791388`*^9, 3.6574965906858106`*^9, 3.6574969555671444`*^9, 3.6574970052994595`*^9, 3.657497229965989*^9, 3.6575210205295067`*^9, 3.6575532393442764`*^9, 3.6575533016041822`*^9, 3.6575812708658276`*^9, 3.6575851314545603`*^9, 3.6575904657804337`*^9, 3.65759291259464*^9, 3.65801438264843*^9, 3.749088481758355*^9, 3.7817825816905465`*^9, 3.783285829094343*^9, 3.7950382499849396`*^9, 3.795038720976636*^9, 3.7950452825915527`*^9, 3.795047539225176*^9, 3.7950476305246506`*^9, 3.795047707945391*^9}] }, Open ]], Cell[TextData[{ "The ", Cell[BoxData[ FormBox[ RowBox[{"-", "2"}], TraditionalForm]]], " is the only term that\[CloseCurlyQuote]s different. 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The last parameter (", StyleBox["f", FontSlant->"Italic"], ") optionally allows a caller to specify the Base", StyleBox["F", FontSlant->"Italic"], "orm." }], "Text", CellChangeTimes->{{3.4902222980950003`*^9, 3.4902223196429996`*^9}, { 3.561927483658*^9, 3.5619274859849997`*^9}, {3.588176264637044*^9, 3.588176265807111*^9}, {3.6541472600883985`*^9, 3.6541473115619345`*^9}, { 3.654150190187974*^9, 3.6541502429396725`*^9}, {3.654150284469946*^9, 3.6541503022562046`*^9}, {3.6574209549967976`*^9, 3.657420956490988*^9}, { 3.6574240056581826`*^9, 3.657424163280198*^9}, {3.6574941985625496`*^9, 3.6574942026180644`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"TriRow", "[", RowBox[{"b_", ",", " ", "r_", ",", " ", "z_", ",", RowBox[{"f_:", "10"}]}], "]"}], ":=", RowBox[{"baseForm", "[", RowBox[{ RowBox[{"FullSimplify", "[", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"x", "=", "0"}], RowBox[{"mi", "[", RowBox[{"z", ",", RowBox[{"-", "1"}]}], "]"}]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"leftedge", 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Cell["We\[CloseCurlyQuote]ll unset this so we can do it more elegantly...", \ "Text", CellChangeTimes->{{3.6541474022959566`*^9, 3.654147416959819*^9}}], Cell[BoxData[ RowBox[{"TriRowSimple", "=."}]], "Input", CellChangeTimes->{{3.6541473611257286`*^9, 3.654147397256817*^9}, 3.6574224477818575`*^9, 3.6574225209586496`*^9}], Cell[TextData[{ "Manually running the quadratic equation on ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["x", "2"], "+", "x"}], "2"], "=", "y"}], TraditionalForm]]], " gives: ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ RowBox[{"-", "1"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["1", "2"], "-", RowBox[{"(", RowBox[{"4", "*", "1", "*", "2", "y"}], ")"}]}]]}], RowBox[{"2", "*", "1"}]], TraditionalForm]]], ". The positive result is: ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SqrtBox[ RowBox[{"1", "+", RowBox[{"8", " ", "y"}]}]]}], ")"}]}], TraditionalForm]]], ". Since we already made the ", Cell[BoxData[ FormBox[ RowBox[{"qtri", "[", "x", "]"}], TraditionalForm]]], " function earlier to compute the quadratic value of ", Cell[BoxData[ FormBox[ RowBox[{"tri", "[", "x", "]"}], TraditionalForm]]], ". 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So if we sum the entire thing we get ...\ \>", "Text", CellChangeTimes->{ 3.490230559885*^9, {3.5890704988517904`*^9, 3.589070501061917*^9}, { 3.589071659800193*^9, 3.589071671021835*^9}, {3.654150554788272*^9, 3.6541505549577937`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"x", "=", "1"}], "3"], RowBox[{"TriRow", "[", RowBox[{"6", ",", "3", ",", "x"}], "]"}]}]], "Input", CellChangeTimes->{{3.490223876366*^9, 3.490223880764*^9}, 3.6574220128276253`*^9, 3.6574222978258157`*^9}], Cell[BoxData["35"], "Output", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmVkYGAwAuJ578MfCjG+cozb9XHmWiB9fXHCChB9RKFgG4j24ona d63zleOCXe+tb6585ShypXrX092vHBkCC92fAWnZ/tdhIFqKV/8DiNbQkk15 DqSVLK8GvwTSBv4difP2vnI8W7Rn6XwgbffrWIXjvleOaqJJd0F0FO9X8S2W rx0tMnlObgXSVRtkJXYA6bMnNugX2r92/JjVOr0GSL9t1dgIorXy9/79CaRt tbr9fgHpNwHJT0D0orbl30D0ieinYb+B9DmBhXtA9MO7xQ9kHV47nsp/zSgP pJ2yWvVB9IzHu01B9DO+y4Eg2tj43HstIG0l9X2TFZCODv8Y5QGk1Rd22XoB 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Though it would seem the baseform plays an important part in whatever the \ relationship is. I tried summing various combinations to see if anything \ beneath base 55 would sum to a series of ", Cell[BoxData[ FormBox[ SuperscriptBox["primes", "2"], TraditionalForm]]], " (i.e. ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"x", "=", "1"}], "28424"], SuperscriptBox[ RowBox[{"Prime", "[", "x", "]"}], "2"]}], "=", "971511442813339"}], ")"}], TraditionalForm]]], ", but in all instances it didn't work out. " }], "Text", CellChangeTimes->{{3.4909272569560003`*^9, 3.490927406343*^9}, { 3.490927445951*^9, 3.4909274701949997`*^9}, {3.490927634289*^9, 3.4909276447679996`*^9}, {3.4909277262349997`*^9, 3.490927770922*^9}, { 3.5810802106421556`*^9, 3.5810802155464363`*^9}, {3.6541505965680776`*^9, 3.65415064430614*^9}, {3.6571645558444376`*^9, 3.6571646786265287`*^9}, { 3.6571647217540054`*^9, 3.6571647595603065`*^9}, {3.6571660464087152`*^9, 3.657166069934703*^9}, {3.65716610393252*^9, 3.657166135205491*^9}, { 3.657171687067988*^9, 3.657171696355667*^9}, {3.6574203183709564`*^9, 3.6574203183709564`*^9}, {3.657422483898944*^9, 3.6574224864142632`*^9}}], Cell[TextData[{ StyleBox["Conjecture:", FontWeight->"Bold", FontVariations->{"Underline"->True}], " That aside, this seems to hint that a triangular number, when summing its \ rows will result in another tri- number. However it must be the value in its \ baseform? At a later point it may be worthwhile to attempt to figure out how \ to use the difference between the base-10 and the base-x form to determine \ the integer digits in the native base that when transformed to base-10 \ without a conversion would then result in a tri- number (i.e. ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["55", "6"], "=", SubscriptBox["35", "10"]}], ";"}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ SubscriptBox["55", "10"], TraditionalForm]]], " is a triangular number but ", Cell[BoxData[ FormBox[ SubscriptBox["35", "10"], TraditionalForm]]], " is not). This would then allow me to use an arbitrary number base and \ still be able to find tri- numbers. 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compressible to 7 digits, one less than 8. As a caveat since \ base-10 ", StyleBox["isn't", FontWeight->"Bold", FontVariations->{"Underline"->True}], " dense enough to contain ", Cell[BoxData[ FormBox["rgbbfg", TraditionalForm]]], " there is probably ", StyleBox["not", FontWeight->"Bold"], " a reverse map. Unless we can convert ", Cell[BoxData[ FormBox[ SubscriptBox["b", "36"], TraditionalForm]]], " to something like 11 and concatenate that way (i.e. / bb \[Implies] 1111) \ the best we can probably do is test in an upwards direction. 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I\[CloseCurlyQuote]m doubtful (how would this be \ expressed?), but maybe there is something that could done algorithmically. \ For now we can at least test whether ", Cell[BoxData[ FormBox[ SubscriptBox["21251029660", "10"], TraditionalForm]]], " is triangular when we drop the ", Cell[BoxData[ FormBox[ SubscriptBox["base", "10"], TraditionalForm]]], " and convert to ", Cell[BoxData[ FormBox[ SubscriptBox["base", "36"], TraditionalForm]]], ". 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represented as two base 6 digits." }], "Text", CellChangeTimes->{{3.4902261600439997`*^9, 3.490226189365*^9}, { 3.6571827858434887`*^9, 3.657182830326033*^9}, {3.657185187592861*^9, 3.6571852013046455`*^9}, {3.6571953050111074`*^9, 3.657195318083267*^9}, { 3.6574220226793766`*^9, 3.6574220226793766`*^9}, {3.6574223040391045`*^9, 3.6574223040391045`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"BaseForm", "[", RowBox[{ RowBox[{"TriRow", "[", RowBox[{"36", ",", "8", ",", "x"}], "]"}], ",", "6"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "1", ",", " ", "8"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{ 3.490225615433*^9, 3.657422023874028*^9, 3.6574223064979167`*^9, { 3.657424607073553*^9, 3.657424615761156*^9}, {3.6575896365846395`*^9, 3.6575896416182785`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ InterpretationBox[ SubscriptBox["\<\"55\"\>", "\<\"6\"\>"], 35, Editable->False], BaseForm[#, 6]& ], ",", TagBox[ InterpretationBox[ 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Since a base 666 system could contain 665 in 1 character at most this \ would be 15 characters long natively ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["log", "666"], "(", SubscriptBox["p", "o"], ")"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.490222729519*^9, 3.490222737273*^9}, {3.490231807742*^9, 3.490231839836*^9}, {3.5891170943469005`*^9, 3.589117094662918*^9}, { 3.5891171408225584`*^9, 3.5891171410485716`*^9}, {3.589117408075845*^9, 3.589117409779942*^9}, {3.5891181748066993`*^9, 3.589118179351959*^9}, { 3.591316706686952*^9, 3.5913167365236588`*^9}, 3.5921723430824623`*^9, { 3.6571829356110554`*^9, 3.657182939102255*^9}, {3.7807324749330297`*^9, 3.7807324756077566`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SubscriptBox["p", "0"], "=", "370054437526114837142163851665508571813290862848927310785932329262799778945\ 81784762614450464857290"}]], "Input", CellChangeTimes->{{3.4902236306610003`*^9, 3.490223632722*^9}}], Cell[BoxData[\ "37005443752611483714216385166550857181329086284892731078593232926279977894581\ 784762614450464857290"], "Output", CellChangeTimes->{3.6580143841394324`*^9, 3.749088484022353*^9, 3.780732631520132*^9, 3.781782582781601*^9, 3.783285831088371*^9, 3.7950382514854465`*^9, 3.795038722498564*^9, 3.7950390941291103`*^9, 3.795045284047682*^9, 3.7950475406105003`*^9, 3.795047632001972*^9, 3.795047709496244*^9}] }, Open ]], Cell["\<\ One of the major questions is does the sum of the rows of a tri-sequence \ always result in another triangular pattern?\ \>", "Text", CellChangeTimes->{{3.490224242068*^9, 3.490224251957*^9}, {3.490225848459*^9, 3.49022586237*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"n", FractionBox[ RowBox[{"(", RowBox[{"n", "+", "1"}], ")"}], "2"]}], "\[Equal]", SubscriptBox["p", "0"]}], ",", "n"}], "]"}]], "Input", CellChangeTimes->{{3.490224253837*^9, 3.490224273605*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"n", "\[Rule]", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "-", RowBox[{ "\[Sqrt]", "29604355002089186971373108133240685745063269027914184862874586341023\ 9823156654278100915603718858321"}]}], ")"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"n", "\[Rule]", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ "\[Sqrt]", "29604355002089186971373108133240685745063269027914184862874586341023\ 9823156654278100915603718858321"}]}], ")"}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.7807326345296807`*^9, 3.7817825828005486`*^9, 3.783285831144417*^9, 3.795038251516362*^9, 3.7950387225314755`*^9, 3.795039098111453*^9, 3.7950452840795703`*^9, 3.7950475406423845`*^9, 3.795047632035881*^9, 3.795047709529154*^9}] }, Open ]], Cell[TextData[{ "In this case no. However as established earlier it does appear to give a \ tri-number when in baseform. Recall, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"x", "=", "1"}], "3"], RowBox[{"TriRow", "[", RowBox[{"6", ",", "3", ",", "x"}], "]"}]}], " ", "=", " ", "35"}], TraditionalForm]]], ". However when the number is expressed in base-6 form we get, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"BaseForm", "[", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"x", "=", "1"}], "3"], RowBox[{"TriRow", "[", RowBox[{"6", ",", "3", ",", "x"}], "]"}]}], ",", "6"}], "]"}], " ", "=", " ", SubscriptBox["55", "6"]}], TraditionalForm]]], ". And ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"tri", "[", "10", "]"}], "=", "55"}], TraditionalForm]]], ". So even though ", Cell[BoxData[ FormBox[ SubscriptBox["35", "10"], TraditionalForm]]], " isn't triangular ", Cell[BoxData[ FormBox[ SubscriptBox["55", "6"], TraditionalForm]]], " (dropping the 6 and treating it as though it were natively base-10) does \ give us a triangular value. This may be purely coincidental however. Also \ recall that it reveals a problem. Base-10 can encapsulate base-6. \ Unfortunately though base-10 cannot encapsulate base-666. So there doesn\ \[CloseCurlyQuote]t appear to be an easy way to show ", Cell[BoxData[ FormBox[ SubscriptBox["p", "0"], TraditionalForm]]], " is triangular when converted to some other number base (see: ", StyleBox["\[OpenCurlyDoubleQuote]Sum of tri[n] rows in base form is equal to \ tri[n + 1]?\[CloseCurlyDoubleQuote]", FontWeight->"Bold", FontVariations->{"Underline"->True}], ")." }], "Text", CellChangeTimes->{{3.490224301144*^9, 3.4902243023059998`*^9}, { 3.5619263017609997`*^9, 3.5619263391619997`*^9}, {3.561926396332*^9, 3.561926660524*^9}, {3.5619282117860003`*^9, 3.5619282528570004`*^9}, { 3.5787102868815994`*^9, 3.5787103492371655`*^9}, {3.57910657358423*^9, 3.579106575125318*^9}, 3.5894114907655153`*^9, {3.589411897248765*^9, 3.5894118974467764`*^9}, {3.589415973740927*^9, 3.589415976741099*^9}, { 3.5894188698005724`*^9, 3.589418889593704*^9}, {3.5894189217135415`*^9, 3.589418938536504*^9}, {3.657182198449892*^9, 3.657182198892917*^9}, { 3.6571932678114157`*^9, 3.6571932683939896`*^9}, {3.6572291468938956`*^9, 3.6572291510161314`*^9}, {3.65742034981995*^9, 3.657420352914843*^9}, { 3.6574220292102056`*^9, 3.657422030378354*^9}, {3.6574223108519697`*^9, 3.657422311920105*^9}}], Cell[TextData[{ "Trying to figure out if the ", Cell[BoxData[ FormBox[ SubscriptBox["p", "0"], TraditionalForm]]], " value is a sum of square primes is going to take a long time to calculate \ ..." }], "Text", CellChangeTimes->{{3.4902241964110003`*^9, 3.490224200842*^9}, { 3.5619282799230003`*^9, 3.5619283368929996`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"x", "=", "1"}], "7"], SuperscriptBox[ RowBox[{"Prime", "[", "x", "]"}], "2"]}]], "Input", CellChangeTimes->{{3.4902234669709997`*^9, 3.490223469281*^9}}], Cell[BoxData["666"], "Output", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmVkYGAwBOI/KzeuFWJ85Vhg+23+WiC9oejFOhAtsu7yHhAd8pj1 yLXOV46qZ37Y3Vz5ynFX1fpdT3e/coxa3ej+DEjn5XwJA9FLvph9ANFcZSop z4H0gcV3g18CaTffiYnz9gLNqzy6dD6QPpVwscJx3yvHG7pZd0F0X+If8S2W rx39OsRPbgXScaVqEjuAdMSGHfqF9q8dWRinT68B0j/PGm8E0QfKT/79CaR3 Mc70+wWSFyt6AqIlObZ+A9HyzZ/CfgPpGqeNe0D0R/mWB7IOrx1n+f9llAfS At+m6oPootKTpiB694uXgSD6Y86t91pAesJkts1WQFrf8U+UB5CeqDLV1gtI i7tE3vAB0plP9bn8gPS/bNZr0o6vHdmNWudcfPfacVLWrTZd3TeOi88+7eA2 eON4ZXYha5nbG8fz8kEOIFp3VXNJJZC+sGVqXxWQFvJhmgWi5WU3LQbRAFPR yrY= "]] }, Open ]], Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"x", "=", "1"}], RowBox[{"10", "^", "7"}]], SuperscriptBox[ RowBox[{"Prime", "[", "x", "]"}], "2"]}]], "DisplayFormula", CellChangeTimes->{{3.490222742849*^9, 3.490222830013*^9}, 3.490223326633*^9, {3.7950475996200485`*^9, 3.7950476099554596`*^9}}], Cell[BoxData["103158861357874372432083"], "DisplayFormula", CellChangeTimes->{{3.490222752425*^9, 3.490222827007*^9}, 3.49022332301*^9, 3.49022336977*^9, 3.4905407996470003`*^9, 3.49054091919*^9, 3.490541030481*^9, 3.561928224116*^9, 3.578710519737918*^9, 3.579114312171852*^9, 3.588173293016077*^9, 3.588174420000537*^9, 3.588174543957627*^9, 3.5881757696837344`*^9, 3.5881766973047915`*^9, 3.588180662805605*^9, 3.589075748746067*^9, 3.589078342265408*^9, 3.5894097580554104`*^9, 3.5894105634574766`*^9, 3.6541319344782934`*^9, 3.654135399193256*^9, 3.65414012738266*^9, 3.6571406350832024`*^9, 3.6571639950072203`*^9, 3.657164199338667*^9, 3.6574208117166033`*^9, 3.6574214595383663`*^9, 3.6574226581735735`*^9, 3.657422803927082*^9, 3.657423573795343*^9, 3.657424386960602*^9, 3.657494305565137*^9, 3.657496623526981*^9, 3.657496987775234*^9, 3.657497037661569*^9, 3.6574972617200212`*^9, 3.6575210523495474`*^9, 3.6575533334202228`*^9, 3.6575813021603017`*^9, 3.657585163166587*^9, 3.6575904966228504`*^9, 3.6575929436450825`*^9, 3.6580144114604707`*^9, 3.7490885305013514`*^9, 3.781782605617051*^9, 3.783285862285323*^9, 3.7950382780108557`*^9, 3.7950387485333357`*^9, 3.795045309180667*^9, 3.795047566453154*^9}], Cell["\<\ Running 10^7 takes almost half a minute. 10^10 takes hours upon hours. I\ \[CloseCurlyQuote]ll have to try to come up with a way to optimize this \ process. \ \>", "Text", CellChangeTimes->{{3.57911431930326*^9, 3.5791143600855923`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Other Possible Connections", "Subtitle", CellChangeTimes->{{3.589070242171109*^9, 3.5890702496585374`*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Summing rows results in next tri numbers first row? ", StyleBox["NO", FontWeight->"Bold", FontSlant->"Italic"] }], "Section", CellChangeTimes->{{3.5890791229950633`*^9, 3.589079150956662*^9}, { 3.74909081073479*^9, 3.749090841301793*^9}}], Cell[TextData[{ "Interestingly enough, it seems there is some degree of further \ compressibility. The sum of the rows from ", Cell[BoxData[ FormBox[ RowBox[{"TriRowSimple", "[", RowBox[{ RowBox[{"tri", "[", "n", "]"}], ",", "x"}], "]"}], TraditionalForm]]], " in baseform of ", Cell[BoxData[ FormBox[ RowBox[{"tri", "[", "n", "]"}], TraditionalForm]]], " where the digits are transposed to base ", Cell[BoxData[ FormBox[ RowBox[{"tri", "[", RowBox[{"n", "+", "1"}], "]"}], TraditionalForm]]], " equal the bottom row of ", Cell[BoxData[ FormBox[ RowBox[{"tri", "[", RowBox[{"n", "+", "1"}], "]"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmViYGAQAWIQ/UrtK4Mw4yvHBVLszCCageGEshiQvqjErgaiWZ7n zGbqfeUYdXkOmJ735bTF7L2vHCe4XATTK4oyQ0D0KznLZBD9Lci4CEQLXTgN potY95eDaHefJ9Ug+tj0+CYQbaB+qA1EZ/N294JoschHE0F0nMOLrSD63vF/ YNpB+PErEP2HI/c12B6xvB8gekuB8G8QzeA/2W4OkO6ps7UH0UlbCruc9r1y PLVhMpjWMv198qf9a8dlaYdOgeiM9YaXfyHRT5uvrvsDpHP834LpWyWRF30d XjsmzMoF049mFHJJO752lHrTKAGii5p+JILoT+uvpYFoAH8GqmQ= "]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ 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So if this pattern holds we should see that\n0123 + 456 + 78 + 9 when \ converted to base-15 and after summing the rows should equal 1234 in base 10. \ \n\nFirst let\[CloseCurlyQuote]s convert the ", Cell[BoxData[ FormBox[ RowBox[{"tri", "[", "4", "]"}], TraditionalForm]]], " row 4 to base-15 (", Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{"tri", "[", "5", "]"}]}], TraditionalForm]]], ")..." }], "Text", CellChangeTimes->{{3.589069242335922*^9, 3.5890692609149847`*^9}, { 3.5890703675352793`*^9, 3.589070412199834*^9}, {3.5890708999957347`*^9, 3.5890709090372515`*^9}, {3.5890714099279013`*^9, 3.589071413913129*^9}, { 3.589071579092577*^9, 3.5890715800336304`*^9}, {3.6574203833892126`*^9, 3.6574203833892126`*^9}, {3.658008666010182*^9, 3.658008729476271*^9}, { 3.6580087669943237`*^9, 3.6580087725443316`*^9}, {3.6580098553845096`*^9, 3.6580099319588895`*^9}, {3.6580101118371778`*^9, 3.6580101474432144`*^9}, {3.6580102301929474`*^9, 3.6580102573284993`*^9}, 3.6580106444366407`*^9, 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We get:" }], "Text", CellChangeTimes->{{3.623807644513976*^9, 3.623807657273706*^9}, { 3.6238076967119617`*^9, 3.623807740319456*^9}, {3.623813283522509*^9, 3.6238133199505925`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"BaseForm", "[", RowBox[{ RowBox[{"SumTriRows", "[", RowBox[{"tri", "[", "2", "]"}], "]"}], ",", " ", RowBox[{"tri", "[", "2", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.6238076587557907`*^9, 3.6238076838012233`*^9}, { 3.6574205940274606`*^9, 3.657420595309123*^9}, 3.657422599614638*^9, 3.6574234932821193`*^9, 3.6574235291556745`*^9, {3.657425097973389*^9, 3.6574251006412277`*^9}, {3.657590615700471*^9, 3.6575906244440813`*^9}}], Cell[BoxData[ TagBox[ InterpretationBox[ SubscriptBox["\<\"10\"\>", "\<\"3\"\>"], 3, Editable->False], BaseForm[#, 3]& ]], "Output", CellChangeTimes->{{3.6238076603438816`*^9, 3.6238076845132637`*^9}, 3.6541319346568155`*^9, 3.6541353994257855`*^9, 3.6541401276221905`*^9, 3.6571406352912145`*^9, 3.6571639954292736`*^9, 3.6571641996382055`*^9, 3.657420811997139*^9, 3.6574214597988997`*^9, 3.65742265846011*^9, 3.6574228041671124`*^9, 3.6574235741013823`*^9, 3.657424387288644*^9, 3.6574251012648067`*^9, 3.657494306005693*^9, 3.657496623741008*^9, 3.657496988046769*^9, 3.657497037975109*^9, 3.657497262174079*^9, 3.6575210533796782`*^9, 3.6575533337852693`*^9, 3.6575813023848305`*^9, 3.6575851633701134`*^9, 3.6575904968283763`*^9, 3.6575906250676603`*^9, 3.6575929439531217`*^9, 3.658014411720471*^9, 3.7490885312823544`*^9, 3.7817826060349345`*^9, 3.7832858630701885`*^9, 3.795038278649191*^9, 3.7950387491885824`*^9, 3.79504530975813*^9, 3.7950475667294188`*^9, 3.795047632714116*^9}] }, Open ]], Cell["\<\ For comparison purposes lets look at all the possible ways of \ \[OpenCurlyDoubleQuote]viewing\[CloseCurlyDoubleQuote] this number in number \ bases 2 to 10.\ \>", "Text", CellChangeTimes->{{3.6238157559449234`*^9, 3.6238158035386457`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"BaseForm", "[", RowBox[{ RowBox[{"SumTriRows", "[", RowBox[{"tri", "[", "2", "]"}], "]"}], ",", "x"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "2", ",", " ", "10"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.623815805479757*^9, 3.623815830068163*^9}, 3.6238158641721134`*^9, 3.6574205963337536`*^9, 3.657422601658397*^9, 3.657423494400261*^9, 3.6574235300157833`*^9, {3.657425115835657*^9, 3.657425117793406*^9}, {3.657590637658759*^9, 3.6575906414407396`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ InterpretationBox[ SubscriptBox["\<\"11\"\>", "\<\"2\"\>"], 3, Editable->False], BaseForm[#, 2]& ], ",", TagBox[ InterpretationBox[ SubscriptBox["\<\"10\"\>", "\<\"3\"\>"], 3, Editable->False], BaseForm[#, 3]& ], ",", TagBox[ InterpretationBox[ SubscriptBox["\<\"3\"\>", "\<\"4\"\>"], 3, Editable->False], BaseForm[#, 4]& ], ",", TagBox[ InterpretationBox[ SubscriptBox["\<\"3\"\>", "\<\"5\"\>"], 3, Editable->False], BaseForm[#, 5]& ], ",", TagBox[ InterpretationBox[ SubscriptBox["\<\"3\"\>", "\<\"6\"\>"], 3, Editable->False], BaseForm[#, 6]& ], ",", TagBox[ InterpretationBox[ SubscriptBox["\<\"3\"\>", "\<\"7\"\>"], 3, Editable->False], BaseForm[#, 7]& ], ",", TagBox[ InterpretationBox[ SubscriptBox["\<\"3\"\>", "\<\"8\"\>"], 3, Editable->False], BaseForm[#, 8]& ], ",", TagBox[ InterpretationBox[ SubscriptBox["\<\"3\"\>", "\<\"9\"\>"], 3, Editable->False], BaseForm[#, 9]& ], ",", TagBox[ InterpretationBox["\<\"3\"\>", 3, Editable->False], BaseForm[#, 10]& ]}], "}"}]], "Output", CellChangeTimes->{3.623815831882267*^9, 3.623815866097224*^9, 3.6541319346703176`*^9, 3.654135399443288*^9, 3.654140127652694*^9, 3.6571406353062153`*^9, 3.657163995469279*^9, 3.6571641996632085`*^9, 3.657420812017642*^9, 3.6574214598199024`*^9, 3.6574226584806128`*^9, 3.657422804186615*^9, 3.6574235741198845`*^9, 3.657424387310646*^9, 3.657425118168954*^9, 3.6574943060361967`*^9, 3.6574966237585106`*^9, 3.657496988066272*^9, 3.657497037999112*^9, 3.6574972622095833`*^9, 3.65752105339618*^9, 3.6575533338132725`*^9, 3.6575813024033327`*^9, 3.6575851633886156`*^9, 3.6575904968543797`*^9, 3.657590642214838*^9, 3.6575929439896264`*^9, 3.658014411735471*^9, 3.7490885313363533`*^9, 3.781782606056901*^9, 3.7832858631180325`*^9, 3.7950382786801085`*^9, 3.7950387492264814`*^9, 3.7950453097890477`*^9, 3.7950475667503595`*^9, 3.795047632746996*^9}] }, Open ]], Cell[TextData[{ "So if we were to drop the number base of 3 from ", Cell[BoxData[ FormBox[ SubscriptBox["10", "3"], TraditionalForm]]], " and increment ", "tri(", "n) where ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", "2"}], TraditionalForm]]], " to ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", "4"}], TraditionalForm]]], ". We get:" }], "Text", CellChangeTimes->{{3.6238076936637874`*^9, 3.623807693827797*^9}, { 3.6238077456317596`*^9, 3.6238077517601104`*^9}, {3.6238078192079678`*^9, 3.6238078287395134`*^9}, 3.623813090744483*^9, {3.623813903263956*^9, 3.6238139544788857`*^9}, {3.6238149458955913`*^9, 3.623814951218896*^9}, { 3.623816183776394*^9, 3.623816213316084*^9}, 3.6574209603704805`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"tri", "[", "4", "]"}]], "Input", CellChangeTimes->{{3.623813646513271*^9, 3.623813677595049*^9}, { 3.623813749313151*^9, 3.6238137609338155`*^9}, 3.657420597185362*^9}], Cell[BoxData["10"], "Output", CellChangeTimes->{{3.6238136469942985`*^9, 3.6238136779900713`*^9}, { 3.6238137568745832`*^9, 3.6238137613348384`*^9}, 3.6541319346838193`*^9, 3.65413539946129*^9, 3.654140127674697*^9, 3.657140635320216*^9, 3.6571639955087843`*^9, 3.657164199687211*^9, 3.657420812038144*^9, 3.6574214598404045`*^9, 3.6574226585016155`*^9, 3.6574228042051177`*^9, 3.6574235741383867`*^9, 3.657424387332649*^9, 3.657494306069701*^9, 3.6574966237755127`*^9, 3.657496988085774*^9, 3.657497038024115*^9, 3.657497262241087*^9, 3.6575210534126825`*^9, 3.6575533338377757`*^9, 3.6575813024208345`*^9, 3.657585163407118*^9, 3.657590496874382*^9, 3.657592944004628*^9, 3.658014411750471*^9, 3.7490885313773537`*^9, 3.7817826060778217`*^9, 3.783285863146285*^9, 3.795038278710029*^9, 3.7950387492594337`*^9, 3.795045309817957*^9, 3.7950475667573414`*^9, 3.7950476327809057`*^9}] }, Open ]], Cell[TextData[{ "In summary, the sum of the rows of tri[2] works out to be ", Cell[BoxData[ FormBox[ SubscriptBox["10", "3"], TraditionalForm]]], " which maps to ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["10", "10"], "=", RowBox[{"tri", "[", "4", "]"}]}], TraditionalForm]]], " (n-diff: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"4", "-", "2"}], "=", StyleBox["2", FontWeight->"Bold"]}], TraditionalForm]]], ", base-diff: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"10", "-", "3"}], "=", StyleBox["7", FontWeight->"Bold"]}], TraditionalForm]]], "). Next the sum of the rows of ", "tri[", "3] works out to be ", Cell[BoxData[ FormBox[ SubscriptBox["55", "6"], TraditionalForm]]], " which maps to ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["55", "10"], "=", RowBox[{"tri", "[", "10", "]"}]}], TraditionalForm]]], " (n-diff: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"10", "-", "3"}], "=", StyleBox["7", FontWeight->"Bold"]}], TraditionalForm]]], ", base-diff: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"10", "-", "6"}], "=", StyleBox["4", FontWeight->"Bold"]}], TraditionalForm]]], ")." }], "Text", CellChangeTimes->{{3.6238151401787033`*^9, 3.623815242136535*^9}, { 3.6238152751124215`*^9, 3.6238152849549847`*^9}, {3.6238154518205285`*^9, 3.623815467858446*^9}, {3.623815499596261*^9, 3.6238155555974646`*^9}, { 3.623816569862477*^9, 3.6238165764088516`*^9}, {3.6238166129569416`*^9, 3.6238166567314453`*^9}, {3.6574205981564846`*^9, 3.657420602614051*^9}}], Cell[CellGroupData[{ Cell["\<\ Failed Explanation Number #1 \[LongDash] Summed tri[n] rows in base form of \ tri[n+1] is equal to tri[tri[n + 1]]?\ \>", "Subsection", CellChangeTimes->{{3.6238185717379775`*^9, 3.6238186271381464`*^9}, { 3.623819133121087*^9, 3.62381915625241*^9}, {3.623819187536199*^9, 3.623819191658435*^9}, {3.6238195145349026`*^9, 3.623819515152938*^9}, { 3.657420603773198*^9, 3.657420606515046*^9}}], Cell["So since the sum of the rows of tri[4] equals: ", "Text", CellChangeTimes->{{3.5890798405721064`*^9, 3.589079886649742*^9}, { 3.623816623362537*^9, 3.6238166240175743`*^9}, 3.65742060803974*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"BaseForm", "[", RowBox[{ RowBox[{"SumTriRows", "[", "10", "]"}], ",", "10"}], "]"}]], "Input", CellChangeTimes->{{3.589079463493539*^9, 3.589079494904335*^9}, { 3.5890797034912653`*^9, 3.5890797052723675`*^9}, {3.5890797857219687`*^9, 3.5890797905062428`*^9}, 3.6574226034716277`*^9, 3.657423496372512*^9, 3.6574235314614673`*^9}], Cell[BoxData[ TagBox[ InterpretationBox["\<\"666\"\>", 666, Editable->False], BaseForm[#, 10]& ]], "Output", CellChangeTimes->{3.589079496775442*^9, 3.5890797059894085`*^9, 3.5890797911342783`*^9, 3.5894097582364206`*^9, 3.589410563604485*^9, 3.6541319346968207`*^9, 3.6541353994782925`*^9, 3.6541401276911993`*^9, 3.6571406353372173`*^9, 3.6571639955497894`*^9, 3.657164199712714*^9, 3.6574208120581465`*^9, 3.6574214598614073`*^9, 3.657422658521118*^9, 3.6574228042256203`*^9, 3.6574235741573887`*^9, 3.6574243873541517`*^9, 3.657494306101205*^9, 3.6574966237915144`*^9, 3.6574969881057763`*^9, 3.6574970380476184`*^9, 3.6574972622760916`*^9, 3.6575210534291844`*^9, 3.657553333863779*^9, 3.657581302440337*^9, 3.65758516342562*^9, 3.657590496894885*^9, 3.6575929440391326`*^9, 3.658014411770471*^9, 3.749088531421356*^9, 3.7817826061226993`*^9, 3.7832858631871243`*^9, 3.7950382787379537`*^9, 3.795038749291308*^9, 3.79504530984785*^9, 3.7950475667633276`*^9, 3.795047632815812*^9}] }, Open ]], Cell["And:", "Text", CellChangeTimes->{{3.623815303968072*^9, 3.623815307418269*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"tri", "[", "36", "]"}]], "Input", CellChangeTimes->{{3.623815309103366*^9, 3.623815310041419*^9}, 3.657420608883347*^9}], Cell[BoxData["666"], "Output", CellChangeTimes->{3.623815310501446*^9, 3.6541319347098227`*^9, 3.6541353994957943`*^9, 3.6541401277097015`*^9, 3.6571406353512177`*^9, 3.6571639955887938`*^9, 3.6571641997362175`*^9, 3.65742081207815*^9, 3.65742145988241*^9, 3.657422658542121*^9, 3.657422804244622*^9, 3.657423574175891*^9, 3.6574243873766546`*^9, 3.65749430614021*^9, 3.657496623808017*^9, 3.657496988125279*^9, 3.6574970380716214`*^9, 3.657497262306596*^9, 3.6575210534456863`*^9, 3.657553333887282*^9, 3.6575813024583397`*^9, 3.6575851634436226`*^9, 3.6575904969113865`*^9, 3.6575929440541344`*^9, 3.658014411785471*^9, 3.7490885314603567`*^9, 3.7817826061436434`*^9, 3.783285863220518*^9, 3.795038278768871*^9, 3.795038749323263*^9, 3.795045309874818*^9, 3.7950475667683105`*^9, 3.795047632848748*^9}] }, Open ]], Cell[TextData[{ "The pattern seems to hold. \n\nSo for our first naive attempt to explain \ what\[CloseCurlyQuote]s happening. Lets assume that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"SumTriRows", "[", RowBox[{"tri", "[", "n", "]"}], "]"}], "=", StyleBox["s", FontWeight->"Bold"]}], TraditionalForm]]], " gives a number ", Cell[BoxData[ FormBox[ StyleBox["s", FontWeight->"Bold"], TraditionalForm]]], " that in baseform of ", "tri[", "n+1] will equal some other ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"tri", "[", "x", "]"}], "=", StyleBox["s", FontWeight->"Bold"]}], TraditionalForm]]], " (possibly as ", Cell[BoxData[ FormBox[ RowBox[{"tri", "[", RowBox[{"tri", "[", RowBox[{"n", "+", "1"}], "]"}], "]"}], TraditionalForm]]], "). So, lets convert the number as though it was previously in base-15." }], "Text", CellChangeTimes->{{3.6238186777610416`*^9, 3.623818690940796*^9}, { 3.623818741504688*^9, 3.623818768180214*^9}, {3.623818830229762*^9, 3.623818873931262*^9}, {3.6238189065211263`*^9, 3.6238189325326138`*^9}, { 3.6238189725969057`*^9, 3.6238190293731527`*^9}, {3.623819166430992*^9, 3.6238191666330037`*^9}, 3.6238192087354116`*^9, {3.657420609725954*^9, 3.657420613025873*^9}, {3.6574226051633425`*^9, 3.6574226051633425`*^9}, { 3.6574234974831524`*^9, 3.6574234974831524`*^9}, {3.657423533073172*^9, 3.657423533073172*^9}}], Cell[CellGroupData[{ Cell[BoxData["15^^666"], "Input", CellChangeTimes->{{3.589079819903924*^9, 3.589079825127223*^9}, { 3.623819240641237*^9, 3.6238192607253857`*^9}}], Cell[BoxData["1446"], "Output", CellChangeTimes->{ 3.589079825867265*^9, 3.5894097582524214`*^9, 3.589410563619486*^9, { 3.623819241786302*^9, 3.6238192612324142`*^9}, 3.6541319347233243`*^9, 3.654135399512797*^9, 3.654140127726204*^9, 3.6571406353662186`*^9, 3.657163995628299*^9, 3.6571641997597203`*^9, 3.657420812097652*^9, 3.657421459902913*^9, 3.6574226585621233`*^9, 3.657422804263625*^9, 3.657423574194894*^9, 3.6574243873991575`*^9, 3.6574943061687136`*^9, 3.657496623828019*^9, 3.657496988144781*^9, 3.6574970380956244`*^9, 3.6574972623466005`*^9, 3.6575210534621887`*^9, 3.6575533339107847`*^9, 3.6575813024763417`*^9, 3.657585163461625*^9, 3.657590496928389*^9, 3.657592944089139*^9, 3.6580144118004713`*^9, 3.7490885315033536`*^9, 3.781782606164624*^9, 3.783285863260335*^9, 3.7950382787997894`*^9, 3.7950387493531823`*^9, 3.79504530990574*^9, 3.7950475667842712`*^9, 3.795047632879671*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"tri", "[", "15", "]"}]], "Input", CellChangeTimes->{{3.5890797960095577`*^9, 3.589079805532102*^9}, 3.6574206139594917`*^9}], Cell[BoxData["120"], "Output", CellChangeTimes->{3.5890798059731274`*^9, 3.5894097582674227`*^9, 3.5894105636334867`*^9, 3.654131934736326*^9, 3.654135399530799*^9, 3.6541401277427053`*^9, 3.65714063538122*^9, 3.6571639956648035`*^9, 3.657164199781723*^9, 3.6574208121151543`*^9, 3.657421459923415*^9, 3.6574226585821257`*^9, 3.6574228042836275`*^9, 3.6574235742143965`*^9, 3.6574243874211607`*^9, 3.6574943061907167`*^9, 3.6574966238450212`*^9, 3.657496988164784*^9, 3.6574970381196275`*^9, 3.657497262373104*^9, 3.657521053479191*^9, 3.6575533339317875`*^9, 3.6575813024958444`*^9, 3.657585163479127*^9, 3.657590496945391*^9, 3.6575929441051407`*^9, 3.658014411815471*^9, 3.749088531580357*^9, 3.7817826061825404`*^9, 3.783285863294758*^9, 3.795038278831703*^9, 3.7950387493850975`*^9, 3.7950453099356174`*^9, 3.7950475667952414`*^9, 3.795047632910599*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"BaseForm", "[", RowBox[{ RowBox[{"tri", "[", "15", "]"}], ",", "15"}], "]"}]], "Input", CellChangeTimes->{{3.589079975199806*^9, 3.589079978081971*^9}, 3.6574206146295767`*^9}], Cell[BoxData[ TagBox[ InterpretationBox[ SubscriptBox["\<\"80\"\>", "\<\"15\"\>"], 120, Editable->False], BaseForm[#, 15]& ]], "Output", CellChangeTimes->{3.5890799783789883`*^9, 3.589409758282423*^9, 3.5894105636474876`*^9, 3.654131934748828*^9, 3.654135399548301*^9, 3.6541401277592077`*^9, 3.65714063539622*^9, 3.6571639957013083`*^9, 3.657164199803226*^9, 3.6574208121331563`*^9, 3.657421459943918*^9, 3.657422658602128*^9, 3.6574228043021297`*^9, 3.6574235742328987`*^9, 3.657424387442663*^9, 3.65749430621472*^9, 3.657496623861023*^9, 3.657496988184286*^9, 3.6574970381436305`*^9, 3.6574972624001074`*^9, 3.6575210534956927`*^9, 3.6575533339537907`*^9, 3.6575813025158467`*^9, 3.657585163496629*^9, 3.6575904969653935`*^9, 3.657592944120143*^9, 3.658014411825471*^9, 3.74908853162541*^9, 3.7817826062004914`*^9, 3.783285863326547*^9, 3.7950382788616233`*^9, 3.7950387494170036`*^9, 3.795045309965567*^9, 3.795047566812196*^9, 3.7950476329464617`*^9}] }, Open ]], Cell[TextData[{ "Sadly ", Cell[BoxData[ RowBox[{ RowBox[{"tri", "[", "15", "]"}], "\[NotEqual]", "1446"}]], "Input"], " nor does it equal 666. However since ", Cell[BoxData[ FormBox[ RowBox[{"tri", "[", "n", "]"}], TraditionalForm]]], " will always equal a number that has a whole ", Cell[BoxData[ FormBox[ RowBox[{"qtri", "[", RowBox[{"tri", "[", "n", "]"}], "]"}], TraditionalForm]]], ". 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Meaning sum of primes (3+4+5) cubed or (3+4+5+6) to the fourth.\ \>", "Text", CellChangeTimes->{{3.7490904444677916`*^9, 3.74909055047279*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"PrimePowerSum", "[", RowBox[{"x_", ",", RowBox[{"p_:", "1"}]}], "]"}], ":=", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"a", "=", "1"}], "x"], SuperscriptBox[ RowBox[{"Prime", "[", "a", "]"}], "p"]}]}]], "Input", CellChangeTimes->{{3.7490894258526783`*^9, 3.7490895468596773`*^9}, { 3.7490895783006773`*^9, 3.749089578599682*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PrimePowerSum", "[", "3", "]"}]], "Input", CellChangeTimes->{{3.749089603401677*^9, 3.7490896105696783`*^9}}], Cell[BoxData["10"], "Output", CellChangeTimes->{3.7490896109886827`*^9, 3.7490907133877883`*^9, 3.7817826063042145`*^9, 3.7832858635066423`*^9, 3.7950382789763174`*^9, 3.7950387495356855`*^9, 3.7950453100812254`*^9, 3.7950475668391237`*^9, 3.7950476330591607`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", 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", StyleBox["No Answer Yet", FontWeight->"Bold"] }], "Section", CellChangeTimes->{{3.749090378880789*^9, 3.74909044050779*^9}, { 3.7490905567627907`*^9, 3.749090652839789*^9}, {3.74909077024579*^9, 3.7490907706837935`*^9}, {3.79503141452387*^9, 3.795031464466117*^9}, { 3.795032769480091*^9, 3.7950327826388807`*^9}}], Cell[TextData[{ "On Imgur I stumbled on ", StyleBox[ButtonBox["https://imgur.com/gallery/on6zVM6/comment/1834371379", BaseStyle->"Hyperlink", ButtonData->{ URL["https://imgur.com/gallery/on6zVM6/comment/1834371379"], None}, ButtonNote->"https://imgur.com/gallery/on6zVM6/comment/1834371379"], FontVariations->{"Underline"->True}], " which resulted in a ", ButtonBox["brief dialogue with Greg", BaseStyle->"Hyperlink", ButtonData->{ URL["https://discordapp.com/channels/@me/440404991233228810/\ 696053898993205320"], None}, ButtonNote-> "https://discordapp.com/channels/@me/440404991233228810/\ 696053898993205320"], " and later again in ", ButtonBox["2022-08-16", BaseStyle->"Hyperlink", ButtonData->{ URL["https://discord.com/channels/@me/440404991233228810/\ 1009177831521140736"], None}, ButtonNote-> "https://discord.com/channels/@me/440404991233228810/1009177831521140736"],\ ". This made me realize Pascal\[CloseCurlyQuote]s triangle (", ButtonBox["https://www.geogebra.org/m/hcmgsfpu", BaseStyle->"Hyperlink", ButtonData->{ URL["https://www.geogebra.org/m/hcmgsfpu"], None}, ButtonNote->"https://www.geogebra.org/m/hcmgsfpu"], ") is fundamentally tied to ", Cell[BoxData[ FormBox[ SuperscriptBox["11", "n"], TraditionalForm]]], " and how ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["11", "5"], "=", StyleBox[ RowBox[{"1", StyleBox["610", FontWeight->"Bold"], "51"}]]}], TraditionalForm]]], " shows how triangular values bridge from base-10 to potentially \ base-infinity (", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{"1", StyleBox["5", FontWeight->"Bold"]}]], StyleBox[ RowBox[{ StyleBox["AA", FontWeight->"Bold"], "51"}]]}], TraditionalForm]]], "). In other words the pattern of ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["11", "n"], "=", RowBox[{"PascalRow", "(", "n", ")"}]}], TraditionalForm]]], " for ", Cell[BoxData[ FormBox[ RowBox[{"n", "<", "5"}], TraditionalForm]]], " as shown below, but Pascal\[CloseCurlyQuote]s triangle easily handles any \ number of rows with perfect symmetry forever.\n\n", Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJzsnQecVNXZh7ew9CK9o4BdmvSmIoLGhjWJiWKNsUaDX0ysIBFbYkP6Lr0X DVhARUSNPUYNFhBEZXen7M6W2elz6/nee96Zw7AFd5eyC/6f3/F6587ts8x9 5j3nvKfnjXdf/seMtLS0/2tMk8tv+NvZ99xzw/1XNKAXY+//y803ZtLMzfT+ 5TTjzAsAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAABxvDMGgaDodpGgqFLMuimXg8rt7VNC0SiaRu ous6rUBTXpngGZqapsmbR6NRtT7tnPdsS1I3IWKxGJ8DzRzC6wQAAAAA+MVD AkbeRapGU7I1djDyt0AgQDPsaeR4wWCQvY5gZ6O3aDltwi9pOe2E16clbHGl 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Meaning we could possibly disentangle this to get it in a format like ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{"1", StyleBox["5", FontWeight->"Bold"]}]], StyleBox[ RowBox[{ StyleBox["AA", FontWeight->"Bold"], "51"}]]}], TraditionalForm]]], " by using the factors. Basically we know the front and end are correct \ because we have 1s there. The middle we know is somehow messed up because it \ doesn\[CloseCurlyQuote]t mirror on the left and right as ", Cell[BoxData[ FormBox[ RowBox[{"5", "AA5"}], TraditionalForm]]], ". So the idea is we could use the expansion of ", Cell[BoxData[ SuperscriptBox[ RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], "5"]], CellChangeTimes->{{3.795009988547777*^9, 3.795010009109855*^9}, { 3.795012184943037*^9, 3.7950121870702963`*^9}}], " to decompose the middle piece." }], "Text", CellChangeTimes->{{3.795033969948369*^9, 3.79503397152813*^9}, { 3.7950340135090046`*^9, 3.795034015629332*^9}, {3.7950341318666315`*^9, 3.795034568733473*^9}, {3.7950349516387153`*^9, 3.795034982870658*^9}, { 3.7950418711858406`*^9, 3.7950418772635736`*^9}, {3.795048989796714*^9, 3.7950490366076365`*^9}, {3.7950490879692307`*^9, 3.7950490971679125`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Expand", "[", SuperscriptBox[ RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], "5"], "]"}]], "Input"], Cell[BoxData[ RowBox[{ SuperscriptBox["a", "5"], "+", RowBox[{"5", " ", SuperscriptBox["a", "4"], " ", "b"}], "+", RowBox[{"10", " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["b", "2"]}], "+", RowBox[{"10", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "3"]}], "+", RowBox[{"5", " ", "a", " ", SuperscriptBox["b", "4"]}], "+", SuperscriptBox["b", "5"]}]], "Output", CellChangeTimes->{3.795034573274324*^9, 3.7950382794612017`*^9, 3.795038750040304*^9, 3.795045310374441*^9, 3.7950475669828677`*^9, 3.7950476333623495`*^9}] }, Open ]], Cell[TextData[{ "Given this expansion we can take the middle terms and set it to the \ difference removing ", Cell[BoxData[ FormBox[ SuperscriptBox["a", "5"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SuperscriptBox["b", "5"], TraditionalForm]]], " from the entire thing combined as ", Cell[BoxData[ FormBox[ SuperscriptBox["11", "5"], TraditionalForm]]], ". 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So since we know how to \ go from 5AA5 to 6105 (i.e. ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"5", "+", "1"}], ")"}], RowBox[{"(", RowBox[{"0", "+", "1"}], ")"}], "05"}], "=", "6105"}], TraditionalForm]]], ") we can ask if we can go from 6105 to 5AA5?\n\nI suspect we would have to \ know the number-base we are already in (guessing base-11 here for ", Cell[BoxData[ FormBox[ SuperscriptBox["11", "5"], TraditionalForm]]], ") and the numberbase we\[CloseCurlyQuote]re going to. This gets to the \ problem from earlier:" }], "Text", CellChangeTimes->{{3.795034639478627*^9, 3.795034640128886*^9}, { 3.7950346778200855`*^9, 3.7950346783696146`*^9}, {3.7950347569651794`*^9, 3.7950347873073683`*^9}, {3.7950352253681855`*^9, 3.7950352861499934`*^9}, { 3.795041266799179*^9, 3.795041439014346*^9}, {3.7950491403489933`*^9, 3.79504914122864*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"x", "=", "1"}], "8"], RowBox[{"TriRow", "[", RowBox[{"36", ",", "8", ",", "x"}], "]"}]}], "//", RowBox[{"b", "@", "36"}]}]], "Input"], Cell[BoxData[ TagBox[ InterpretationBox[ SubscriptBox["\<\"9rgbbfg\"\>", "\<\"36\"\>"], 21251029660, Editable->False], BaseForm[#, 36]& ]], "Output", CellChangeTimes->{ 3.795041432136771*^9, 3.7950453104462166`*^9, {3.7950475670157814`*^9, 3.795047584040973*^9}, 3.795047633439175*^9}] }, Open ]], Cell[TextData[{ "If we look at the left-most ", Cell[BoxData[ FormBox["r", TraditionalForm]], FormatType->"TraditionalForm"], " it would equal ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"r", "*", SuperscriptBox["36", "5"]}], "//", RowBox[{"b", "@", "36"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " and r is the 18th character so we start counting 9+18." }], "Text", CellChangeTimes->{{3.795041446934601*^9, 3.7950414820560503`*^9}, { 3.795041521914897*^9, 3.795041533656146*^9}, {3.7950420703387947`*^9, 3.795042073128645*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"9", "+", "18"}], ")"}], "*", SuperscriptBox["36", "5"]}], "//", RowBox[{"b", "@", "36"}]}]], "Input", CellChangeTimes->{{3.795041498982577*^9, 3.7950415108581266`*^9}}], Cell[BoxData[ TagBox[ InterpretationBox[ SubscriptBox["\<\"r00000\"\>", "\<\"36\"\>"], 1632586752, Editable->False], BaseForm[#, 36]& ]], "Output", CellChangeTimes->{{3.7950415040061646`*^9, 3.7950415115821915`*^9}, 3.7950453104771357`*^9, 3.795047567032735*^9, 3.7950476334721003`*^9, 3.7950477980834665`*^9}] }, Open ]], Cell[TextData[{ "The problem is this value in base-10 isn\[CloseCurlyQuote]t just a simple \ 10+1 scenario where we carry a single digit to the left and leave a single \ digit in the right column. There are 18 1\[CloseCurlyQuote]s (from ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"9", "+", "18"}], "=", "27"}], TraditionalForm]]], ") that need to carry to the left in base-10. Meaning if there was a 0 to \ the left of our ", Cell[BoxData[ FormBox[ RowBox[{"r", "*", SuperscriptBox["36", "5"]}], TraditionalForm]], FormatType->"TraditionalForm"], " it would become 18 in base-10. This is further complicated because our \ positions in this string represent multiples of ", Cell[BoxData[ FormBox[ SuperscriptBox["36", "x"], TraditionalForm]]], ". This means 10 is really 37. Having 18 extra 37s gives 666 (funny). To \ make matters more difficult ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["36", "5"], "=", "60466176"}], TraditionalForm]], FormatType->"TraditionalForm"], ". This indicates our 18 1\[CloseCurlyQuote]s are multiples of this and \ generates integers all the way up to ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["10", "9"], "."}], TraditionalForm]]], " So going from base-10 to base-36 in-situ for middle values is probably \ pretty hard because it ends up affecting numbers in every term of the \ expansion." }], "Text", CellChangeTimes->{{3.795041555865749*^9, 3.795041656019425*^9}, { 3.7950421082696557`*^9, 3.79504219382125*^9}, {3.7950422344821224`*^9, 3.7950423536857457`*^9}, {3.7950427418361444`*^9, 3.7950428643301363`*^9}, 3.795042936828494*^9, {3.7950430237293186`*^9, 3.795043024637888*^9}, { 3.795043085501281*^9, 3.7950430866392355`*^9}, {3.7950431840547576`*^9, 3.7950432102147164`*^9}, {3.7950435003313665`*^9, 3.795043520122549*^9}, { 3.7950435537551727`*^9, 3.795043602044551*^9}, {3.795043779089821*^9, 3.795043808649462*^9}, {3.7950438449122567`*^9, 3.795043845010992*^9}, { 3.7950491965606728`*^9, 3.7950491985313997`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"9", "+", "18"}], ")"}], "*", SuperscriptBox["36", "5"]}], "//", RowBox[{"b", "@", "10"}]}]], "Input", CellChangeTimes->{{3.7950415471779084`*^9, 3.7950415479677773`*^9}}], Cell[BoxData["1632586752"], "Output", CellChangeTimes->{3.795041548320833*^9, 3.7950453105070534`*^9, 3.7950475285409994`*^9, 3.7950475670486927`*^9, 3.7950476335029726`*^9}] }, Open ]], Cell[TextData[{ "11 may be a unique case that helps abstract this so we can handle the \ carrying more cleanly as we go from base-11 (?) to base-10 (as the simple \ case that we see with 5AA5 and 6105) to more complicated operations. \n\n\ Since we have a way to go from 5AA5 to 6105 (i.e. ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"5", "+", "1"}], ")"}], RowBox[{"(", RowBox[{"0", "+", "1"}], ")"}], "05"}], "=", "6105"}], TraditionalForm]]], ") let\[CloseCurlyQuote]s see how we reverse it using ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], "5"], TraditionalForm]], FormatType->"TraditionalForm"], " as ", Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"5", " ", SuperscriptBox["a", "4"], " ", "b"}], ")"}], "+", RowBox[{"(", RowBox[{"10", " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["b", "2"]}], ")"}], "+", RowBox[{"(", RowBox[{"10", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "3"]}], ")"}], "+", RowBox[{"(", RowBox[{"5", " ", "a", " ", SuperscriptBox["b", "4"]}], ")"}]}]], CellChangeTimes->{{3.7950439662140694`*^9, 3.7950439806840086`*^9}}], ". Here we have ", Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"5", " ", SuperscriptBox["a", "4"], " ", "b"}], ")"}], "=", "60000"}]], CellChangeTimes->{{3.7950439662140694`*^9, 3.7950439806840086`*^9}}], " or ", Cell[BoxData[ RowBox[{"=", "50000"}]], CellChangeTimes->{{3.7950439662140694`*^9, 3.7950439806840086`*^9}}], ". In the case of it being ", Cell[BoxData[ FormBox[ RowBox[{"5", "*", SuperscriptBox["10", "4"]}], TraditionalForm]], FormatType->"TraditionalForm"], " this means ", Cell[BoxData[ FormBox["a", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["b", TraditionalForm]], FormatType->"TraditionalForm"], " have to be 10 and 1. 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So the ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ SuperscriptBox["a", "4"], SuperscriptBox["a", "4"]], "=", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], " meaning ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"5", "*", "12000"}], "=", "60000"}], TraditionalForm]], FormatType->"TraditionalForm"], ". In the case of ", Cell[BoxData[ FormBox[ RowBox[{"5", "*", SuperscriptBox["10", "4"]}], TraditionalForm]], FormatType->"TraditionalForm"] }], "Text", CellChangeTimes->{{3.7950461545140667`*^9, 3.795046208211294*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"5", " ", SuperscriptBox["a", "4"], " ", "b"}], ")"}], "\[Equal]", "50000"}], ",", "b"}], "]"}]], "Input", CellChangeTimes->{3.7950462220587845`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"b", "\[Rule]", FractionBox["10000", SuperscriptBox["a", "4"]]}], "}"}], "}"}]], "Output", CellChangeTimes->{3.7950462226890693`*^9, 3.7950475671344643`*^9, 3.795047633708698*^9}] }, Open ]], Cell[TextData[{ "What changes is the 10,000 from 12,000. So somehow I need to know how to go \ from ", Cell[BoxData[ FormBox[ RowBox[{"12", "*", SuperscriptBox["10", "4"]}], TraditionalForm]], FormatType->"TraditionalForm"], "\[Rule]", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"10", "*", SuperscriptBox["10", "4"]}], "=", SuperscriptBox["10", "5"]}], TraditionalForm]], FormatType->"TraditionalForm"], ". The leading 5 is obviously the giveaway (", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["5", FontWeight->"Bold"], SuperscriptBox["a", "4"], "b"}], TraditionalForm]], FormatType->"TraditionalForm"], "). The problem is we don\[CloseCurlyQuote]t have that expression by \ default. It\[CloseCurlyQuote]s constructed by hand and we want to get this by \ doing something like ", Cell[BoxData[ FormBox[ SuperscriptBox["11", "5"], TraditionalForm]], FormatType->"TraditionalForm"], ". We could use the previous row to build it. However, it feels like we \ should be able to do this just by looking at a single row alone. I suppose if \ ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["a", "4"], "b"}], "\[NotEqual]", SuperscriptBox["10", "5"]}], TraditionalForm]], FormatType->"TraditionalForm"], " then we know we\[CloseCurlyQuote]re already up or down 1 in the current \ column. The problem is we still need to know what to divide by. Given 12000 I \ can see the value is 5 ", Cell[BoxData[ FormBox[ RowBox[{"(", FractionBox["60000", "12000"]}], TraditionalForm]], FormatType->"TraditionalForm"], "). Therefore we\[CloseCurlyQuote]re +1 and need to subtract that value and \ add it to the column to the right. ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}], "*", SuperscriptBox["10", "3"]}], TraditionalForm]], FormatType->"TraditionalForm"], ". Maybe we can look at its mirror in the ", Cell[BoxData[ FormBox[ SuperscriptBox["10", "1"], TraditionalForm]], FormatType->"TraditionalForm"], "? No because after 1,2,3,4,5,6,7,8,9, we get a 10 and then it would carry \ to the left and need to be reverted.\n\nBasically we\[CloseCurlyQuote]re in a \ bind. If I have Pascal\[CloseCurlyQuote]s triangle computing the coefficients \ I don\[CloseCurlyQuote]t need ", Cell[BoxData[ FormBox[ SuperscriptBox["11", "x"], TraditionalForm]], FormatType->"TraditionalForm"], ". If I do have Pascal\[CloseCurlyQuote]s triangle in play I can use it to \ do the conversion because " }], "Text", CellChangeTimes->{{3.795046229626505*^9, 3.7950463572384977`*^9}, { 3.7950463983800964`*^9, 3.7950464152508707`*^9}, {3.7950464519615455`*^9, 3.7950465012632*^9}, {3.795046571065362*^9, 3.7950467011689696`*^9}, { 3.7950467690189495`*^9, 3.795046817891182*^9}, {3.7950469299621134`*^9, 3.795046937213229*^9}, {3.7950470189120016`*^9, 3.795047123734914*^9}, { 3.7950484363456745`*^9, 3.7950484605957894`*^9}, {3.795049707396208*^9, 3.7950497555976524`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", FractionBox["60000", "12000"], "]"}]], "Input", CellChangeTimes->{{3.79504650267409*^9, 3.795046508256572*^9}, { 3.795046723110262*^9, 3.7950467264323726`*^9}, {3.795046826302677*^9, 3.795046870284614*^9}, {3.795049766188051*^9, 3.7950497764477835`*^9}}], Cell[BoxData["5.`"], "Output", CellChangeTimes->{3.7950497783635826`*^9}] }, Open ]], Cell["\<\ However, really all that ever does is just create the 0s. So I could just use \ Log to find the length for a particular position in the number. \ \>", "Text", CellChangeTimes->{{3.7950498117111044`*^9, 3.795049844065291*^9}, { 3.7950500311301064`*^9, 3.795050038690406*^9}, {3.795050103058414*^9, 3.7950501060184917`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{"Ceiling", "[", RowBox[{"Log", "[", RowBox[{"10", ",", RowBox[{"60000", "+", ".0000001"}]}], "]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.795049845596193*^9, 3.79504990596941*^9}, { 3.7950499573697224`*^9, 3.7950500124029293`*^9}, 3.7950501144103303`*^9}], Cell[BoxData["5.`"], "Output", CellChangeTimes->{{3.7950498502926807`*^9, 3.795049906385298*^9}, { 3.7950499578314877`*^9, 3.795050012746973*^9}, 3.7950501148759885`*^9}] }, Open ]], Cell[TextData[{ "So all I really need is the difference ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"6", "-", "5"}], "=", "1"}], TraditionalForm]]], ". That means we add the value back to the right. " }], "Text", CellChangeTimes->{{3.795050128928096*^9, 3.7950501613195577`*^9}, { 3.7951049684712*^9, 3.7951049733910403`*^9}, {3.7951050426931477`*^9, 3.795105056233917*^9}, {3.795105586476472*^9, 3.795105671967461*^9}, { 3.795105706198906*^9, 3.7951057091988697`*^9}, {3.7951057531689296`*^9, 3.7951058295314875`*^9}, {3.795105903533845*^9, 3.795105973698408*^9}, { 3.7951060111367826`*^9, 3.7951060223071156`*^9}, {3.7951060605869865`*^9, 3.795106079259019*^9}, {3.7951061791909037`*^9, 3.7951061795010777`*^9}, { 3.795106222768032*^9, 3.7951063025320644`*^9}}], Cell[CellGroupData[{ Cell["\<\ This won\[CloseCurlyQuote]t work as a standard number-base conversion because \ the outer characters maintain a lower base from the factors in the center. \ \>", "Item", CellChangeTimes->{{3.795106194352044*^9, 3.7951062678211207`*^9}}], Cell["\<\ Finding a way to convert from nonstandard numbers to other numbers of bases \ (i.e. converting 17 base-10 to base-2) is likely a way to understand all of \ this at a deeper level because it\[CloseCurlyQuote]s not base-specific then\ \>", "Item", CellChangeTimes->{{3.795106194352044*^9, 3.7951062086399927`*^9}, { 3.7951062702809153`*^9, 3.7951062824165096`*^9}}], Cell["\<\ Trying to use formulas like TriRow could help to think about how to properly \ represent these values\ \>", "Item", CellChangeTimes->{{3.795106194352044*^9, 3.7951062086399927`*^9}, { 3.7951062702809153`*^9, 3.79510629598538*^9}}], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["11", "5"], "=", SubscriptBox["10000", "11"]}], TraditionalForm]]], " shows that converting to 15AA51 isn\[CloseCurlyQuote]t a simple matter of \ base conversion. I think each factor has its own number base or the pairs \ have the same number base. i.e ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"BaseForm", "[", RowBox[{"10", ",", "11"}], "]"}], "=", SubscriptBox["a", "11"]}], TraditionalForm]]] }], "Item", CellChangeTimes->{{3.795106194352044*^9, 3.7951062086399927`*^9}, { 3.7951062702809153`*^9, 3.79510629647803*^9}, {3.7951076759068193`*^9, 3.7951077259203157`*^9}, {3.795183299065548*^9, 3.795183299080269*^9}}], Cell[TextData[{ "Note: ", Cell[BoxData[ RowBox[{ SuperscriptBox["A", "5"], "+", RowBox[{"5", " ", SuperscriptBox["A", "4"], " ", "B"}], "+", RowBox[{"10", " ", SuperscriptBox["A", "3"], " ", SuperscriptBox["B", "2"]}], "+", RowBox[{"10", " ", SuperscriptBox["A", "2"], " ", SuperscriptBox["B", "3"]}], "+", RowBox[{"5", " ", "A", " ", SuperscriptBox["B", "4"]}], "+", SuperscriptBox["B", "5"]}]], CellChangeTimes->{{3.79504525069493*^9, 3.7950452585947604`*^9}, 3.795045970222347*^9, {3.795047518694649*^9, 3.795047523973193*^9}, { 3.7950478399215217`*^9, 3.7950478958339157`*^9}, {3.795047927466359*^9, 3.79504793299781*^9}, 3.7950480255571327`*^9, 3.7950493374588795`*^9}], " , in this situation A represents the number-base (i.e. ", Cell[BoxData[ FormBox[ SuperscriptBox["11", "X"], TraditionalForm]], FormatType->"TraditionalForm"], " shown in ", Cell[BoxData[ FormBox[ SubscriptBox["base", "infinity"], TraditionalForm]], FormatType->"TraditionalForm"], "). 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", "Text", CellChangeTimes->{{3.5619355655220003`*^9, 3.5619355694040003`*^9}, { 3.5619357244040003`*^9, 3.561935725832*^9}, {3.561936009176*^9, 3.56193601342*^9}, {3.589412043909154*^9, 3.5894121464810205`*^9}, { 3.589413321925252*^9, 3.589413326752528*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Intersection", "[", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"tri", "[", "x", "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "1", ",", " ", "69"}], "}"}]}], "]"}], ",", RowBox[{"Table", "[", RowBox[{ RowBox[{"p", "[", RowBox[{"x", ",", "2"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "1", ",", " ", "13"}], "}"}]}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.589413329216669*^9, 3.58941336088048*^9}, 3.657420640370345*^9}], Cell[BoxData[ RowBox[{"{", "666", "}"}]], "Output", CellChangeTimes->{{3.5894133450495744`*^9, 3.589413361486515*^9}, 3.6541319350248623`*^9, 3.6541353997733297`*^9, 3.654140127975235*^9, 3.6571406356202335`*^9, 3.657163996206873*^9, 3.657164200096263*^9, 3.65742081239669*^9, 3.657421460219953*^9, 3.6574226588701625`*^9, 3.6574228045581627`*^9, 3.657423574486931*^9, 3.6574243877357006`*^9, 3.6574943065762653`*^9, 3.657496624094553*^9, 3.6574969884348183`*^9, 3.6574970384506693`*^9, 3.6574972628631663`*^9, 3.65752105379123*^9, 3.657553334398847*^9, 3.6575813029038963`*^9, 3.6575851637326593`*^9, 3.657590497197923*^9, 3.6575929443426714`*^9, 3.6580144120314713`*^9, 3.7490885323703523`*^9, 3.781782606642336*^9, 3.783285864281604*^9, 3.7950382799630175`*^9, 3.7950387505629063`*^9, 3.7950453112144165`*^9, 3.795047567347891*^9, 3.7950476342039433`*^9}] }, Open ]], Cell["\<\ So it\[CloseCurlyQuote]s not exactly a common occurrence. What\ \[CloseCurlyQuote]s interesting though is, again, we see 10 and 666 seem to \ have something in common. Now let\[CloseCurlyQuote]s try to sum lots of different number bases and see \ whether those numbers align with the above prime lists (these numbers should \ always be triangular numbers). Obviously only triangular numbers will give whole \[DoubleStruckCapitalZ]\ \[CloseCurlyQuote]s, as the command below shows ...\ \>", "Text", CellChangeTimes->{{3.5619360661140003`*^9, 3.561936146274*^9}, { 3.561936359918*^9, 3.5619363860620003`*^9}, {3.5894133231063194`*^9, 3.589413323907365*^9}, {3.5894133928853106`*^9, 3.589413414762562*^9}, { 3.589413463135329*^9, 3.5894134768111105`*^9}, 3.6575928271832933`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"N", "[", RowBox[{"SumTriRows", "[", "x", "]"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "2", ",", " ", "12"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.589413065286573*^9, 3.589413065603591*^9}, 3.6574226065585194`*^9, 3.657423499564417*^9, 3.6574235341558094`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ "1.`", ",", "3.`", ",", "9.`", ",", "17.`", ",", "35.`", ",", "105.`", ",", "209.`", ",", "353.`", ",", "666.`", ",", "2395.`", ",", "5026.`"}], "}"}]], "Output", CellChangeTimes->{3.5894130667126546`*^9, 3.654131935054366*^9, 3.654135399792332*^9, 3.654140127994238*^9, 3.6571406356392345`*^9, 3.6571639962448773`*^9, 3.657164200119766*^9, 3.6574208124166927`*^9, 3.657421460241456*^9, 3.657422658890665*^9, 3.657422804578165*^9, 3.6574235745064335`*^9, 3.6574243877582035`*^9, 3.6574943066417737`*^9, 3.657496624113055*^9, 3.657496988454321*^9, 3.6574970384751725`*^9, 3.6574972629001713`*^9, 3.6575210539697533`*^9, 3.6575533344283504`*^9, 3.657581302925399*^9, 3.6575851637526617`*^9, 3.657590497216426*^9, 3.6575929443596735`*^9, 3.6580144120464716`*^9, 3.749088532640354*^9, 3.7817826068138533`*^9, 3.7832858645774155`*^9, 3.7950382801086273`*^9, 3.795038750640697*^9, 3.7950453113351336`*^9, 3.7950475673678365`*^9, 3.7950476342408752`*^9}] }, Open ]], Cell["So just putting in tri\[CloseCurlyQuote]s ...", "Text", CellChangeTimes->{{3.5894134819094024`*^9, 3.5894134878487425`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"SumTriRows", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"{", RowBox[{ "3", ",", "6", ",", "10", ",", "15", ",", "21", ",", "28", ",", "36", ",", "45", ",", "55"}], "}"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.5894124132492785`*^9, 3.5894124290351815`*^9}, 3.657422607814179*^9, 3.657423501469159*^9, 3.6574235351954412`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ "3", ",", "35", ",", "666", ",", "24605", ",", "1564690", ",", "152843733", ",", "21251029660", ",", "3988218576606", ",", "971559132683085"}], "}"}]], "Output", CellChangeTimes->{{3.589412415614414*^9, 3.589412429476207*^9}, 3.654131935078369*^9, 3.6541353998118343`*^9, 3.6541401280102396`*^9, 3.657140635656235*^9, 3.657163996283882*^9, 3.6571642001432695`*^9, 3.657420812436695*^9, 3.6574214602624583`*^9, 3.6574226589126673`*^9, 3.657422804597167*^9, 3.6574235745254354`*^9, 3.657424387780706*^9, 3.6574943066707773`*^9, 3.6574966241335583`*^9, 3.6574969884743233`*^9, 3.657497038500676*^9, 3.657497262940176*^9, 3.6575210539887557`*^9, 3.6575533345053606`*^9, 3.657581302945401*^9, 3.6575851637731643`*^9, 3.657590497235428*^9, 3.6575929443766756`*^9, 3.658014412061472*^9, 3.749088532768356*^9, 3.781782606829809*^9, 3.7832858647169223`*^9, 3.7950382803769107`*^9, 3.795038750750404*^9, 3.7950453113749866`*^9, 3.7950475673867874`*^9, 3.7950476343476*^9}] }, Open ]], Cell["The set of hits includes {666}. 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(", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["1", FontColor->RGBColor[0, 0, 1]], StyleBox["*", FontColor->RGBColor[1, 0, 0]], StyleBox[".5", FontColor->RGBColor[0, 0, 1]]}], "=", StyleBox[ FractionBox["1", "2"], FontColor->RGBColor[1, 0, 0]]}], TraditionalForm]]], ") where 5 mimics 2, and vice versa, on the flipside of the radix point ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ StyleBox[ FractionBox["5", "2"], FontColor->RGBColor[1, 0, 0]], "=", StyleBox["2.5", FontColor->RGBColor[0, 0, 1]]}], ")"}], TraditionalForm]]], " that appears to show a connection between ", StyleBox["operations", FontColor->RGBColor[1, 0, 0]], " versus ", StyleBox["values", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ StyleBox[ SuperscriptBox["5", "2"], FontColor->RGBColor[1, 0, 0]], "=", StyleBox["25", FontColor->RGBColor[0, 0, 1]]}], ")"}], TraditionalForm]]], ". Almost as though the numbers somehow intrinsically flip about each other. \ This \[OpenCurlyDoubleQuote]flipping\[CloseCurlyDoubleQuote] is possibly \ reflected as a property of the number-base (mirroring half of the \ rowreverse[], leftedge[], and the rightedge[] functions as discussed in the \ preamble establishing the initial ", "SumTriRows[]", "):" }], "Text", CellChangeTimes->{{3.57871005592939*^9, 3.578710065938962*^9}, { 3.5787106434869957`*^9, 3.578710726638752*^9}, {3.5787108269544897`*^9, 3.5787108850768137`*^9}, {3.578711022739688*^9, 3.578711036607481*^9}, { 3.5787148214139595`*^9, 3.578714841439105*^9}, {3.5787149329173374`*^9, 3.578714934203411*^9}, {3.578715045635784*^9, 3.5787150933835154`*^9}, { 3.578715277185028*^9, 3.5787152784210987`*^9}, {3.5787170242549553`*^9, 3.5787170408649054`*^9}, {3.5852244313242707`*^9, 3.585224447780212*^9}, { 3.780160634595677*^9, 3.780160821416703*^9}, {3.7801610797118826`*^9, 3.78016108645988*^9}, {3.7801612647947197`*^9, 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