(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 42665, 1365]*) (*NotebookOutlinePosition[ 43829, 1402]*) (* CellTagsIndexPosition[ 43697, 1395]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Finding a/N with partial quotients all 1 or 2", "Title"], Cell["\<\ An algorithm to determine whether such an a exists, given N, and to find it \ if it does.\ \>", "Subtitle"], Cell[TextData[{ "The Zaremba conjecture says that the partial quotients need never be \ greater than 5. But perhaps for N sufficiently large, even 2 suffices. One \ check to speculation has been that it is hard to tell of an arbitrary given \ N, whether or not there exists an integer a, 1"Italic"], " notebook uses a method significantly better than naive checking, to \ accelerate the search. The method is far from polynomial time in the digital \ length of N, but its reach significantly exceeds that of direct search. \n" }], "Subsubtitle"], Cell["\<\ The basic idea is that there are intervals of values of a that are doomed \ from the outset to failure. Thus, for instance, any a sufficiently near N/2 \ will either give an expansion of a/N of the form 1/(1+1/(l+1/large)), or \ 1/(2+1/large)). The algorithm finds all fractions p/q that have all partial \ quotients of 1 or 2 and that have denominators comparable to Sqrt[N]. The \ search may safely be limited to short intervals on either side of these p/q. \ The aggregate length of these intervals is comparable to N^(0.5312805), as is \ the number of p/q which must be considered, so we gain a factor of nearly \ N^(1/2) in speed, compared to the basic strategy of checking all values a, \ 1", "Text"], Cell[BoxData[ \(\(\(backtrackstep[vthing_] := Module[{qm, q, v, a}, {qm, q, v} = vthing; \[IndentingNewLine]If[ Length[v] \[Equal] 1, Return[{0, 1, {}}]]; \[IndentingNewLine]a = v[\([\(-1\)]\)]; {q - a\ qm, qm, Drop[v, \(-1\)]}]\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell["\<\ The next bit of code, nextv, finds the next leaf on the tree, given that \ we're starting at a leaf. The input has two entries, the leaf we're on, and \ the parameter x which gives the upper limit for q. Leaves themselves have \ three entries: the previous q, called qm, for our string of 1s and 2s less \ its last entry, the current q, the denominator of the expansion of v as a \ continued fraction, and the list v itself of 1s and 2s. \ \>", "Text"], Cell[BoxData[ \(\(\(nextv[vthing_, x_] := Module[{vv, a, qm, q, v, r, t}, \[IndentingNewLine]Print["\", vthing, "\<,\>", x]; vv = vthing; {qm, q, v} = vv; a = v[\([\(-1\)]\)]; r = Length[v]; \[IndentingNewLine]If[ a \[Equal] 1 && q + qm \[LessEqual] x, Return[{qm, q + qm, Join[Drop[v, \(-1\)], {2}]}]]; \[IndentingNewLine]If[ r \[Equal] 1, Return[False]]; \[IndentingNewLine]v = Drop[v, \(-1\)]; t = q - a\ qm; q = qm; qm = t; \[IndentingNewLine]If[Union[v] \[Equal] {2}, Return[False]]; \[IndentingNewLine]a = v[\([\(-1\)]\)]; Print["\<{qm,q,v}=\>", {qm, q, v}]; \[IndentingNewLine]While[ a \[Equal] 2, \[IndentingNewLine]t = q - a\ qm; q = qm; qm = t; v = Drop[v, \(-1\)]; \[IndentingNewLine]Print["\<{qm,q,v}=\>", {qm, q, v}]; a = v[\([\(-1\)]\)]]; \[IndentingNewLine]q += qm; v = Join[Drop[v, \(-1\)], {2}]; Print["\<{qm,q,v}=\>", {qm, q, v}]; \[IndentingNewLine]t = q + qm; \[IndentingNewLine]While[q + qm \[LessEqual] x, t = q + qm; qm = q; q = t; AppendTo[v, 1]; \[IndentingNewLine]Print["\<{qm,q,v}=\>", {qm, q, v}]]; \[IndentingNewLine]{qm, q, v}]\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(nextv[{19, 27, {2, 1, 2, 2, 1}}, 40]\)], "Input"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{8, 19, {2, 1, 2, 2}}\), SequenceForm[ "{qm,q,v}=", {8, 19, {2, 1, 2, 2}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{3, 8, {2, 1, 2}}\), SequenceForm[ "{qm,q,v}=", {3, 8, {2, 1, 2}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{2, 3, {2, 1}}\), SequenceForm[ "{qm,q,v}=", {2, 3, {2, 1}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{2, 5, {2, 2}}\), SequenceForm[ "{qm,q,v}=", {2, 5, {2, 2}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{5, 7, {2, 2, 1}}\), SequenceForm[ "{qm,q,v}=", {5, 7, {2, 2, 1}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{7, 12, {2, 2, 1, 1}}\), SequenceForm[ "{qm,q,v}=", {7, 12, {2, 2, 1, 1}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{12, 19, {2, 2, 1, 1, 1}}\), SequenceForm[ "{qm,q,v}=", {12, 19, {2, 2, 1, 1, 1}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{19, 31, {2, 2, 1, 1, 1, 1}}\), SequenceForm[ "{qm,q,v}=", {19, 31, {2, 2, 1, 1, 1, 1}}], Editable->False]], "Print"], Cell[BoxData[ \({19, 31, {2, 2, 1, 1, 1, 1}}\)], "Output"] }, Open ]], Cell["\<\ Here is the same code, but with the print commands rendered inoperative. \ \>", "Text"], Cell[BoxData[ \(\(\(nextv[vthing_, x_] := Module[{vv, a, qm, q, v, r, t}, \[IndentingNewLine] (*\(Print["\", vthing, "\<,\>", x];\)*) vv = vthing; {qm, q, v} = vv; a = v[\([\(-1\)]\)]; r = Length[v]; \[IndentingNewLine]If[ a \[Equal] 1 && q + qm \[LessEqual] x, Return[{qm, q + qm, Join[Drop[v, \(-1\)], {2}]}]]; \[IndentingNewLine]If[ r \[Equal] 1, Return[False]]; \[IndentingNewLine]v = Drop[v, \(-1\)]; t = q - a\ qm; q = qm; qm = t; \[IndentingNewLine]If[Union[v] \[Equal] {2}, Return[False]]; \[IndentingNewLine]a = v[\([\(-1\)]\)]; (*\(Print["\<{qm,q,v}=\>", {qm, q, v}];\)*) \[IndentingNewLine]While[ a \[Equal] 2, \[IndentingNewLine]t = q - a\ qm; q = qm; qm = t; v = Drop[ v, \(-1\)]; \[IndentingNewLine] (*\(Print["\<{qm,q,v}=\>", {qm, q, v}];\)*) a = v[\([\(-1\)]\)]]; \[IndentingNewLine]q += qm; v = Join[ Drop[v, \(-1\)], {2}]; (*\(Print["\<{qm,q,v}=\>", {qm, q, v}];\)*) \[IndentingNewLine]t = q + qm; \[IndentingNewLine]While[q + qm \[LessEqual] x, t = q + qm; qm = q; q = t; AppendTo[v, 1]\[IndentingNewLine] (*Print["\<{qm,q,v}=\>", {qm, q, v}]*) ]; \[IndentingNewLine]{qm, q, v}]\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[BoxData[""], "Input"], Cell["\<\ The procedure firstvee takes as input x, and returns a string of 1s just long \ enough that the resulting q is less than or equal to x, but appending another \ 1 would put q over the line. \ \>", "Text"], Cell[BoxData[ \(firstvee[x_] := Module[{u, v, w, qm, q}, v = {}; qm = 0; q = 1; While[q + qm \[LessEqual] x, {qm, q} = {q, q + qm}; AppendTo[v, 1]]; {qm, q, v}]\)], "Input"], Cell["\<\ vmatrix finds the fractions p/q for v and for the predecessor of v. \ \>", "Text"], Cell[BoxData[ \(vmatrix[v_] := Module[{m, n, k}, p = 0; q = 1; n = Length[v]; k = 1; m = {{1, 0}, {0, 1}}; While[k \[LessEqual] n, m = m . {{0, 1}, {1, v[\([\(-k\)]\)]}}; \(k++\)]; \[IndentingNewLine]m]\)], \ "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(alpha = 1/3\)], "Input"], Cell[BoxData[ \(1\/3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(beta = 1/\((1 + alpha)\)\)], "Input"], Cell[BoxData[ \(3\/4\)], "Output"] }, Open ]], Cell["\<\ vinterval finds the interval on either side of p/q such that multiplying that \ interval by N and checking the a's in it captures all feasible candidates for \ which the expansion of a/N would kick off with v. \ \>", "Text"], Cell[BoxData[ \(vinterval[v_] := Module[{r, qm, p, q}, {{r, qm}, {p, q}} = vmatrix[v]; \[IndentingNewLine]If[ EvenQ[Length[ v]], {\((p + r\ alpha)\)/\((q + qm\ alpha)\), \((p + r\ beta)\)/\((q + qm\ beta)\)}, {\((p + r\ beta)\)/\((q + qm\ beta)\), \((p + r\ alpha)\)/\((q + qm\ alpha)\)}]]\)], "Input"], Cell["\<\ \"good\" checks whether a candidate \"a\" for a/N really works. \ \>", "Text"], Cell[BoxData[ \(good[a_, b_] := Module[{p, q, k}, p = a; q = b; While[p \[NotEqual] 0, k = Floor[q/p]; If[k > 2, Return[False]]; \[IndentingNewLine]{p, q} = {q - k\ p, p}]; q \[Equal] 1]\)], "Input"], Cell[BoxData[ \(\(\(zarembacheck[n_] := Module[{v, vv, a, b, c, r, s, t}, x = Floor[Sqrt[n]]; vv = firstvee[ x]; (*\(Print["\", vv];\)*) \[IndentingNewLine]While[ ListQ[vv], \[IndentingNewLine]{a, b} = n\ vinterval[ vv[\([\(-1\)]\)]]; (*\(Print["\<{a,b}=\>", {a, b}];\)*) a = Ceiling[a]; b = Floor[ b]; (*\(Print["\<{a,b}=\>", {a, b}];\)*) \[IndentingNewLine]c = a; While[c \[LessEqual] b, If[good[c, n], Return[{c, n, True}]]; \(c++\)]; \[IndentingNewLine]vv = nextv[vv, x]]; False]\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell["\<\ For comparison, we include code that implements the naive approach of trying \ everything. It is, as one might expect, slow when given large inputs. Thus, \ for instance, in the tests executed below, it took longer with the basic \ version to get an answer for N=10^7+1, than it took with the main algorithm \ to get an answer for N=10^12+1. \ \>", "Text"], Cell[BoxData[ \(basiczarembacheck[n_] := Module[{a = 1}, While[a < n, If[good[a, n], Return[{a, n, True}]]; \(a++\)]; False]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(firstvee[10]\)], "Input"], Cell[BoxData[ \({5, 8, {1, 1, 1, 1, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nextv[%, 10]\)], "Input"], Cell[BoxData[ \({3, 8, {1, 1, 1, 2}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nextv[%, 10]\)], "Input"], Cell[BoxData[ \({5, 7, {1, 1, 2, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nextv[%, 10]\)], "Input"], Cell[BoxData[ \({4, 7, {1, 2, 1, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nextv[%, 10]\)], "Input"], Cell[BoxData[ \({7, 10, {1, 2, 2, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nextv[%, 10]\)], "Input"], Cell[BoxData[ \({5, 8, {2, 1, 1, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nextv[%, 10]\)], "Input"], Cell[BoxData[ \({3, 8, {2, 1, 2}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nextv[%, 10]\)], "Input"], Cell[BoxData[ \({5, 7, {2, 2, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nextv[%, 10]\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(zarembacheck[104]\)], "Input"], Cell[BoxData[ InterpretationBox[\("vv="\[InvisibleSpace]{5, 8, {1, 1, 1, 1, 1}}\), SequenceForm[ "vv=", {5, 8, {1, 1, 1, 1, 1}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{a,b}="\[InvisibleSpace]{3016\/47, 1872\/29}\), SequenceForm[ "{a,b}=", { Rational[ 3016, 47], Rational[ 1872, 29]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{a,b}="\[InvisibleSpace]{65, 64}\), SequenceForm[ "{a,b}=", {65, 64}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{a,b}="\[InvisibleSpace]{1768\/27, 2704\/41}\), SequenceForm[ "{a,b}=", { Rational[ 1768, 27], Rational[ 2704, 41]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{a,b}="\[InvisibleSpace]{66, 65}\), SequenceForm[ "{a,b}=", {66, 65}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{a,b}="\[InvisibleSpace]{60, 2600\/43}\), SequenceForm[ "{a,b}=", {60, Rational[ 2600, 43]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{a,b}="\[InvisibleSpace]{60, 60}\), SequenceForm[ "{a,b}=", {60, 60}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{a,b}="\[InvisibleSpace]{1872\/25, 377\/5}\), SequenceForm[ "{a,b}=", { Rational[ 1872, 25], Rational[ 377, 5]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{a,b}="\[InvisibleSpace]{75, 75}\), SequenceForm[ "{a,b}=", {75, 75}], Editable->False]], "Print"], Cell[BoxData[ \({75, 104, True}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nextv[%, 40]\)], "Input"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{13, 21, {1, 1, 1, 1, 1, 1, 1, 1}}\), SequenceForm[ "{qm,q,v}=", {13, 21, {1, 1, 1, 1, 1, 1, 1, 1}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{13, 34, {1, 1, 1, 1, 1, 1, 1, 2}}\), SequenceForm[ "{qm,q,v}=", {13, 34, {1, 1, 1, 1, 1, 1, 1, 2}}], Editable->False]], "Print"], Cell[BoxData[ \({13, 34, {1, 1, 1, 1, 1, 1, 1, 2}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nextv[%, 40]\)], "Input"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{8, 13, {1, 1, 1, 1, 1, 1, 1}}\), SequenceForm[ "{qm,q,v}=", {8, 13, {1, 1, 1, 1, 1, 1, 1}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{8, 21, {1, 1, 1, 1, 1, 1, 2}}\), SequenceForm[ "{qm,q,v}=", {8, 21, {1, 1, 1, 1, 1, 1, 2}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{21, 29, {1, 1, 1, 1, 1, 1, 2, 1}}\), SequenceForm[ "{qm,q,v}=", {21, 29, {1, 1, 1, 1, 1, 1, 2, 1}}], Editable->False]], "Print"], Cell[BoxData[ \({21, 29, {1, 1, 1, 1, 1, 1, 2, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nextv[%, 40]\)], "Input"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{8, 21, {1, 1, 1, 1, 1, 1, 2}}\), SequenceForm[ "{qm,q,v}=", {8, 21, {1, 1, 1, 1, 1, 1, 2}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{5, 8, {1, 1, 1, 1, 1, 1}}\), SequenceForm[ "{qm,q,v}=", {5, 8, {1, 1, 1, 1, 1, 1}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{5, 13, {1, 1, 1, 1, 1, 2}}\), SequenceForm[ "{qm,q,v}=", {5, 13, {1, 1, 1, 1, 1, 2}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{13, 18, {1, 1, 1, 1, 1, 2, 1}}\), SequenceForm[ "{qm,q,v}=", {13, 18, {1, 1, 1, 1, 1, 2, 1}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{18, 31, {1, 1, 1, 1, 1, 2, 1, 1}}\), SequenceForm[ "{qm,q,v}=", {18, 31, {1, 1, 1, 1, 1, 2, 1, 1}}], Editable->False]], "Print"], Cell[BoxData[ \({18, 31, {1, 1, 1, 1, 1, 2, 1, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(alpha = 1/\((2 + 1/\((1 + 1/\((2 + 1/\((1 + 1/3)\))\))\))\)\)], "Input"], Cell[BoxData[ \(15\/41\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(zarembacheck[104]\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(vmatrix[{1, 2, 3}]\)], "Input"], Cell[BoxData[ \({{2, 3}, {7, 10}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(1/\((1 + 1/\((2 + 1/3)\))\)\)], "Input"], Cell[BoxData[ \(7\/10\)], "Output"] }, Open ]], Cell[BoxData[ \(vnorm\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(vinterval[{2}]\)], "Input"], Cell[BoxData[ \({56\/153, 41\/97}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[%]\)], "Input"], Cell[BoxData[ \({0.3660130718954248`, 0.422680412371134`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(vinterval[{1, 2}]\)], "Input"], Cell[BoxData[ \({97\/138, 153\/209}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[%]\)], "Input"], Cell[BoxData[ \({0.7028985507246377`, 0.7320574162679426`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(firstvee[10]\)], "Input"], Cell[BoxData[ \({5, 8, {1, 1, 1, 1, 1, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(firstvee[10]\)], "Input"], Cell[BoxData[ \({5, 8, {1, 1, 1, 1, 1, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nextv[%, 10]\)], "Input"], Cell[BoxData[ InterpretationBox[\("nextv has been called on inputs"\[InvisibleSpace]{5, 8, {1, 1, 1, 1, 1, 1}}\[InvisibleSpace]","\[InvisibleSpace]10\), SequenceForm[ "nextv has been called on inputs", {5, 8, {1, 1, 1, 1, 1, 1}}, ",", 10], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{3, 5, {1, 1, 1, 1, 1}}\), SequenceForm[ "{qm,q,v}=", {3, 5, {1, 1, 1, 1, 1}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("{qm,q,v}="\[InvisibleSpace]{3, 8, {1, 1, 1, 1, 2}}\), SequenceForm[ "{qm,q,v}=", {3, 8, {1, 1, 1, 1, 2}}], Editable->False]], "Print"], Cell[BoxData[ \({3, 8, {1, 1, 1, 1, 2}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(good[73, 100]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ContinuedFraction[73/100]\)], "Input"], Cell[BoxData[ \({0, 1, 2, 1, 2, 2, 1, 2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(good[7, 11]\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(good[8, 13]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(zarembacheck[104]\)], "Input"], Cell[BoxData[ \({75, 104, True}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(basiczarembacheck[200]\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(basiczarembacheck[350]\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(basiczarembacheck[481]\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[basiczarembacheck[n], {n, 100, 200}]\)], "Input"], Cell[BoxData[ \({{73, 100, True}, {71, 101, True}, False, False, {75, 104, True}, {74, 105, True}, {75, 106, True}, False, {79, 108, True}, {45, 109, True}, False, {41, 111, True}, {41, 112, True}, False, False, {81, 115, True}, {45, 116, True}, {43, 117, True}, False, {44, 119, True}, False, {46, 121, True}, False, {89, 123, True}, False, False, False, False, {47, 128, True}, {49, 129, True}, False, {50, 131, True}, False, False, False, False, False, False, False, False, False, False, 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