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Cell[BoxData[
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Cell[CellGroupData[{

Cell["Miller-Rabin-Selfridge Probabilistic Primality Test", "Title"],

Cell[TextData[{
  "The Miller-Rabin-Selfridge test is described in the following steps:\n\n\
Given a natural number ",
  Cell[BoxData[
      \(TraditionalForm\`n = 1 + \(2\^t\) m\)]],
  " where ",
  StyleBox["m",
    FontSlant->"Italic"],
  " is and odd natural number and ",
  StyleBox["t",
    FontSlant->"Italic"],
  "\[Element]\[DoubleStruckCapitalN], and a parameter r\[Element]\
\[DoubleStruckCapitalN],\n\n(1) Choose a random integer ",
  StyleBox["a",
    FontSlant->"Italic"],
  " with 2\[LessEqual]",
  StyleBox["a",
    FontSlant->"Italic"],
  "\[LessEqual]",
  StyleBox["n",
    FontSlant->"Italic"],
  "-2.\n(2) If ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(a\^m\)\(\[Congruent]\)\)\)]],
  "1(mod ",
  StyleBox["n",
    FontSlant->"Italic"],
  ") then return to step (1).\n(3) Compute ",
  Cell[BoxData[
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  " for 0\[LessEqual]",
  StyleBox["j",
    FontSlant->"Italic"],
  "\[LessEqual]",
  StyleBox["t",
    FontSlant->"Italic"],
  "-1. If none of the congruences equals -1, ",
  "then ",
  StyleBox["n",
    FontSlant->"Italic"],
  " is composite.\n(4) Decrement ",
  StyleBox["r",
    FontSlant->"Italic"],
  " and return to (1)."
}], "Text"],

Cell[CellGroupData[{

Cell["Example 1", "Section"],

Cell["\<\
The integer 3539 is prime so it should pass the \
Miller-Rabin-Selfridge test.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(PrimeQ[n = 3539]\)], "Input"],

Cell[BoxData[
    \(True\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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    While[EvenQ[m], m = m/2; t = t + 1]; {t, m}\)], "Input"],

Cell[BoxData[
    \({1, 1769}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(n \[Equal] 1 + m*2^t\)], "Input"],

Cell[BoxData[
    \(True\)], "Output"]
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Cell[CellGroupData[{

Cell["Step (1)", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(a = Random[Integer, {2, n - 2}]\)], "Input"],

Cell[BoxData[
    \(3206\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Step (2)", "Subsection"],

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Cell[BoxData[
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Cell[BoxData[
    \(1\)], "Output"]
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Cell["Step (3)", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    \({1}\)], "Output"]
}, Open  ]],

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  "Since this list contains no element congruent to 1 modulo ",
  StyleBox["n",
    FontSlant->"Italic"],
  ", then ",
  StyleBox["n",
    FontSlant->"Italic"],
  " is probably prime (in fact it is prime)."
}], "Text"]
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Cell[CellGroupData[{

Cell["Example 2", "Section"],

Cell["\<\
The integer 11137 is composite so it may fail the \
Miller-Rabin-Selfridge test.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(PrimeQ[n = 11137]\)], "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[BoxData[
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Cell["Step (1)", "Subsection"],

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  StyleBox["n",
    FontSlant->"Italic"],
  ", then ",
  StyleBox["n",
    FontSlant->"Italic"],
  " is composite."
}], "Text"]
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Cell["Example 3", "Section"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell["Example 4", "Section"],

Cell["\<\
The integer 48613 is composite. Can the Miller-Rabin-Selfridge test \
declare it to be prime?\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(PrimeQ[48613]\)], "Input"],

Cell[BoxData[
    \(False\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

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Cell[BoxData[
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      Editable->False]], "Print"],

Cell[BoxData[
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Cell[BoxData[
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      Editable->False]], "Print"],

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      Editable->False]], "Print"],

Cell[BoxData[
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      SequenceForm[ "Step 2: ", 40613],
      Editable->False]], "Print"],

Cell[BoxData[
    InterpretationBox[\("Step 3: "\[InvisibleSpace]{40613, 25292}\),
      SequenceForm[ "Step 3: ", {40613, 25292}],
      Editable->False]], "Print"],

Cell[BoxData[
    \({"Composite", {12153, 2}, {11087, 11548, 24240, 6944, 
        6947}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(MillerRabinSelfridge[48613, {17, 19, 23}, 
      Verbose \[Rule] True]\)], "Input"],

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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      Editable->False]], "Print"],

Cell[BoxData[
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      Editable->False]], "Print"],

Cell[BoxData[
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      Editable->False]], "Print"],

Cell[BoxData[
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Cell[BoxData[
    \({"Composite", {12153, 2}}\)], "Output"]
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}, Closed]],

Cell[CellGroupData[{

Cell["Example 5", "Section"],

Cell["\<\
Let's see if we can find some composite numbers which pass the \
Miller-Rabin-Selfridge test.\
\>", "Text"],

Cell[BoxData[
    \(\(facts = Prime[Range[200000, 200099]];\)\)], "Input"],

Cell[BoxData[
    \(\(n = Union[Flatten[Outer[Times, facts, facts]]];\)\)], "Input"],

Cell[BoxData[
    \(\(data = Map[MillerRabinSelfridge[#, 1] &, n];\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]]
}, Closed]]
}, Open  ]]
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