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Notebook[{

Cell[CellGroupData[{
Cell["The Euler \[Phi] function", "Subtitle"],

Cell["\<\
I. Cnop
Vrije Universiteit Brussel
icnop@vub.ac.be\
\>", "Subsubtitle"],

Cell[CellGroupData[{

Cell["Introduction", "Section"],

Cell[TextData[{
  "The Euler Phi function counts for each natural number   n  the number of \
natural numbers relatively prime and smaller  than  n  .  \nThis important \
function from number theory is included in the standard  ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " commands."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(EulerPhi[20]\)], "Input"],

Cell[BoxData[
    \(8\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "In order to get more insight we want to program our own. \nIt shows \
programming style in  ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " and therefore is a useful exercise. Programming will involve lists such \
as"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Range[10]\)], "Input"],

Cell[BoxData[
    \({1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\)], "Output"]
}, Open  ]],

Cell["and operations on them: making their product", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(\ \)\(Times @@ %\)\)\)], "Input"],

Cell[BoxData[
    \(3628800\)], "Output"]
}, Open  ]],

Cell["or generating a set of special fractions", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Map[\((1 - 1/#)\) &, Range[10]]\)], "Input"],

Cell[BoxData[
    \({0, 1\/2, 2\/3, 3\/4, 4\/5, 5\/6, 6\/7, 7\/8, 8\/9, 9\/10}\)], "Output"]
}, Open  ]],

Cell["\<\
As a bonus we will be able to obtain a symbolic result by our \
programming.\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Writing our own program", "Section"],

Cell[CellGroupData[{

Cell["Start by reasoning from an example", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(n = 30\)], "Input"],

Cell[BoxData[
    \(30\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(FactorInteger[n]\)], "Input"],

Cell[BoxData[
    \({{2, 1}, {3, 1}, {5, 1}}\)], "Output"]
}, Open  ]],

Cell["\<\
Being relatively prime only depends on (not) having a prime factor \
in common with  n . \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(primes[n_] := First[Transpose[FactorInteger[n]]]\)], "Input"],

Cell[BoxData[
    \(General::"spell1" \(\(:\)\(\ \)\) 
      "Possible spelling error: new symbol name \"\!\(primes\)\" is similar \
to existing symbol \"\!\(Primes\)\"."\)], "Message"]
}, Open  ]],

Cell[BoxData[
    \(relPrime[k_] := \ GCD[k, n] == 1\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Select[Range[n], relPrime]\)], "Input"],

Cell[BoxData[
    \({1, 7, 11, 13, 17, 19, 23, 29}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(commonPrime[k_] := \ GCD[k, n] != 1\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Select[Range[n], commonPrime]\)], "Input"],

Cell[BoxData[
    \({2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 
      27, 28, 30}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "We write out all prime factor of  n  and for each prime factor  p  , we \
delete all multiples  j p   of this prime factor. \nThe amount of numbers we \
delete for  p  is  ",
  Cell[BoxData[
      \(n\/p\)]],
  " .\nTaking into account doubles (multiples of two different   p ), triples \
(multiples of three different   p ), ....  , we obtain by the \
inclusion-exclusion principle a formula where the ratio of not deleted \
numbers for all  p  is  the product of all  (1 - ",
  Cell[BoxData[
      \(1\/p\)]],
  "). \n[If you do not know this inclusion-exclusion principle, do this by \
reasoning on sets of multiples obtained by] "
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Table[\(primes[n]\)[\([j]\)]\ Range[n/\(primes[n]\)[\([j]\)]], {j, 
          Length[primes[n]]}] // TableForm\)], "Input"],

Cell[BoxData[
    InterpretationBox[GridBox[{
          {"2", "4", "6", "8", "10", "12", "14", "16", "18", "20", "22", "24",
             "26", "28", "30"},
          {"3", "6", "9", "12", "15", "18", "21", "24", "27", 
            "30", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \
"\<\"\"\>"},
          {"5", "10", "15", "20", "25", 
            "30", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \
"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}
          },
        RowSpacings->1,
        ColumnSpacings->3,
        RowAlignments->Baseline,
        ColumnAlignments->{Left}],
      TableForm[ {{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}, {
        3, 6, 9, 12, 15, 18, 21, 24, 27, 30}, {5, 10, 15, 20, 25, 
        30}}]]], "Output"]
}, Open  ]],

Cell["So finally:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FactorInteger[n]\)], "Input"],

Cell[BoxData[
    \({{2, 1}, {3, 1}, {5, 1}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(First[Transpose[%]]\)], "Input"],

Cell[BoxData[
    \({2, 3, 5}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\((1 - 1/#)\) &\)[%]\)], "Input"],

Cell[BoxData[
    \({1\/2, 2\/3, 4\/5}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(n\ Times @@ %\)], "Input"],

Cell[BoxData[
    \(8\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Collect these last lines into a program", "Subsection"],

Cell[BoxData[{
    \(\(primes[n_] := 
        First[Transpose[FactorInteger[n]]];\)\), "\[IndentingNewLine]", 
    \(\(mult[n_] := 
        Map[\((1 - 1/#)\) &, primes[n]];\)\), "\[IndentingNewLine]", 
    \(ourPhi[n_] := n\ Times @@ mult[n]\)}], "Input"],

Cell[CellGroupData[{

Cell["Testing", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(n = 96\)], "Input"],

Cell[BoxData[
    \(96\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(primes[n]\)], "Input"],

Cell[BoxData[
    \({2, 3}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(mult[n]\)], "Input"],

Cell[BoxData[
    \({1\/2, 2\/3}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(ourPhi[n]\)], "Input"],

Cell[BoxData[
    \(32\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(EulerPhi[n]\)], "Input"],

Cell[BoxData[
    \(32\)], "Output"]
}, Open  ]]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["The case  n  is prime", "Subsection"],

Cell["\<\
Question: what is the  \[Phi](n) for a prime number n?\
\>", "Text",
  FontWeight->"Bold"]
}, Closed]],

Cell[CellGroupData[{

Cell["The case  n   is a product of different primes", "Subsection"],

Cell["First answer the following questions:", "Text"],

Cell["\<\
Question: what is the \[Phi](n) for   n = p q    (both prime \
numbers)?\
\>", "Text",
  FontWeight->"Bold"],

Cell["\<\
Question: what is the  \[Phi](n) for   n = p q r   (all prime \
numbers)?\
\>", "Text",
  FontWeight->"Bold"],

Cell[CellGroupData[{

Cell["This can be handled symbolically too:", "Subsubsection"],

Cell["\<\
If we tell the computer what are the different primes involved - \
how could it know the factorisation of an unspecified number\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(primes = {p, q, r}\)], "Input"],

Cell[BoxData[
    \({p, q, r}\)], "Output"]
}, Open  ]],

Cell["and tell it   n   is their product", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(n = \ Times @@ %\)], "Input"],

Cell[BoxData[
    \(p\ q\ r\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Map[\((1 - 1/#)\) &, primes]\)], "Input"],

Cell[BoxData[
    \({1 - 1\/p, 1 - 1\/q, 1 - 1\/r}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(n\ Times @@ %\)], "Input"],

Cell[BoxData[
    \(\((1 - 1\/p)\)\ p\ \((1 - 1\/q)\)\ q\ \((1 - 1\/r)\)\ r\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Expand[Together[%]]\)], "Input"],

Cell[BoxData[
    \(\(-1\) + p + q - p\ q + r - p\ r - q\ r + p\ q\ r\)], "Output"]
}, Open  ]],

Cell["A correct answer!", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Questions", "Subsubsection",
  CellDingbat->None],

Cell[TextData[{
  "Try this for two and four primes as well.\nNext try it for   n =",
  " ",
  Cell[BoxData[
      \(p\^3\ q\^2\ r\ \ and\ check\ your\ result\ for\ \ \ n\  = \ \ 360\)],
    FontFamily->"Times"],
  "."
}], "Text"]
}, Closed]]
}, Closed]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Other issues", "Section"],

Cell[CellGroupData[{

Cell["Plotting  \[Phi]", "Subsection"],

Cell["\<\
Plotting  \[Phi]   for related numbers also yields some unexpected \
results. Perform\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(ListPlot[Table[ourPhi[k], {k, 2, 120}]]\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
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