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Documentation for order, orders, primitive-root, exists-primitive-root?, primitive-roots

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Jens Axel Søgaard 2012-10-16 19:17:41 +02:00 committed by Neil Toronto
parent 0af1cb034e
commit 2178282fd6
1 changed files with 82 additions and 1 deletions
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83
collects/math/scribblings/math-number-theory.scrbl
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@ -633,4 +633,85 @@ Returns a list of all natural solutions to the equation @math-style{a x^2 + b x
(quadratic-natural-solutions 1 0 -2)]
}
@(close-eval untyped-eval)
@; ----------------------------------------
@section[#:tag "primitive_roots"]{The group Zn and Primitive Roots}
The numbers @math-style{0, 1, ..., n-1} with addition and multiplication
modulo @racket[n] is a ring called @math-style{Zn}.
The group of units in @math-style{Zn} with respect to multiplication
modulo @racket[n] is called @math-style{Un}.
The order of an element @math-style{x} in @math-style{Un}
is the least @math-style{k>0} such that @math-style{x^k=1 mod n}.
A generator the group @math-style{Un} is called a @emph{primitive root} modolo @racket[n].
Note that @math-style{g} is a primitive root if and only if @math-style{order(g)=phi(n)},
where @math-style{phi} is Eulers totient. A group with a generator is called @emph{cyclic}.
@defproc[(unit-group [n Natrual]) (listof Natural)]{
Returns a list of all elements of @math-style{Un}, the unit group modulo @racket[n].
@interaction[(require math)
(unit-group 5)
(unit-group 6)]
}
@defproc[(order [x Natural] [n Natural]) natural?]{
Returns the order of @racket[x] in the group @math-style{Un}.
Note: @racket[x] is in @math-style{Un} if and only if @racket[(gcd x n)] is 1.
@interaction[(require math)
(order 2 5)
(order 2 6)]
}
@defproc[(orders [n Natural]) (listof natural?)]{
Returns a list @racket[(list (order x0) (order x1) ...)] where
@racket[x0], @racket[x1], ... are the elements of @math-style{Un}.
@interaction[(require math racket)
(orders 5)
(map (curryr order 5) (unit-group 5))]
}
@defproc[(primitive-root? [x Natural] [n Natural]) boolean?]{
Returns @racket[#t] if the element @racket[x] in @math-style{Un} is a primitive root modulo @racket[n],
otherwise @racket[#f] is returned. An error is signaled if @racket[x] is not a member of @math-style{Un}.
@interaction[(require math)
(primitive-root? 1 5)
(primitive-root? 2 5)
(primitive-root? 5 5)]
}
@defproc[(exists-primitive-root? [n Natural]) boolean?]{
Returns @racket[#t] if the group @math-style{Un} has a primitive root (i.e. it is cyclic),
otherwise @racket[#f] is returned.
In other words, @racket[#t] is returned if @racket[n] is one of
@math-style{1, 2, 4, p^e, 2*p^e} where @math-style{p} is an odd prime,
and @racket[#f] otherwise.
@interaction[(require math)
(exists-primitive-root? 5)
(exists-primitive-root? 6)
(exists-primitive-root? 12)]
}
@defproc[(primitive-root [n Natural]) (Union natural? #f)]{
Returns a primitive root of @math-style{Un} if one exists,
otherwise @racket[#f] is returned.
@interaction[(require math)
(primitive-root 5)
(primitive-root 6)]
}
@defproc[(primitive-roots [n Natural]) (Listof natural?)]{
Returns a list of all primitive roots of @math-style{Un}.
@interaction[(require math)
(primitive-roots 3)
(primitive-roots 5)
(primitive-roots 6)]
}
@(close-eval untyped-eval)