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Fixed a few errors caught by Pierpaolo.

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This commit is contained in:
Jens Axel Søgaard 2013-12-08 22:37:45 +01:00 committed by Vincent St-Amour
parent 0909acc473
commit 6c8eddb6a5
1 changed files with 8 additions and 10 deletions
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18
pkgs/math-pkgs/math-doc/math/scribblings/math-number-theory.scrbl
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@ -77,7 +77,7 @@
@margin-note{Wikipedia: @hyperlink["http://en.wikipedia.org/wiki/Pairwise_coprime"]{Pairwise Coprime}}
@defproc[(pairwise-coprime? [a Integer] [b Integer] ...) Boolean]{
Returns @racket[#t] if the integers @racket[a b ...] are @italic{pairwise} coprime, meaning
that each adjacent pair of integers is coprime.
that each pair of integers is coprime.
The numbers 2, 6 and 15 are coprime, but not @italic{pairwise} coprime, because 2 and 6 share the
factor 3:
@ -521,7 +521,7 @@ square, otherwise @racket[#f].
The functions in this section are @deftech{multiplicative}.
In number theory, a multiplicative function is a function @racket[f] such that
@racket[(f a b) = (* (f a) (f b))] for all coprime natural numbers @racket[a] and @racket[b].
@racket[(f (* a b)) = (* (f a) (f b))] for all coprime natural numbers @racket[a] and @racket[b].
@margin-note{Wikipedia: @hyperlink["http://en.wikipedia.org/wiki/Euler%27s_totient_function"]{Euler's Totient}}
@defproc[(totient [n Natural]) Natural]{
@ -538,8 +538,8 @@ This function is known as Eulers totient or phi function.
@margin-note{Wikipedia: @hyperlink["http://en.wikipedia.org/wiki/M%C3%B6bius_function"]{Moebius Function}}
@defproc[(moebius-mu [n Natural]) (U -1 0 1)]{
Returns:
@itemlist[@item{@racket[1] if @racket[n] is a product of an even number of primes}
@item{@racket[-1] if @racket[n] is a product of an odd number of primes}
@itemlist[@item{@racket[1] if @racket[n] is a square-free product of an even number of primes}
@item{@racket[-1] if @racket[n] is a square-free product of an odd number of primes}
@item{@racket[0] if @racket[n] has a multiple prime factor}]
@interaction[#:eval untyped-eval
@ -563,8 +563,6 @@ all divisors of @racket[n].
@defproc[(prime-omega [n Natural]) natural?]{
Counting multiplicities the number of prime factors of @racket[n] is returned.
Note: The function @racket[prime-omega] is multiplicative.
@interaction[#:eval untyped-eval
(prime-omega (* 2 2 2 3 3 5))]
}
@ -815,9 +813,9 @@ 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}.
A generator the group @math-style{Un} is called a @emph{primitive root} modulo @racket[n].
Note that @math-style{g} is a primitive root if and only if @math-style{order(g)=totient(n)}.
A group with a generator is called @emph{cyclic}.
@defproc[(unit-group [n Integer]) (Listof Positive-Integer)]{
@ -836,7 +834,7 @@ If @racket[x] and @racket[n] are not coprime, @racket[(unit-group-order x n)] ra
(unit-group-order 2 6)]
}
@defproc[(unit-group-orders [n Integer]) (Listf Positive-Integer)]{
@defproc[(unit-group-orders [n Integer]) (Listof Positive-Integer)]{
Returns a list @racket[(list (unit-group-order x0 n) (unit-group-order x1 n) ...)] where
@racket[x0], @racket[x1], ... are the elements of @math-style{Un}.
The modulus @racket[n] must be positive.