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Documentation of tangent-number, quadratic-solutions, quadratic-residue and quadratic-character

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Jens Axel Søgaard 2012-10-16 17:48:30 +02:00 committed by Neil Toronto
parent 63b5f47989
commit 0af1cb034e
1 changed files with 71 additions and 11 deletions
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82
collects/math/scribblings/math-number-theory.scrbl
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@ -20,7 +20,7 @@
(parameterize ([sandbox-output 'string]
[sandbox-error-output 'string])
(make-evaluator 'racket)))
;(interaction-eval #:eval math-eval (require racket math))
@;(interaction-eval #:eval math-eval (require racket math))
@title[#:tag "number-theory" #:style '(toc)]{Number Theory}
@ -59,10 +59,6 @@
@; http://en.wikipedia.org/wiki/B%C3%A9zout's_identity
}
@defproc[(bezout-binary [a Integer] [b Integer]) (Listof Integer)]{
Same as @racket[bezout] but restricted to two arguments.
}
@defproc[(coprime? [a Integer] [b Integer] ...) boolean?]{
Returns @racket[#t] if the integers @racket[a],@racket[b],... are coprime.
@ -128,6 +124,35 @@
@; http://en.wikipedia.org/wiki/Chinese_remainder_theorem
}
@defproc[(quadratic-residue? [a natural?] [n natural?]) boolean?]{
Returns @racket[#t] if @racket[a] is a quadratic residue modulo @racket[n],
otherwise @racket[#f] is returned.
A number @racket[a] is a quadratic residue modulo @racket[n], if there
exists a number @racket[x] such that @math-style{x^2=a mod n}.
@interaction[(require math)
(quadratic-residue? 0 4)
(quadratic-residue? 1 4)
(quadratic-residue? 2 4)
(quadratic-residue? 3 4)]
}
@defproc[(quadratic-character [a natural?] [p prime?]) (union -1 1)]{
Returns the values of the quadratic character modulo the prime @racket[p].
That is, for a non-zero @racket[a] the number @racket[1] is returned when
@racket[a] is a quadratic residue,
and @racket[-1] is returned when @racket[a] is a non-residue.
If @racket[a] is zero, then @racket[0] is returned.
This function is also known as the @emph{Legendre Symbol}.
@interaction[(require math)
(quadratic-character 0 4)
(quadratic-character 1 4)
(quadratic-character 2 4)
(quadratic-character 3 4)]
}
@; ----------------------------------------
@section[#:tag "primes"]{Primes}
@ -448,8 +473,9 @@ Note: The function @racket[divisor-sum] is multiplicative.
@defproc[(fibonacci [n Natural]) natural?]{
Returns the @racket[n]th Fibonacci number.
Definition:
@url{http://en.wikipedia.org/wiki/Fibonacci_number}.
See
@url{http://en.wikipedia.org/wiki/Fibonacci_number}
for a definition.
The ten first Fibonacci numbers.
@interaction[(require math racket)
@ -469,14 +495,20 @@ Returns a list of the numbers in the @racket[n]th Farey sequence.
The @racket[n]th Farey sequence is the sequence of all
completely reduced rational numbers from 0 to 1 which denominators
are less than or equal to @racket[n].
@interaction[(require math racket)
@interaction[(require math)
(farey 1)
(farey 2)
(farey 3)]
}
@; ----------------------------------------
@section[#:tag "quadratic-residues"]{Quadratic Residues}
@defproc[(tangent-number [n integer?]) integer?]{
Returns the @racket[n]th tangent number.
See @url{http://mathworld.wolfram.com/TangentNumber.html} for a definition.
@interaction[(require math)
(tangent-number 1)
(tangent-number 2)
(tangent-number 3)]
}
@; ----------------------------------------
@ -563,6 +595,8 @@ type of polygonal number.
}
@; ----------------------------------------
@section[#:tag "fractions"]{Fractions}
@defproc[(mediant [x Rational] [y Rational]) rational?]{
Computes the @racket[mediant] of the numbers @racket[x] and @racket[y].
@ -572,5 +606,31 @@ lowest term is the number @math-style{(p+r)/(q+s)}.
(mediant 1/2 5/6)]
}
@(close-eval untyped-eval)
@; ----------------------------------------
@section[#:tag "quadratics"]{The Quadratic Equation}
@defproc[(quadratic-solutions [a Real] [b Real] [c Real]) (listof Real)]{
Returns a list of all real solutions to the equation @math-style{a x^2 + b x +c = 0}.
@interaction[(require math)
(quadratic-solutions 1 0 -1)
(quadratic-solutions 1 2 1)
(quadratic-solutions 1 0 1)]
}
@defproc[(quadratic-integer-solutions [a Real] [b Real] [c Real]) (listof Integer)]{
Returns a list of all integer solutions to the equation @math-style{a x^2 + b x +c = 0}.
@interaction[(require math)
(quadratic-integer-solutions 1 0 -1)
(quadratic-integer-solutions 1 0 -2)]
}
@defproc[(quadratic-natural-solutions [a Real] [b Real] [c Real]) (listof Natural)]{
Returns a list of all natural solutions to the equation @math-style{a x^2 + b x +c = 0}.
@interaction[(require math)
(quadratic-natural-solutions 1 0 -1)
(quadratic-natural-solutions 1 0 -2)]
}
@(close-eval untyped-eval)