1e52736089
Fixes after merge weirdness from pull request (specifically, removed `bfrandom' from "mpfr.rkt" again) Removed dependence of math/flonum on math/bigfloat (better build parallelization) Changed `divides?' to return #f when its first argument is 0 Made return type of `quadratic-character' more precise Made argument types more permissive: * second argument to `solve-chinese' * second argument to `next-primes' * second argument to `prev-primes'
40 lines
1.5 KiB
Racket
40 lines
1.5 KiB
Racket
#lang typed/racket/base
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(require racket/list
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"divisibility.rkt"
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"modular-arithmetic.rkt"
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"number-theory.rkt")
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(provide quadratic-character quadratic-residue?)
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; DEFINITION (Quadratic residue)
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; a in Un is a quadratic residue,
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; if there exists an s such that a=s^2 (mod n)
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; The number s is called a squre root of a modulo n.
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; p is prime
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(: quadratic-character : Integer Integer -> (U -1 0 1))
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(define (quadratic-character a p)
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(cond [(a . < . 0) (raise-argument-error 'quadratic-character "Natural" 0 a p)]
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[(p . <= . 0) (raise-argument-error 'quadratic-character "Positive-Integer" 1 a p)]
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[else (let ([l (modular-expt a (quotient (- p 1) 2) p)])
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(cond [(or (eqv? l 0) (eqv? l 1)) l]
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[else -1]))]))
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(: quadratic-residue? : Integer Integer -> Boolean)
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(define (quadratic-residue? a n)
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(cond [(a . < . 0) (raise-argument-error 'quadratic-residue? "Natural" 0 a n)]
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[(n . <= . 0) (raise-argument-error 'quadratic-residue? "Positive-Integer" 1 a n)]
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[else
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(let* ([ps (prime-divisors n)]
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[odd-ps (if (= (first ps) 2)
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(rest ps)
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ps)])
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(and (andmap (λ: ([p : Natural])
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(= (quadratic-character a p) 1))
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odd-ps)
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(cond
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[(divides? 8 n) (= (modulo a 8) 1)]
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[(divides? 4 n) (= (modulo a 4) 1)]
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[else #t])))]))
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