/primality.rkt

(require racket/include)
(include "small-primes.rkt")

(define (pow x y) (cond
                    ((= y 0) 1)
                    ((= y 1) x)
                    (else (* x (pow x (- y 1))))))
(define (member? x l)
  (cond
    ((null? l) #f)
    ((equal? (car l) x) #t)
    (else (member? x (cdr l)))
    ))
(define (l-product a)
  (list-product-helper a 0))
(define (list-product-helper a r)
  (cond
    ((null? a) r)
    (else (list-product-helper (cdr a) (* (car a) r)))
    ))

(define (prime? n k)
  (cond
    ((< n 5) (member? n '(2 3)))
    ((= (remainder n 2) 0) #f)
    (else
     (let ((ch (call-helper n)))
       (prime-tester k (+ (random (- n 4)) 2) n (cadr (call-helper n)) (car (call-helper n)))))
    ))
(define (call-helper n)
  (prime-helper 0 (- n 1)))
  
(define (prime-helper s d)
  (cond
    ((= (remainder d 2) 0) (prime-helper  (+ s 1) (/ d 2)))
    (else (list s d))
   ))
(define (not-witness a d n s)
  (cond
    ((= (remainder (pow a d) n) 1) #t)
    (else (not-witness-helper s (remainder (pow a d) n) n))
   ))
(define (not-witness-helper r x n)
  (cond
    ((= r 0) #f)
    ((= x (- n 1)) #t)
    (else (not-witness-helper (- r 1) (remainder (pow x 2) n) n))
    ))
(define (prime-tester k a n d s)
  (cond 
    ((> (gcd (+ 2 (random (- n 4))) n) 1) #f)
    ((not (not-witness a d n s)) #f)
    ((= k 0) #t)
    (else (prime-tester (- k 1) a n d s))
    ))
(define (gen-prime-helper p)
  (let ((k (random (/(- p 1) 2))))
    (cond
      ((prime? (+ (* 2 k p) 1) 50) (+ (* 2 k p) 1))
      (else (gen-prime-helper p))
      )))
(define (gen-prime p n)
  (cond
    ((= n 0) p)
    (else (gen-prime (gen-prime-helper p) (- n 1)))
    ))