Tuesday's Solutions
--------------------

Exercise 1:

(define beautiful-rectangle
  (lambda (width)
    (* width (/ (+ (sqrt 5) 1) 2))))

Exercise 2:
plan: check numbers starting with 1 to see if their square is n.  If
      the square ever exceeds n, know n is not a square.

(define check-squares
  (lambda (n s)
    (if (= (* s s) n)
        s
        (if (> (* s s) n)
            #f
            (check-squares n (+ s 1))))))

(define is-square?
  (lambda (n)
    (check-squares n 1)))

Exercise 3:
plan: use prime? from before

(define remainder
  (lambda (x y)
    (if (< x y)
        x
        (remainder (- x y) y))))

(define divisible?
  (lambda (x y)
    (= (remainder x y) 0)))

(define smallest-factor
  (lambda (n f)
    (if (>= f n)      ;No factors larger than n (there's a better bound)
        #f
        (if (divisible? n f)
            f
            (smallest-factor n (+ f 1))))))

(define prime?
  (lambda (n)
    (not (smallest-factor n 2))))

(define nth-prime-bigger-than
  (lambda (n p)
    (if (= n 0)
        p
        (if (prime? (+ p 1))
            (nth-prime-bigger-than (- n 1) (+ p 1))
            (nth-prime-bigger-than n (+ p 1))))))

(define nth-prime
  (lambda (n)
    (nth-prime-bigger-than n 2)))