#lang at-exp scheme
(require (planet soegaard/infix))
#|
Each new term in the Fibonacci sequence is generated by adding the 
previous two terms. By starting with 1 and 2, the first 10 terms will be:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Find the sum of all the even-valued terms in the sequence which do not 
exceed four million.
|#

(require (planet "while.scm" ("soegaard" "control.plt" 2 0))) ; while
(define-values (f g t) (values 1 2 0))
(define sum f)
@${ 
while[ g< 4000000,
  when[ even?[g], sum:=sum+g];
  t := f + g;
  f := g;
  g := t];
sum}


#|
The sum of the squares of the first ten natural numbers is,
    1^2 + 2^2 + ... + 10^2 = 385
The square of the sum of the first ten natural numbers is,
   (1 + 2 + ... + 10)^2 = 552 = 3025
Hence the difference between the sum of the squares of the first ten natural 
numbers and the square of the sum is 3025 - 385 = 2640.

Find the difference between the sum of the squares of the first one hundred 
natural numbers and the square of the sum.|#

(define n 0)
(define ns 0)
(define squares 0)
@${
sum:=0;
while[ n<100,
  n := n+1;
  ns := ns+n;
  squares := squares + n^2];
ns^2-squares
}



#|
A Pythagorean triplet is a set of three natural numbers, a,b,c for which,
   a^2 + b^2 = c^2
For example, 3^2 + 4^2 = 9 + 16 = 25 = 5^2.

There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.
|#

(let-values ([(a b c) (values 0 0 0)])
  (let/cc return
    (for ([k (in-range 1 100)])
      (for ([m (in-range 2 1000)])
        (for ([n (in-range 1 m)])
          @${ a := k* 2*m*n;
              b := k* (m^2 - n^2);
              c := k* (m^2 + n^2);
              when[ a+b+c = 1000, 
                 display[{{k,m,n}, {a,b,c}}]; 
                 newline[];
                 return[a*b*c] ]})))))


#| Primality testing |#

(define (factor2 n)
  ; return r and s, s.t n = 2^r * s where s odd
  ; invariant: n = 2^r * s
  (let loop ([r 0] [s n])
    (let-values ([(q r) (quotient/remainder s 2)])
      (if (zero? r)
          (loop (+ r 1) q)
          (values r s)))))

(require srfi/27) ; random-integer

(define (miller-rabin n)
  ; Input: n odd
  (define (mod x) (modulo x n))
  (define (expt x m)
    (cond [(zero? m) 1]
          [(even? m) @${mod[sqr[x^(m/2)] ]}]
          [(odd? m)  @${mod[x*x^(m-1)]}]))
  (define (check? a)
    (let-values ([(r s) (factor2 (sub1 n))])
      ; is a^s congruent to 1 or -1 modulo n ?
      (and @${member[a^s,{1,mod[-1]}]} #t)))
  (andmap check? 
          (build-list 50 (λ (_) (+ 2 (random-integer (- n 3)))))))

(define (prime? n)
  (cond [(< n 2) #f]
        [(= n 2) #t]
        [(even? n) #f]
        [else (miller-rabin n)]))

(prime? @${2^89-1})