N+1 Primality Prover

February 7, 2014

We use the lucas and selfridge functions from the previous exercise:

(define (nplus1-prime? n)
  (let-values (((d p q) (selfridge n)))
    (if (< 1 (gcd d n)) #f
      (if (positive? (u p q n (+ n 1))) #f
        (let f-loop ((fs (facts (+ n 1))))
          (if (null? fs) #t
            (let d-loop ((p p) (q q))
              (if (= (gcd (u p q n (/ (+ n 1) (car fs))) n) 1)
                  (f-loop (cdr fs))
                  (d-loop (+ p 2) (+ p q 1))))))))))

Our facts function finds some of the factors of n using trial division, stopping when the factored portion is greater than the remaining unfactored portion. Here, f is the current trial factor, fs is the current list of factors, without duplicates, fp is the current product of the factors, and r is the remaining unfactored portion of n:

(define (facts n)
  (let loop ((f 2) (fs (list)) (fp 1) (r n))
    (cond ((< n (* fp fp)) (reverse fs))
          ((< r (* f f)) (reverse (cons r fs)))
          ((zero? (modulo r f))
            (loop f (if (member f fs) fs (cons f fs))
                  (* fp f) (/ r f)))
          (else (loop (+ f 1) fs fp r)))))

Facts can be made to run faster, but if you use a method other than trial division, you will have to recursively prove prime all of factors that you find.

Here are some examples:

> (nplus1-prime? (+ #e1e12 39))
#t
> (nplus1-prime? (+ #e1e12 61))
#t

You can run the program at http://programmingpraxis.codepad.org/J9oc7Q3B.

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