Ferma Test for determining prime numbers

; Fermat's little theorem:

; If n is a prime number and A is any positive integer less than N,

; Then a raised to the n-th power is congruent to a modulo n


; Two numbers are said to be congruent modulo n if they both
; Have the same remainder when divided by N


; The Fermat test:
; Given a number N, pick a random number A <n
; And compute the remainder of a ^ n modulo n.
; If the result is not equal to A, then n is certainly not prime
; If it is a, then chances are good that N is prime.
; Now pick anthor random number A and test it with the same method.
; If it also satisfies the equation,
; Then we can be even more confident that N is prime.
; By trying more and more values of,
; We can increase our confidence in the result.
; This algorithm is known as the Fermat test.


(Define (Bad-exp base exp)

(Cond (= exp 0) 1)

(Even? Exp)

(Square (Bad-exp base (/Exp 2 ))))

(Else (* base (Bad-exp base (-Exp 1 ))))))

(Define (Bad-expmod base exp m)

(Remainder (Bad-exp base exp) m ))

(Define (expmod base exp m)

(Cond (= exp 0) 1)

(Even? Exp)

(Remainder

(Square (expmod base (/Exp 2) M ))M ))

(Else

(Remainder

(* Base (expmod base (-Exp 1) M ))M ))))

; The first process is slower than the second process, and a value that exceeds a certain length cannot be calculated,
; Because the first is to calculate the factorial first and then take the modulo, and the second is to take the modulo in the factorial process.
; 26 mod 4 and (2 mod 4) * (13 mod 4) have the same results
; (Expmod 32 10911110033 10911110033)
; (My-expmod 32 1091111003 1091111003)


(Define (Fermat-Test N)

(Define (try-It)

(= (Expmod a n) ))

(Try-It (+ 1 (random (-N 1 )))))

(Define (fast-Prime? N times)

(Cond (= times 0) True)

(Fermat-Test N)

(Fast-Prime? N (-times 1 )))

(Else false )))

; The Fermat test differs in character from most familiar algorithms,
; In which one computes an answer that is guaranteed to be correct.
; Here, the answer obtained is only probably correct.
; More precisely, if n ever fails the Fermat test,
; We can be certain that N is not prime.
; But the fact that N passes the test,
; While an extremely strong indication,
; Is still not a guarantee that N is prime.
; What we wowould like to say is that for any number N,
; If we perform the test enough times and find that n always passes the test,
; Then the probability of error in our primality test can be made as small as we like.


; Unfortunately, this assertion is not quite correct.
; There do exist numbers that fool the Fermat test:
; Numbers n that are not prime and yet have the property that
; An is congruent to a modulo n for All integers A <n.
; Such numbers are extremely rare,
; So the Fermat test is quite reliable in practice


; There are variations of the Fermat test that cannot be fooled.
; In these tests, as with the Fermat method,
; One tests the primality of an integer n
; By choosing a random integer a <n and
; Checking some condition that depends upon N and.


; On the other hand, in contrast to the Fermat test,
; One can prove that, for any n,
; The condition does not hold for most of the integers A <n
; Unless n is prime.
; Thus, if n passes the test for some random choice of,
; The chances are better than even that N is prime.
; If n passes the test for two random choices of,
; The chances are better than 3 out of 4 that N is prime.
; By running the test with more and more randomly chosen
; Values of a we can make the probability of error as small as we like.
; The existence of tests for which one can prove that
; The chance of error becomes arbitrarily small has sparked interest in
; Algorithms of this type,
; Which have come to be known as Probabilistic algorithms.
; There is a great deal of research activity in this area,
; And probabilistic algorithms have been fruitfully applied to define Fields