With the code of Assignment4.hs we found out that the least composite number to fail Fermat's primality test as implemented with prime_tests_F is 4, which actually is the first composite number.
prime_tests_F uses a random a as base in the formula a^(n-1) mod n. We concluded that this test must always fail when n > 1 and a = 1. Then 1^x will always result in 1 and thus 1 mod n will result in 1.
With the composite number 4 the value of a will be in the range [1..3], leaving a chance of 1 out of 3 that the Fermat test fails.
In our code we use minimumFailingFermat k n to test the Fermat test on composite numbers with a certain k and finding the minimal value after n executions. When we increase n it is more probable that the resulting value is 4.
From this we can also conclude that the chance of a composite number to fail is:
|∀x in N (0 < x < n ^ x^(n-1) mod n == 1)| / (n-1)
Fermat test is likely to fail on composite numbers due to a base value a = 1 (especially on low composites). When k is increased it is less likely to pass the Fermat test with a composite, but still not impossible due do the randomness of a.