Prime Numbers

166 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
3.19. [Lenstra, Granville] Show that if an odd number n be divisible by the
square of some prime, then W (n), the least witness for n, is less than ln 2 n.
(Hint: Use (1.45).) This exercise is re-visited in Exercise 4.28.
3.20. Describe a probabilistic algorithm that gives nontrivial factorizations
of Carmichael numbers in expected polynomial time.
3.21. We say that an odd composite number n is an Euler pseudoprime base
a if a is coprime to n and
a (n−1)/2
a
≡ (mod n), (3.32)
n
where
a
n is the Jacobi symbol (see Definition 2.3.3). Euler’s criterion (see
Theorem 2.3.4) asserts that odd primes n satisfy (3.32). Show that if n is a
strong pseudoprime base a, thennis an Euler pseudoprime base a, and that
if n is an Euler pseudoprime base a, thennis a pseudoprime base a.
3.22. [Lehmer, Solovay–Strassen] Let n be an odd composite. Show that
the set of residues a (mod n) for which n is an Euler pseudoprime is a proper
subgroup of Z ∗ n. Conclude that the number of such bases a is at most ϕ(n)/2.
3.23. Along the lines of Algorithm 3.5.6 develop a probabilistic compositeness
test using Exercise 3.22. (This test is often referred to as the Solovay–Strassen
primality test.) Using Exercise 3.21 show that this algorithm is
majorized by Algorithm 3.5.6.
3.24. [Lenstra, Robinson] Show that if n is odd and if there exists an integer
b with b (n−1)/2 ≡−1(modn), then any integer a with a (n−1)/2 ≡±1(modn)
also satisfies a (n−1)/2 ≡ a
n
(mod n). Using this and Exercise 3.22, show that
if n is an odd composite and a (n−1)/2 ≡±1(modn) for all a coprime to n,
then in fact a (n−1)/2 ≡ 1(modn) for all a coprime to n. Such a number must
be a Carmichael number; see Exercise 3.12. (It follows from the proof of the
infinitude of the set of Carmichael numbers that there are infinitely many odd
composite numbers n such that a (n−1)/2 ≡±1(modn) for all a coprime to
n. The first example is Ramanujan’s “taxicab” number, 1729.)
3.25. Show that there are seven Fibonacci pseudoprimes smaller than 323.
3.26. Show that every composite number coprime to 6 is a Lucas
pseudoprime with respect to x 2 − x +1.
3.27. Show that if (3.12) holds, then so does
(a − x) n
x (mod (f(x),n)),
≡
a − x (mod (f(x),n)),
∆ if n = −1,
if
∆
n =1.
In particular, conclude that a Frobenius pseudoprime with respect to f(x) =
x 2 − ax + b is also a Lucas pseudoprime with respect to f(x).