Prime Numbers

222 Chapter 4 PRIMALITY PROVING
show that if gcd(a, n) = 1, then (x + a) n = x n + a in Zn[x] if and only if n is
prime.
4.26. Using Theorem 4.5.2 prove that Algorithm 4.5.1 correctly decides
whether n is prime or composite.
4.27. Show that the set G in the proof of Theorem 4.5.2 is the union of {0}
and a cyclic multiplicative group.
4.28. By the same method as in Exercise 3.19, show that if a n ≡ a (mod n)
for each positive integer a smaller than ln 2 n,thenn is squarefree. Further
show that the AKS congruence (x + a) n ≡ x n + a (mod x r − 1,n) implies that
(a +1) n ≡ a +1(modn). Conclude that the hypotheses of Theorem 4.5.2
imply that if n is divisible by a prime larger than ϕ(r)lgn, thenn is equal
to this prime. Use this to establish a shorter version of Algorithm 4.5.1, where
Step [Power test] may be skipped entirely.
4.29. Show that if n ≡±3 (mod 8), then the value of r in Step [Setup] in
Algorithm 4.5.1 is bounded above by 8 lg 2 n. Hint: Show that if r2 is the least
power of 2 with the order of n in Z ∗ r2 exceeding lg2 n,thenr2 < 8lg 2 n.
4.30. Using an appropriate generalization of the idea suggested in Exercise
4.29, and Theorem 1.4.7, show that the value of r in Step [Setup] in Algorithm
4.5.1 is bounded above by lg 2 n lg lg n for all but possibly o(π(x)) primes n ≤ x.
Conclude that Algorithm 4.5.1 runs in time Õ(ln6 n) for almost all primes n,
in the sense that the number of exceptional primes n ≤ x is o(π(x)).
4.31. Prove the converse of Lemma 4.5.5; that is, assuming that r, p are
unequal primes, d|r − 1andfr,d(x) is irreducible modulo p, prove that the
order of p (r−1)/d modulo r is d.
4.32. Suppose that r1,r2,...,rk are primes and that d1,d2,...,dk are
positive and pairwise coprime, with di|ri − 1foreachi. Letf(x) bethe
minimal polynomial for ηr1,d1ηr2,d2 ...ηrk,dk over Q. Show that for primes p
unequal to each ri, f(x) is irreducible modulo p if and only if the order of
each p (ri−1)/di modulo ri is di.
4.33. In the text we only sketched the proof of Theorem 4.5.8. Give a
complete proof.
4.7 Research problems
4.34. Design a practical algorithm that rigorously determines primality of
an arbitrary integer n ∈ [2,...,x] for as large an x as possible, but carry out
the design along the following lines.
Use a probabilistic primality test but create a (hopefully minuscule) table
of exceptions. Or use a small combination of simple tests that has no exceptions
up to the bound x. For example, in [Jaeschke 1993] it is shown that no