Prime Numbers

220 Chapter 4 PRIMALITY PROVING
have in hand the final Pepin residue, namely,
r =3 (Fn−1)/2 mod Fn.
Say that someone discovers a factor f of Fn, so that we can write
Prove that if we assign
then
Fn = fG.
x =3 f−1 mod Fn,
gcd(r 2 − x, G) =1
implies that the cofactor G is neither a prime nor a prime power. As in Exercise
4.7, the relatively fast (mod Fn) operation is the reason why we interpose said
operation prior to the implicit (mod G) operation in the gcd. All of this shows
the importance of carefully squirreling away one’s Pepin residues, to be used
again in some future season!
4.9. There is an interesting way to find, rigorously, fairly large primes of the
Proth form p = k2 n +1. Prove this theorem of Suyama [Williams 1998], that if
a p of this form divides some Fermat number Fm,andifk2 n−m−2 < 9·2 m+2 +6,
then p is prime.
4.10. Prove the following theorem of Proth: If n>1, 2 k |n−1, 2 k > √ n,and
a (n−1)/2 ≡−1(modn) for some integer a, thenn is prime.
4.11. In the algorithm based on Theorem 4.1.6, one is asked for the integral
roots (if any) of a cubic polynomial with integer coefficients. As an initial
foray, show how to do this efficiently using a Newton method or a divideand-conquer
strategy. Note the simple Algorithm 9.2.11 for design guidance.
Consider the feasibility of rapidly solving even higher-order polynomials for
possible integer roots.
A hint is in order for the simpler case of polynomials x k − a. To generalize
Algorithm 9.2.11 for finding integer k-th roots, say ⌊N 1/k ⌋, consider
In Step [Initialize], replace B(N)/2 → B(N)/k;
In Step [Perform Newton iteration], make the iteration
y = ⌊((k − 1)x + ⌊N/x k−1 ⌋)/k⌋,
or some similar such reduction formula.
4.12. Prove Theorem 4.2.4.
4.13. If the partial factorization (4.2) is found by trial division on n − 1
up to the bound B, then we have the additional information that R’s prime
factors are all >B. Show that if a satisfies (4.3) and also gcd(a F − 1,n)=1,
then every prime factor of n exceeds BF. In particular, if BF ≥ n 1/2 ,thenn
is prime.