Prime Numbers

528 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
9.44. Implement Algorithm 9.5.19, with a view to proving that p =2 521 − 1
is prime via the Lucas–Lehmer test. The idea is to maintain the peculiar,
variable-base representation for everything, all through the primality test. (In
other words, the output of Algorithm 9.5.19 is ready-made as input for a
subsequent call to the algorithm.) For larger primes, such as the gargantuan
new Mersenne prime discoveries, investigators have used run lengths such
that q/D, the typical bit size of a variable-base digit, is roughly 16 bits or
less. Again, this is to suppress as much as possible the floating-point errors.
9.45. Implement Algorithm 9.5.17 to establish the character of various
Fermat numbers, using the Pepin test, that Fn is prime if and only if
3 (Fn−1)/2 ≡−1(modFn). Alternatively, the same algorithm can be used in
factorization studies [Brent et al. 2000]. (Note: The balanced representation
error reduction scheme mentioned in Exercise 9.55 also applies to this
algorithm for arithmetic with Fermat numbers.) This method has been
employed for the resolution of F22 in 1993 [Crandall et al. 1995] and F24
[Crandall et al. 1999].
9.46. Implement Algorithm 9.5.20 to perform large-integer multiplication
via cyclic convolution of zero-padded signals. Can the DWT methods be
applied to do negacyclic integer convolution via an appropriate CRT prime
set?
9.47. Show that if the arithmetic field is equipped with a cube root of
unity, then for D = 3 · 2 k one can perform a length-D cyclic convolution
by recombining three separate length-2 k convolutions. (See Exercise 9.43 and
consider the symbolic factorization of t D − 1forsuchD.) This technique has
actually been used by G. Woltman in the discovery of new Mersenne primes
(he has employed IBDWTs of length 3 · 2 k ).
9.48. Implement the ideas in [Percival 2003], where Algorithm 9.5.19 is
generalized for arithmetic modulo Proth numbers k ·2n ±1. The essential idea
is that working modulo a number a ± b can be done with good error control,
as long as the prime product
p|ab p is sufficiently small. In the Percival
approach, one generalizes the variable-base representation of Theorem 9.5.18
to involve products over prime powers in the form
x =
D−1
j=0
xj
p k a
⌈kj/D⌉
p
q m b
q ⌈−mj/D⌉+mj/D ,
for fast arithmetic modulo a − b.
Note that the marriage of such ideas with the fast mod operation of
Algorithm 9.2.14 would result in an efficient union for computations that
need to move away from the restricted theme of Mersenne/Fermat numbers.
Indeed, as evidenced in the generalized Fermat number searches described in
[Dubner and Gallot 2002], wedding bells have already sounded.