Prime Numbers

536 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
[Müller 1997], [De Win et al. 1998], [Crandall 1999b] and references therein.
As just one example of attempted refinements, some investigators have
considered exponent expansions in which there is some guaranteed number of
0’s interposed between other digits. Then, too, there is the special advantage
inherent in highly compressible exponents [Yacobi 1999], such study being
further confounded by the possibility of base-dependent compressibility. It is
an interesting research matter to ascertain the precise relation between the
compressibility of an exponent and the optimal efficiency of powering to said
exponent.
9.78. In view of complexity results such as in Exercise 9.37, it would seem
that a large-D version of Toom–Cook could, with recursion, be brought down
to what is essentially an ideal bit complexity O ln 1+ɛ N . However, as we
have intimated, the additions grow rapidly. Work out a theory of Toom–Cook
addition counts, and discuss the tradeoffs between very low multiplication
complexity and overwhelming complexity of additions. Note also the existence
of addition optimizations, as intimated in Exercise 9.38.
This is a difficult study, but of obvious practical value. For example, there
is nothing a priori preventing us from employing different, alternating Toom–
Cook schemes within a single, large recursive multiply. Clearly, to optimize
such a mixed scheme one should know something about the interplay of the
multiply and add counts, as well as other aspects of overhead. Yet another
such aspect is the shifting and data shuttling one must do to break up an
integer into its Toom–Cook coefficients.
9.79. How far should one be able to test numerically the Goldbach
conjecture by considering the acyclic convolution of the signal
G =(1, 1, 1, 0, 1, 1, 0, 1, 1, 0,...)
with itself? (Here, as in the text, the signal element Gn equals 1 if and only if
2n + 3 is prime.) What is the computational complexity for this convolutionbased
approach for the settling of Goldbach’s conjecture for all even numbers
not exceeding x? Note that the conjecture has been settled for all even
numbers up to x = 4 · 10 14 [Richstein 2001]. We note that explicit FFTbased
computations up to 10 8 or so have indeed been performed [Lavenier
and Saouter 1998]. Here is an interesting question: Can one resolve Goldbach
representations via pure-integer convolution on arrays of b-bit integers (say
b = 16 or 32), with prime locations signified by 1 bits, knowing in advance
that two prime bits lying in one integer is a relatively rare occurrence?
9.80. One can employ convolution ideas to analyze certain higher-order
additive problems in rings ZN, and perhaps in more complicated settings
leading into interesting research areas. Note that Exercise 9.41 deals with
sums of squares. But when higher powers are involved, the convolution and
spectral manipulations are problematic.
To embark on the research path intended herein, start by considering a kth
powers exponential sum (the square and cubic versions appear in Exercise