Prime Numbers

9.5 Large-integer multiplication 491
}
zn = v mod B;
carry = ⌊v/B⌋;
7. [Final digit adjustment]
Delete leading zeros, with possible carry > 0 as a high digit of z;
return z;
This algorithm description is intended to be general, conveying only the
principles of FFT multiplication, which are transforming, rounding, carry
adjustment. There are a great many details left unsaid, not to mention a great
number of enhancements, some of which we address later. But beyond these
minutiae there is one very strong caveat: The accuracy of the floating-point
arithmetic must always be held suspect. A key step in the general algorithm
is the elementwise rounding of the z signal. If floating-point errors in the
FFTs are too large, an element of the convolution z could get rounded to an
incorrect value.
One immediate practical enhancement to Algorithm 9.5.12 is to employ
the balanced representation of Definition 9.1.2. It turns out that floatingpoint
errors are significantly reduced in this representation [Crandall and
Fagin 1994], [Crandall et al. 1999]. This phenomenon of error reduction is
not completely understood, but certainly has to do with the fact of generally
smaller magnitude for the digits, plus, perhaps, some cancellation in the DFT
components because the signal of digits has a statistical mean (in engineering
terms, “DC component”) that is very small, due to the balancing.
Before we move on to algorithmic issues such as further enhancements to
the FFT multiply and the problem of pure-integer convolution, we should
mention that convolutions can appear in number-theoretical work quite
outside the large-integer arithmetic paradigm. We give two examples to end
this subsection; namely, convolutions applied to sieving and to regularity
results on primes.
Consider the following theorem, which is reminiscent of (although
obviously much less profound than) the celebrated Goldbach conjecture:
Theorem 9.5.13. Let N =2·3·5 ···pm be a product of consecutive primes.
Then every sufficiently large even n is a sum n = a+b with each of a, b coprime
to N.
It is intriguing that this theorem can be proved, without too much trouble,
via convolution theory. (We should point out that there are also proofs using
CRT ideas, so we are merely using this theorem to exemplify applications of
discrete convolution methods (see Exercise 9.40).) The basic idea is to consider
a special signal y defined by yj =1ifgcd(j, N) = 1, else yj =0,withthe
signal given some cutoff length D. Now the acyclic convolution y ×A y will tell
us precisely which n