Prime Numbers

492 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
As a brief digression, we should note here that the original Goldbach
conjecture is true if a different signal of infinite length, namely
G =(1, 1, 1, 0, 1, 1, 0, 1, 1, 0,...),
where the 1’s occur at indices (p−3)/2 for the odd primes p =3, 5, 7, 11, 13,...,
has the property that the acyclic G×A G has no zero elements. In this case the
n-th element of the acyclic is precisely the number of Goldbach representations
of 2n +6.
Back to Theorem 9.5.13: It is advantageous to study the length-N DFT
Y of the aforementioned signal y. This DFT turns out to be a famous sum:
Yk(N) =cN(k) =
e ±2πijk/N , (9.26)
gcd(j,N)=1
where j is understood to run over those elements in the interval [0,N−1] that
are coprime to N, so the sign choice in the exponent doesn’t matter, while
cN(k) is the standard notation for the Ramanujan sum, which sum is already
known to enjoy intriguing multiplicative properties [Hardy and Wright 1979].
In fact, the appearance of the Ramanujan sum in Section 1.4.4 suggests that
it makes sense for cN also to have some application in discrete convolution
studies. We leave the proof of Theorem 9.5.13 to the reader (see Exercise 9.40),
but wish to make several salient points. First, the sum in relation (9.26) can
itself be thought of as a result of “sieving” out finite sums corresponding to
the divisors of N. This gives rise to interesting series algebra. Second, it is
remarkable that the cyclic length-N convolution of y with itself can be given
a closed form. The result is
(y × y)n = ϕ2(N,n) =
(p − θ(n, p)), (9.27)
where θ(n, p) is1ifp|n, else2.Thus,for0≤ n2 is a regular prime, then Fermat’s last theorem, that
p|N
x p + y p = z p
has no Diophantine solution with xyz = 0, holds. (We note in passing that
FLT is now a genuine theorem of A. Wiles, but the techniques here predated
that work and still have application to such remaining open problems as the
Vandiver conjecture.) Furthermore, p is regular if it does not divide any of
the numerators of the even-index Bernoulli numbers
B2,B4,...,Bp−3.