Prime Numbers

8.8 Research problems 441
for every n = 2, 4, 6,..., is equivalent to the RH. The interesting
computational exercise would be to calculate some vast number of such
derivatives. A single negative derivative would destroy the RH. Yet
another criterion equivalent to the RH is that of [Lagarias 1999]:
′ ξ (s)
Re > 0
ξ(s)
whenever Re(s) > 1/2. Again some graphical or other computational
means of analysis is at least interesting. Then there is the work in [Li
1997], [Bombieri and Lagarias 1999] to the effect that the RH is equivalent
to the positivity property
λn =
1 − 1 − 1
n > 0
ρ
ρ
holding for each n =1, 2, 3,... .Theλn constants can be cast in terms
of derivatives of ln ξ(s), but this time, all such evaluated at s = 1. Again
various computational avenues are of interest.
Further details, some computational explorations of these, and yet other new
RH equivalences appear in [Borwein et al. 2000].
8.35. It is not clear what the search limit is for coprime positive solutions
to the Fermat–Catalan equation x p + y q = z r when 1/p +1/q +1/r ≤ 1. This
search limit certainly encompasses the known 10 solutions mentioned in the
chapter, but maybe it is not much higher. Extend the search for solutions,
where the highest of the powers, namely z r , is allowed to run up to 10 25
or perhaps even higher. To aid in this computation, one should not consider
triples p, q, r where we know there are no solutions. For example, if 2 and
3arein{p, q, r}, then we may assume the third member is at least 10. See
[Beukers 2004] and [Bruin 2003] for an up-to-date report on those exponent
triples for which no search is necessary. Also, see [Bernstein 2004c] for a neat
way to search for solutions in the most populous cases.
8.36. Investigate alternative factoring and discrete-logarithm algorithms for
quantum Turing machines (QTMs). Here are some (unguaranteed) ideas.
The Pollard–Strassen method of Section 5.5 uses fast algorithms to
deterministically uncover factors of N in O(N 1/4 ) operations. However, the
usual approach to the required polynomial evaluations is FFT-like, and in
practice often does involve FFTs. Is there a way to go deeper into the Pollard–
Strassen method, using the inherent massive parallelism of QTMs in order to
effect an interesting deterministic algorithm?
Likewise, we have seen exercises involving parallelization of Pollard-rho,
ECM, QS, NFS factoring, and it is a good rule that whenever parallelism
reveals itself, there is some hope of a QTM implementation.
As for DL problems, the rho and lambda methods admit of parallelism;
indeed, the DL approach in [Shor 1999] is very much like the collision methods