Prime Numbers

8.4 Diophantine analysis 415
[Tezuka 1995], [Veach 1997]. These are point clouds that have “minimal fill”
properties. For example, a set of N = b m points in s dimensions is called a
(t, m, s)-net if every justified box of volume b t−m has exactly b t points. Yet
another intriguing connection between primes and discrepancy appears in the
literature (see [Joe 1999] and references therein). This notion of “number-
theoretical rules” involves approximations of the form
f(x) d D x ≈ 1
p−1
j
f
p
K
,
p
[0,1] D
where here {y} denotes the vector composed of the fractional parts of y,and K
is some chosen constant vector having each component coprime to p. Actually,
composite numbers can be used in place of p, but the analysis of what is called
L2 discrepancy, and the associated typical integration error, goes especially
smoothly for p prime. We have mentioned these new approaches to underscore
the notion that qMC is continually undergoing new development. And who
knows when or where number theory or prime numbers in particular will
appear in qMC theories of the future?
In closing this section, we mention a new result that may explain why
qMC experiments sometimes do “so well.” Take the result in [Sloan and
Wozniakowski 1998], in which the authors remark that some errors (such
as those in Traub’s qMC for finance in D = 360 dimensions) appear to have
O(1/N ) behavior, i.e., independent of dimension D. What the authors actually
prove is that there exist classes of integrand functions for which suitable lowdiscrepancy
sequences provide overall integration errors of order O(1/N ρ )for
some real ρ ∈ [1, 2].
8.4 Diophantine analysis
Herein we discuss Diophantine analysis, which loosely speaking is the practice
of discovering integer solutions to various equations. We have mentioned
elsewhere Fermat’s last theorem (FLT), for which one seeks solutions to
j=0
x p + y p = z p ,
and how numerical attacks alone have raised the lower bound on p into the
millions (Section 1.3.3, Exercise 9.68). This is a wonderful computational
problem—speaking independently, of course, of the marvelous FLT proof
by A. Wiles—but there are many other similar explorations. Many such
adventures involve a healthy mix of theory and computation.
For instance, there is the Catalan equation for p, q prime and x, y positive
integers,
x p − y q =1,
of which the only known solution is the trivial yet attractive
3 2 − 2 3 =1.