Prime Numbers

436 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
8.24. It is a highly instructive exercise to cast Algorithm 8.5.2 into a detailed
form that incorporates our brief overview and the various details from the
literature (including the considerations of Exercise 8.23).
A second task that lives high on the pedagogical ladder is to emulate a
QTM with a standard TM program implementation, in a standard language.
Of course, this will not result in a polynomial-time factorer, but only because
the TM does what a QTM could do, yet the former involves an exponential
slowdown. For testing, you might start with input numbers along the lines of
Exercise 8.23. Note that one still has unmentioned options. For example, one
could emulate very deeply and actually model quantum interference, or one
could just use classical arithmetic and FFTs to perform the algebraic steps of
Algorithm 8.5.2.
8.8 Research problems
8.25. Prove or disprove the claim of physicist D. Broadhurst that the number
P = 29035682 ∞
dx
514269
x906
sin(x ln 2) 1
sinh(πx/2) cosh(πx/5) +8sinh2
(πx/5)
0
is not only an integer, but in fact a prime number. This kind of integral
shows up in the theory of multiple zeta functions, which theory in turn has
application in theoretical physics, in fact in quantum field theory (and we
mean here physical fields, not the fields of algebra!).
Since the 1st printing of the present book, Broadhurst has used a publicly
available primality-proof package to establish that P is indeed prime. One
research extension, then, is to find—with proof—an even larger prime having
this kind of trigonometric-integral representation.
8.26. Here we explore a connection between prime numbers and fractals.
Consider the infinite-dimensional Pascal matrix P with entries
i + j
Pi,j = ,
i
for both i and j running through 0, 1, 2, 3, ...; thus the classical Pascal
triangle of binomial coefficients has its apex packed into the upper-left corner
of P , like so:
⎛
1 1 1 1
⎞
···
⎜ 1
⎜
P = ⎜ 1
⎜
⎝ 1
.
2
3
4
.
3
6
10
.
4
10
20
.
··· ⎟
··· ⎟
··· ⎠
. ..
.
There are many interesting features of this P matrix (see [Higham 1996, p.
520]), but for this exercise we concentrate on its fractal structure modulo
primes.
Define the matrix Qn = P mod n, where the mod operation is taken
elementwise. Now imagine a geometrical object created by coloring each zero