Prime Numbers

8.7 Exercises 435
of positive composite numbers. Next, starting from the Lagrange theorem
that every positive integer is a sum of 4 squares (see Exercise 9.41), exhibit a
polynomial in 8 variables with integer coefficients such that its values at all
integral arguments constitute the set of positive composites.
8.22. Suppose the integer n of Proposition 8.5.1 has the distinct prime
factors p1,...,pk, where2 si pi − 1ands1 ≤···≤sk. Show that the relevant
probability is then
1 − 2 −(s1+···+sk)
1+ 2s1k − 1
2k
− 1
and that this expression is not less than 1 − 2 1−k . (Compare with Exercise
3.15.)
8.23. Complete one of the details for Shor factoring, as follows. We gave as
relation (8.4) the probability Pc,k of finding our QTM in the composite state
| c 〉| xk 〉. Explain quantitatively how the probability (for a fixed k, withc
the running variable) should show spikes corresponding to solutions d to the
Diophantine approximation
c d
−
q r
≤ 1
2q .
Explain, then, how one can find d/r in lowest terms from (measured)
knowledge of appropriate c. Note that if gcd(d, r) happens to be 1, this
procedure gives the exact period r for the algorithm, and we know that two
random integers are coprime with probability 6/π2 .
On the computational side, model (on a classical TM, of course) the
spectral behavior of the QTM occurring at the end of Algorithm 8.5.2, using
the following exemplary input. Take n = 77, so that the [Initialization] step
sets q = 8192. Now choose (we are using hindsight here) x = 3, for which
the period turns out to be r = 30 after the [Detect periodicity ...]step.Of
course, the whole point of the QTM is to measure this period physically, and
quickly! To continue along and model the QTM behavior, use a (classical)
FFT to make a graphical plot of c versus the probability Pc,1 from formula
(8.4). You should see very strong spikes at certain c values. One of these values
is c = 273, for example. Now from the relation
273 d
−
8192 r
≤ 1
2q
one can derive the result r = 30 (the literature explains continued-fraction
methods for finding the relevant approximants d/r). Finally, extract a factor of
n via gcd(x r/2 − 1,n). These machinations are intended show the flavor of the
missing details in the presentation of Algorithm 8.5.2; but beyond that, these
examples pave the way to a more complete QTM emulation (see Exercise 8.24).
Note the instructive phenomenon that even this small-n factoring emulationvia-TM
requires FFT lengths into the thousands; yet a true QTM might
require only a dozen or so qbits.