Prime Numbers

9.7 Exercises 521
where “do” simply means one repeats what is in the braces for some
appropriate total iteration count. Note that the duplication of the y iteration
is
√
intentional! Show that this scheme formally generates the binomial series of
1+a via the variable x. How many correct terms obtain after k iterations
of the do loop?
Next, calculate some real-valued square roots in this way, noting the
important restriction that |a| cannot be too large, lest divergence occur (the
formal correctness of the resulting series in powers of a does not, of course,
automatically guarantee convergence).
Then, consider this question: Can one use these ideas to create an
algorithm for extracting integer square roots? This could be a replacement
for Algorithm 9.2.11; the latter, we note, does involve explicit division. On
this question it may be helpful to consider, for given n to be square-rooted,
such as n/4q =2−q√n or some similar construct, to keep convergence under
control.
Incidentally, it is of interest that the standard, real-domain, Newton
iteration for the inverse square root automatically has division-free form,
yet we appear to be compelled to invoke such as the above coupled-variable
expedient for a positive fractional power.
9.15. The Cullen numbers are Cn = n2 n +1. Write a Montgomery powering
program specifically tailored to find composite Cullen numbers, via relations
such as 2 Cn−1 ≡ 1(modCn). For example, within the powering algorithm
for modulus N = C245 you would be taking say R =2 253 so that R>N.
You could observe, for example, that C141 is a base-2 pseudoprime in this way
(it is actually a prime). A much larger example of a Cullen prime is Wilfrid
Keller’s C18496. For more on Cullen numbers see Exercise 1.83.
9.16. Say that we wish to evaluate 1/3 using the Newton reciprocation of
the text (among real numbers, so that the result will be 0.3333 ...). For initial
guess x0 =1/2, prove that for positive n the n-th iterate xn is in fact
xn = 22n
− 1
3 · 22n ,
in this way revealing the quadratic-convergence property of a successful
Newton loop. The fact that a closed-form expression can even be given for the
Newton iterates is interesting in itself. Such closed forms are rare—can you
find any others?
9.17. Work out the asymptotic complexity of Algorithm 9.2.8, in terms of
a size-N multiply, and assuming all the shifting enhancements discussed in
the text. Then give the asymptotic complexity of the composite operation
(xy) modN, for 0 ≤ x, y < N, in the case that the generalized reciprocal is not
yet known. What is the complexity for (xy) modN if the reciprocal is known?
(This should be asymptotically the same as the composite Montgomery
operation (xy) modN if one ignores the precomputations attendant to the
latter.) Incidentally, in actual programs that invoke the Newton–Barrett ideas,