Prime Numbers

5.8 Research problems 255
5.17. Show that if p is a prime and p ≡ 1 (mod 4), then there is a
probabilistic algorithm to write p as a sum of two squares that is expected
to succeed in polynomial time. In the case that p ≡ 5 (mod 8), show how the
algorithm can be made deterministic. Using the deterministic polynomial-time
method in [Schoof 1985] for taking the square root of −1 modulo p, showhow
in the general case the algorithm can be made deterministic, and still run in
polynomial time.
5.18. Suppose that (a, b, c), (a ′ ,b ′ ,c ′ ) are equivalent quadratic forms, n is
a positive integer, ax 2 + bxy + cy 2 = n, and under the equivalence, x, y gets
taken to x ′ ,y ′ .Letu =2ax + by, u ′ =2a ′ x ′ + b ′ y ′ . Show that uy ′ ≡ u ′ y
(mod 2n).
5.19. Show that if (a, b, c) is a quadratic form, then for each integer b ′ ≡ b
(mod 2a), there is an integer c ′ such that (a, b, c) is equivalent to (a, b ′ ,c ′ ).
5.20. Suppose 〈a, b, c〉 ∈C(D). Prove that 〈a, b, c〉 is the identity 1D in C(D)
ifandonlyif(a, b, c) represents 1. Conclude that 〈a, b, c〉∗〈c, b, a〉 =1D.
5.21. Study, and implement the McKee O(n 1/4+ɛ ) factoring algorithm as
described in [McKee 1999]. The method is probabilistic, and is a kind of
optimization of the celebrated Fermat method.
5.22. On the basis of the Dirichlet class number formula (5.3), derive the
following formulae for π:
π =2
p>2
1+ (−1)(p−1)/2
−1
=4
p
1 −
p>2
(−1)(p−1)/2
−1
.
p
From the mere fact that these formulae are well-defined, prove that there
exist infinitely many primes of each of the forms p =4k + 1 and p =4k +3.
(Compare with Exercise 1.7.) As a computational matter, about how many
primes would you need to attain a reliable value for π to a given number of
decimal places?
5.8 Research problems
5.23. Show that for p = 257, the rho iteration x = x 2 − 1modp has only
three possible cycle lengths, namely 2, 7, 12. For p = 7001, show the iteration
x = x 2 +3modp has only the 8 cycle lengths 3, 4, 6, 7, 19, 28, 36, 67. Find
too the number of distinct connected components in the cycle graphs of these
two iterations. Is it true that the number of distinct cycle lengths, as well as
the number of connected components (which always is at least as large) is
O(ln p)? A similar result has been proved in the case of a random function;
see [Flajolet and Odlyzko 1990].
5.24. If a Pollard-rho iteration be taken not as x = x 2 + a mod N but as
x = x 2K + a mod N,