Prime Numbers

170 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
no primes in the interval [x, x+∆]. Describe how such a criterion could be used
for given x, ∆ to show numerically, but rigorously, whether or not primes exist
in such an interval. Of course, any new theoretical inroads into the analysis of
these “gaps” would be spectacular.
3.48. Suppose T is a probabilistic test that takes composite numbers n and,
with probability p(n), provides a proof of compositeness for n. (Forprime
inputs, the test T reports only that it has not succeeded in finding a proof of
compositeness.) Is there such a test T that has p(n) → 1asn runs to infinity
through the composite numbers, and such that the time to run T on n is no
longer than doing k pseudoprime tests on n, for some fixed k?
3.49. For a positive integer n coprime to 12 and squarefree, define K(n)
depending on n mod 12 according to one of the following equations:
K(n) =#{(u, v) : u>v>0; n = u 2 + v 2 }, for n ≡ 1, 5 (mod 12),
K(n) =#{(u, v) : u>0, v > 0; n =3u 2 + v 2 }, for n ≡ 7 (mod 12),
K(n) =#{(u, v) : u>v>0; n =3u 2 − v 2 }, for n ≡ 11 (mod 12).
Then it is a theorem in [Atkin and Bernstein 2004] that n is prime if and only
if K(n) is odd. First, prove this theorem using perhaps the fact (or related
facts) that the number of representations of (any) positive n as a sum of two
squares is
r2(n) =4
(−1) (d−1)/2 ,
d|n, d odd
where we count all n = u 2 + v 2 including negative u or v representations; e.g.
one has as a check the value r2(25) = 12.
A research question is this: Using the Atkin–Bernstein theorem can one
fashion an efficient sieve for primes in an interval, by assessing the parity of
K for many n at once? (See [Galway 2000].)
Another question is, can one fashion an efficient sieve (or even a primality
test) using alternative descriptions of r2(n), for example by invoking various
connections with the Riemann zeta function? See [Titchmarsh 1986] for a
relevant formula connecting r2 with ζ.
Yet another research question runs like so: Just how hard is it to
“count up” all lattice points (in the three implied lattice regions) within a
given “radius” √ n, and look for representation numbers K(n) asnumerical
discontinuities at certain radii. This technique may seem on the face of it to
belong in some class of brute-force methods, but there are efficient formulae—
arising in analyses for the celebrated Gauss circle problem (how many lattice
points lie inside a given radius?)—that provide exact counts of points in
surprisingly rapid fashion. In this regard, show an alternative lattice theorem,
that if n ≡ 1 (mod 4) is squarefree, then n is prime if and only if r2(n) =8.A
simple starting experiment that shows n = 13 to be prime by lattice counting,
via analytic Bessel formulae, can be found in [Crandall 1994b, p. 68].