Prime Numbers

168 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
3.38. Consider a specific choice for the Lagarias–Odlyzko turn-off function
c(u, x), namely, a straight-line connection between the 1, 0 values. Specifically,
for y = √ x, define c =1, (x − u)/y, 0asu ≤ x − y, u ∈ (x − y, x],u > x,
respectively. Show that the Mellin companion function is
F (s, x) = 1
y
xs+1 − (x − y) s+1
.
s(s +1)
Now derive a bound, as in Exercise 3.37, on proper values of T such that π(x)
will be calculated correctly on the basis of
π ∗ T
(x) ≈ Re F (s, x)lnζ(s) dt.
0
Calculate numerically some correct values of π(x) using this particular turn-off
function c.
3.39. In regard to the Galway functions of which F is defined by (3.31), make
rigorous the notion that even though the Riemann zeta function somehow
embodies, if you will, “all the secrets of the primes,” we need to know ζ only
to an imaginary height of “about” x 1/2 to count all primes not exceeding x.
3.40. Using integration by parts, show that the F defined by (3.31) is indeed
the Mellin transform of the given c.
3.9 Research problems
3.41. Find a number n ≡ ±2 (mod 5) that is simultaneously a base-
2 pseudoprime and a Fibonacci pseudoprime. Pomerance, Selfridge, and
Wagstaff offer $620 for the first example. (The prime factorization must also
be supplied.) The prize money comes from the three, but not equally: Selfridge
offers $500, Wagstaff offers $100 and Pomerance offers $20. However, they also
agree to pay $620, with Pomerance and Selfridge reversing their roles, for a
proof that no such number n exists.
3.42. Find a composite number n, together with its prime factorization, that
is a Frobenius pseudoprime for x2 +5x+5 and satisfies
5
n = −1. J. Grantham
has offered a prize of $6.20 for the first example.
3.43. Consider the least witness function W (n) defined for odd composite
numbers n. It is relatively easy to see that W (n) is never a power; prove this.
Are there any other forbidden numbers in the range of W (n)? If some n exists