Prime Numbers

3.8 Exercises 167
3.28. Show that the definition of Frobenius pseudoprime in Section 3.6.5 for
a polynomial f(x) =x 2 − ax + b reduces to the definition in Section 3.6.2.
3.29. Show that if a, n arepositiveintegerswithn odd and coprime to
a, thenn is a Fermat pseudoprime base a if and only if n is a Frobenius
pseudoprime with respect to the polynomial f(x) =x − a.
3.30. Let a, b be integers with ∆ = a 2 −4b not a square, let f(x) =x 2 −ax+b,
let n be an odd prime not dividing b∆, and let R = Zn[x]/(f(x)). Show that
if (x(a − x) −1 ) 2m =1inR, then(x(a − x) −1 ) m = ±1 inR.
3.31. Show that a Frobenius pseudoprime with respect to x 2 − ax + b is also
an Euler pseudoprime (see Exercise 3.21) with respect to b.
3.32. Prove that the various identities in Section 3.6.3 are correct.
3.33. Prove that the recurrence (3.22) is valid.
3.34. Show that if a = π x 1/3 , then the number of terms in the double
sum in (3.23) is O x 2/3 / ln 2 x .
3.35. Show that with M computers where M1, and describing quantitatively the relative ease with which one
can calculate ζ(ns) for large integers n.
3.37. By establishing theoretical bounds on the magnitude of the real part
of the integral
∞
eitα β + it dt,
T
where T,α,β are positive reals, determine a bound on that portion of the
integral in relation (3.27) that comes from Im(s) >T. Describe, then, how
large T must be for π ∗ (x) to be calculated to within some ±ɛ of the true
value. See Exercises 3.38, 3.39 involving the analogous estimates for much
more efficient prime-counting methods.