Prime Numbers

3.8 Exercises 163
3.3. Prove that if n>1 and gcd(a n − a, n) = 1 for some integer a, thennot
only is n composite, it is not a prime power.
3.4. For each number B ≥ 2, let dB be the asymptotic density of the integers
that have a divisor exceeding B with said divisor composed solely of primes
not exceeding B.Thatis,ifN(x, B) denotes the number of positive integers up
to x that have such a divisor, then we are defining dB = limx→∞ N(x, B)/x.
(1) Show that
dB =1−
p≤B
where the product is over primes.
1 − 1
p
(2) Find the smallest value of B with dB >d7.
·
B
m=1
(3) Using the Mertens Theorem 1.4.2 show that limB→∞ dB =1− e −γ ≈
0.43854, where γ is the Euler constant.
(4) It is shown in [Rosser and Schoenfeld 1962] that if x ≥ 285, then
eγ ln x
p≤x (1 − 1/p) is between 1 − 1/(2 ln2 x)and1+1/(2 ln 2 x). Use
this to show that 0.25 ≤ dB