Prime Numbers

1.1 Problems and progress 7
It follows that
x ≤
pi≤x
1+
ln x
≤ 1+
ln pi
π(x) ln x
,
ln 2
which leads immediately to the fact that for all x ≥ 8,
π(x) >
ln x
2lnlnx .
Though this bound is relatively poor, it does prove the infinitude of primes
directly from the fundamental theorem of arithmetic.
The idea of Euclid in the proof of Theorem 1.1.2 is to generate new primes
from old primes. Can we generate all of the primes this way? Here are a few
possible interpretations of the question:
(1) Inductively define a sequence of primes q1,q2,...,whereq1 =2,andqk+1
is the least prime factor of q1 ···qk + 1. Does the sequence (qi) contain
every prime?
(2) Inductively define a sequence of primes r1,r2,...,wherer1 =2,andrk+1
is the least prime not already chosen that divides some d+1, where d runs
over the divisors of the product r1 ···rk. Does the sequence (ri) contain
every prime?
(3) Inductively define a sequence of primes s1,s2,...,wheres1 =2,s2 =3,
and sk+1 is the least prime not already chosen that divides some sisj +1,
where 1 ≤ i