By Bertrand's postulate, we know that there exists at least one prime number between n and

Jeramiah Campos

Jeramiah Campos

Answered question

2022-06-26

By Bertrand's postulate, we know that there exists at least one prime number between n and 2 n for any n > 1. In other words, we have
π ( 2 n ) π ( n ) 1 ,
for any n > 1. The assertion we would like to prove is that the number of primes between n and 2 n tends to , if n , that is,
lim n π ( 2 n ) π ( n ) = .
Do you see an elegant proof?

Answer & Explanation

grcalia1

grcalia1

Beginner2022-06-27Added 23 answers

By the PNT, we expect
2 n ln ( 2 n ) n ln ( n )
= 2 n ln ( n ) n ln ( 2 n ) ln ( 2 n ) ln ( n )
n ln ( n )
, as n
primes in this region.

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