I want to show that 2 p <mrow class="MJX-TeXAtom-ORD"> n &#x2212;<!-

Alessandra Clarke

Alessandra Clarke

Answered question

2022-05-29

I want to show that 2 p n 2 p n 1...
Bertand's postulate shows us that 4 p n 2 p n but can we improve on this?
any ideas?

Answer & Explanation

Haleigh Vega

Haleigh Vega

Beginner2022-05-30Added 13 answers

There are a lot of improvements of the Bertrand postulate but yes, in this case it's sufficient to use that there is a prime between n and 6 n / 5, n sufficiently large (as you write n 25). So we have
p n 2 < p n 1 < 6 p n 2 5
and
p n 1 < p n < 6 p n 1 5 = p n 1 + p n 1 5
then
p n < 36 p n 2 25 p n 1 < 25 p n 18 < 2 p n 2 .

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