Is there a smallest real number a such that there exist a natural number N so that: n>N->p_(n+1)<=a x p_n?

schnelltcr 2022-08-16 Answered
Is there a smallest real number a such that there exist a natural number N so that:
n > N p n + 1 a p n ?
I believe it can be proved that n > 7 p n + 1 2 p n .
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Answers (2)

Jazmyn Bean
Answered 2022-08-17 Author has 18 answers
There is no smallest such a, but the infimum of the set of such a is 1. In other words, for every ϵ > 0, there is a prime between n and ( 1 + ϵ ) n for all sufficiently large n. This follows from the prime number theorem.
As a concrete example, for all n 25, there is a prime between n and 6 5 n
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Gorlandint
Answered 2022-08-18 Author has 5 answers
Using the prime-number theorem:
p n + 1 a p n p n + 1 p n a (asymptotically:) ( n + 1 ) log ( n + 1 ) n log ( n ) a ( 1 + 1 / n ) 1 log ( n + 1 ) log ( n ) a ( 1 + 1 / n ) ( 1 + log ( 1 + 1 / n ) log ( n ) ) a
and the lhs on the last expression can be made arbitrarily near to 1 by increasing n so also a can be taken arbitrarily small by using appropriate n
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