Is it true that $p_{pi(c_n)}ge c_{n+2}$ for all sufficiently large $ninmathbb{N}$?











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Yesterday I was going through this paper and I found it very interesting (especially Property 2.2 and Proposition 3.2). Motivated by it I conjectured the following,




Conjecture. $p_{pi(c_n)}ge c_{n+2}$ for all sufficiently large $ninmathbb{N}$. Here $pi(n)$ denotes the number of primes less than or equal to $n$, $c_n$ denotes the $n$-th composite and $p_n$ the $n$-th prime.




I have tried to prove this inequality for several hours but couldn't succed. Can anyone give me some hint regarding how to prove it?






analytic-number-theory







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  • What makes you believe this is true?
    – Servaes
    1 hour ago















up vote
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down vote

favorite












Yesterday I was going through this paper and I found it very interesting (especially Property 2.2 and Proposition 3.2). Motivated by it I conjectured the following,




Conjecture. $p_{pi(c_n)}ge c_{n+2}$ for all sufficiently large $ninmathbb{N}$. Here $pi(n)$ denotes the number of primes less than or equal to $n$, $c_n$ denotes the $n$-th composite and $p_n$ the $n$-th prime.




I have tried to prove this inequality for several hours but couldn't succed. Can anyone give me some hint regarding how to prove it?






analytic-number-theory







share|cite|improve this question
























  • What makes you believe this is true?
    – Servaes
    1 hour ago













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Yesterday I was going through this paper and I found it very interesting (especially Property 2.2 and Proposition 3.2). Motivated by it I conjectured the following,




Conjecture. $p_{pi(c_n)}ge c_{n+2}$ for all sufficiently large $ninmathbb{N}$. Here $pi(n)$ denotes the number of primes less than or equal to $n$, $c_n$ denotes the $n$-th composite and $p_n$ the $n$-th prime.




I have tried to prove this inequality for several hours but couldn't succed. Can anyone give me some hint regarding how to prove it?






analytic-number-theory







share|cite|improve this question















Yesterday I was going through this paper and I found it very interesting (especially Property 2.2 and Proposition 3.2). Motivated by it I conjectured the following,




Conjecture. $p_{pi(c_n)}ge c_{n+2}$ for all sufficiently large $ninmathbb{N}$. Here $pi(n)$ denotes the number of primes less than or equal to $n$, $c_n$ denotes the $n$-th composite and $p_n$ the $n$-th prime.




I have tried to prove this inequality for several hours but couldn't succed. Can anyone give me some hint regarding how to prove it?






analytic-number-theory




analytic-number-theory






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  • What makes you believe this is true?
    – Servaes
    1 hour ago


















  • What makes you believe this is true?
    – Servaes
    1 hour ago
















What makes you believe this is true?
– Servaes
1 hour ago




What makes you believe this is true?
– Servaes
1 hour ago










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The inequality fails for all $n$; the number $pi(c_n)$ is the number of primes up to the $n$th composite number, so $p_{pi(c_n)}$ is the largest prime smaller than $c_n$. So it is certainly smaller than $c_{n+2}$.






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    The inequality fails for all $n$; the number $pi(c_n)$ is the number of primes up to the $n$th composite number, so $p_{pi(c_n)}$ is the largest prime smaller than $c_n$. So it is certainly smaller than $c_{n+2}$.






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      The inequality fails for all $n$; the number $pi(c_n)$ is the number of primes up to the $n$th composite number, so $p_{pi(c_n)}$ is the largest prime smaller than $c_n$. So it is certainly smaller than $c_{n+2}$.






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        The inequality fails for all $n$; the number $pi(c_n)$ is the number of primes up to the $n$th composite number, so $p_{pi(c_n)}$ is the largest prime smaller than $c_n$. So it is certainly smaller than $c_{n+2}$.






        share|cite|improve this answer












        The inequality fails for all $n$; the number $pi(c_n)$ is the number of primes up to the $n$th composite number, so $p_{pi(c_n)}$ is the largest prime smaller than $c_n$. So it is certainly smaller than $c_{n+2}$.







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