Inmathematics, aRamanujan prime is aprime number that satisfies a result proven bySrinivasa Ramanujan relating to theprime-counting function.

In 1919, Ramanujan published a new proof ofBertrand's postulate which, as he notes, was first proved byChebyshev.^[1] At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:

${\displaystyle \pi (x)-\pi \left(\frac{x}{2}\right)\ge 1,2,3,4,5,\dots \text{ for all }x\ge 2,11,17,29,41,\dots \text{ respectively}}$    OEIS: A104272

where${\displaystyle \pi (x)}$ is theprime-counting function, equal to the number of primes less than or equal to x.

The converse of this result is the definition of Ramanujan primes:

Thenth Ramanujan prime is the least integerR_n for which${\displaystyle \pi (x)-\pi (x/2)\ge n,}$ for allx ≥R_n.^[2] In other words: Ramanujan primes are the least integersR_n for which there are at leastn primes betweenx andx/2 for allx ≥R_n.

The first five Ramanujan primes are thus 2, 11, 17, 29, and 41.

Note that the integerR_n is necessarily a prime number:${\displaystyle \pi (x)-\pi (x/2)}$ and, hence,${\displaystyle \pi (x)}$ must increase by obtaining another prime atx =R_n. Since${\displaystyle \pi (x)-\pi (x/2)}$ can increase by at most 1,

${\displaystyle \pi (R_n)-\pi \left(\frac{R_n}{2}\right)=n.}$

For all${\displaystyle n\ge 1}$, the bounds

hold. If${\displaystyle n>1}$, then also

wherep_n is thenth prime number.

Asn tends to infinity,R_n isasymptotic to the 2nth prime, i.e.,

R_n ~p_2n (n → ∞).

All these results were proved by Sondow (2009),^[3] except for the upper boundR_n <p_3n which was conjectured by him and proved by Laishram (2010).^[4] The bound was improved by Sondow, Nicholson, and Noe (2011)^[5] to

${\displaystyle R_n\le \frac{41}{47}{p}_{3n}}$

which is the optimal form ofR_n ≤c·p_3n since it is an equality forn = 5.

• Srinivasa Ramanujan
• Classes of prime numbers

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