In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.
In 1919, Ramanujan published a new proof of Bertrand's postulate which, as he notes, was first proved by Chebyshev.
[1] At the end of the two-page published paper, Ramanujan derived a generalized result, and that is: where
Note that the integer Rn is necessarily a prime number:
As n tends to infinity, Rn is asymptotic to the 2nth prime, i.e., All these results were proved by Sondow (2009),[3] except for the upper bound Rn < p3n which was conjectured by him and proved by Laishram (2010).
[4] The bound was improved by Sondow, Nicholson, and Noe (2011)[5] to which is the optimal form of Rn ≤ c·p3n since it is an equality for n = 5.