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. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:
where is the prime-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:
The first five Ramanujan primes are thus 2, 11, 17, 29, and 41.
Note that the integer Rn is necessarily a prime number: and, hence, must increase by obtaining another prime at x = Rn. Since can increase by at most 1,
For all , the bounds
hold. If , then also
where pn is the nth prime number.
As n tends to infinity, Rn is asymptotic to the 2nth prime, i.e.,
All these results were proved by Sondow (2009), except for the upper bound Rn < p3n which was conjectured by him and proved by Laishram (2010). The bound was improved by Sondow, Nicholson, and Noe (2011) to
which is the optimal form of Rn ≤ c·p3n since it is an equality for n = 5.