Ramanujan prime

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 ${\displaystyle \pi (x)}$ 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. Equivalently.

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

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