Ramanujan prime

In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan. It relates to the prime counting function.

Origins and definition

In 1919, Ramanujan published a new proof of Bertrand's postulate (which had already been proven by Pafnuty Chebyshev).

Ramanujan's result at the end of the paper was:

[math]\displaystyle{ \pi(x) - \pi(x/2) }[/math] ≥ 1, 2, 3, 4, 5, ... for all x ≥ 2, 11, 17, 29, 41, ... Template:OEIS

where [math]\displaystyle{ \pi }[/math](x) is the prime counting function. The prime counting function is the number of primes less than or equal to x.

The numbers 2, 11, 17, 29, 41 are first few Ramanujan primes. In other words:

Ramanujan primes are the integers R_n that are the smallest to satisfy the condition

[math]\displaystyle{ \pi(x) - \pi(x/2) }[/math] ≥ n, for all x ≥ R_n