Ramanujan prime

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This article is about prime numbers related to the prime counting function. For numbers expressible as sums of cubes in two ways, see Hardy-Ramanujan number.

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

[edit] Origins and definition

In 1919, Ramanujan published a new proof[1] of Bertrand's postulate which, unbeknownst to him, had already been proven by Chebyshev. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:

π(x) − π(x / 2) ≥ 1, 2, 3, 4, 5, ... for all x ≥ 2, 11, 17, 29, 41, ...

where π(x) is the prime counting function, that is, the number of primes less than or equal to x.

The converse of this result is the definition of Ramanujan primes, and the numbers 2, 11, 17, 29, 41 are first few such primes. In other words:

Ramanujan primes are the integers Rn that are the smallest to satisfy the condition
π(x) − π(x / 2) ≥ n, for all x ≥ Rn

Another way to put this is:

Ramanujan primes are the integers Rn that are the smallest to guarantee there would be n primes between x and x/2 for all x ≥ Rn

Since Rn is the smallest such number, it must be a prime: π(x) − π(x / 2) must increase by obtaining another prime.

[edit] References

  1. ^ S. Ramanujan (2000). Collected papers of Srinivasa Ramanujan. American Mathematical Society. 
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