Ramanujan prime

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

Origins and definition[edit]

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:

    OEISA104272

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 nth Ramanujan prime is the least integer Rn for which for all xRn.[2] In other words: Ramanujan primes are the least integers Rn for which there are at least n primes between x and x/2 for all xRn.

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,

Bounds and an asymptotic formula[edit]

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.,

Rn ~ p2n (n → ∞).

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 Rnc·p3n since it is an equality for n = 5.

References[edit]

  1. ^ Ramanujan, S. (1919), "A proof of Bertrand's postulate", Journal of the Indian Mathematical Society, 11: 181–182
  2. ^ Jonathan Sondow. "Ramanujan Prime". MathWorld.
  3. ^ Sondow, J. (2009), "Ramanujan primes and Bertrand's postulate", Amer. Math. Monthly, 116 (7): 630–635, arXiv:0907.5232, doi:10.4169/193009709x458609
  4. ^ Laishram, S. (2010), "On a conjecture on Ramanujan primes" (PDF), International Journal of Number Theory, 6 (8): 1869–1873, CiteSeerX 10.1.1.639.4934, doi:10.1142/s1793042110003848.
  5. ^ Sondow, J.; Nicholson, J.; Noe, T.D. (2011), "Ramanujan primes: bounds, runs, twins, and gaps" (PDF), Journal of Integer Sequences, 14: 11.6.2, arXiv:1105.2249, Bibcode:2011arXiv1105.2249S