#jsDisabledContent { display:none; } My Account | Register | Help

# Ramanujan prime

Article Id: WHEBN0007011111
Reproduction Date:

 Title: Ramanujan prime Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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.

## Contents

• Origins and definition 1
• Bounds and an asymptotic formula 2
• Generalized Ramanujan primes 3
• Ramanujan prime corollary 4
• References 5

## Origins and definition

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:

\pi(x) - \pi(x/2) ≥ 1, 2, 3, 4, 5, ... for all x ≥ 2, 11, 17, 29, 41, ...  A104272 respectively,

where \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 nth Ramanujan prime is the least integer Rn for which \pi(x) - \pi(x/2)n, for all xRn.[2]

The first five Ramanujan primes are thus 2, 11, 17, 29, and 41. Equivalently,

Ramanujan primes are the least integers Rn for which there are at least n primes between x and x/2 for all xRn.

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

\pi(Rn) - \pi(Rn/2) = n.

## Bounds and an asymptotic formula

For all n ≥ 1, the bounds

2n ln 2n < Rn < 4n ln 4n

hold. If n > 1, then also

p2n < Rn < p3n

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

R_n \le \frac{41}{47} \ p_{3n}

which is the optimal form of Rn < c·p3n since it is an equality for n = 5.

In a different direction, Axler[6] showed that

R_n < p_{\lceil t\cdot n \rceil}

is optimal for t > 48/19, where \lceil\cdot \rceil is the ceiling function.

## Generalized Ramanujan primes

Given a constant c between 0 and 1, the nth c-Ramanujan prime is defined as the smallest integer Rc,n with the property that for any integer x ≥ Rc,n there are at least n primes between cx and x, that is, \pi(x) - \pi(cx) \ge n. In particular, when c = 1/2, the nth 1/2-Ramanujan prime is equal to the nth Ramanujan prime: R0.5,n = Rn.

For c = 1/4 and 3/4, the sequence of c-Ramanujan primes begins

R0.25,n = 2, 3, 5, 13, 17, ...  A193761,
R0.75,n = 11, 29, 59, 67, 101, ...  A193880.

It is known[7] that, for all n and c, the nth c-Ramanujan prime Rc,n exists and is indeed prime. Also, as n tends to infinity, Rc,n is asymptotic to pn/(1 − c)

Rc,n ~ pn/(1 − c) (n → ∞)

where pn/(1 − c) is the \lfloorn/(1 − c)\rfloor th prime and \lfloor .\rfloor is the floor function.

## Ramanujan prime corollary

2p_{i-n} > p_i \text{ for } i>k \text{ where } k=\pi(p_k)=\pi(R_n)\, ,

i.e. pk is the kth prime and the nth Ramanujan prime.

This is very useful in showing the number of primes in the range [pk, 2*pi-n] is greater than or equal to 1. By taking into account the size of the gaps between primes in [pin,pk], one can see that the average prime gap is about ln(pk) using the following Rn/(2n) ~ ln(Rn).

Proof of Corollary:

If pi > Rn, then pi is odd and pi − 1 ≥ Rn, and hence π(pi − 1) − π(pi/2) = π(pi − 1) − π((pi − 1)/2) ≥ n. Thus pi − 1 ≥ pi−1 > pi−2 > pi−3 > ... > pin > pi/2, and so 2pin > pi.

An example of this corollary:

With n = 1000, Rn = pk = 19403, and k = 2197, therefore i ≥ 2198 and in ≥ 1198. The smallest i − n prime is pin = 9719, therefore 2pin = 2 × 9719 = 19438. The 2198th prime, pi, is between pk = 19403 and 2pin = 19438 and is 19417.

The left side of the Ramanujan Prime Corollary is the  A168421; the smallest prime on the right side is  A168425. The sequence  A165959 is the range of the smallest prime greater than pk. The values of \pi(R_n)\, are in the  A179196.

The Ramanujan Prime Corollary is due to John Nicholson.

Srinivasan's Lemma [8] states that pk-n < pk/2 if Rn = pk and n > 1. Proof: By the minimality of Rn, the interval (pk/2,pk] contains exactly n primes and hence pk-n < pk/2.

## References

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: 630–635,
4. ^ Laishram, S. (2010), "On a conjecture on Ramanujan primes" (PDF), International Journal of Number Theory 6: 1869–1873, .
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
6. ^ Axler, Christian. "On generalized Ramanujan primes". Retrieved 16 April 2014.
7. ^ Amersi, N.; Beckwith, O.; Miller, S.J.; Ronan, R.; Sondow, J. (2011), Generalized Ramanujan primes,
8. ^ Srinivasan, Anitha (2014), "An upper bound for Ramanujan primes" (PDF), Integers 19
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.