Ramanujan prime

Not to be confused with 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.

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)\geq 1,2,3,4,5,\ldots {\text{ for all }}x\geq 2,11,17,29,41,\ldots {\text{ respectively}}

A104272

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 R_n for which

\pi (x)-\pi (x/2)\geq n,

for all x ≥ R_n.^[2] In other words: Ramanujan primes are the least integers R_n for which there are at least n primes between x and x/2 for all x ≥ R_n.

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: $\pi (x)-\pi (x/2)$ and, hence, $\pi (x)$ must increase by obtaining another prime at x = R_n. Since $\pi (x)-\pi (x/2)$ can increase by at most 1,

\pi (R_{n})-\pi \left({\frac {R_{n}}{2}}\right)=n.

Bounds and an asymptotic formula

For all $n\geq 1$ , the bounds

2n\ln 2n<R_{n}<4n\ln 4n

hold. If $n>1$ , then also

p_{2n}<R_{n}<p_{3n}

where p_n is the nth prime number.

As n tends to infinity, R_n is asymptotic to the 2nth prime, i.e.,

R_n ~ p_2n (n → ∞).

All these results were proved by Sondow (2009),^[3] except for the upper bound R_n < p_3n 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}\leq {\frac {41}{47}}\ p_{{3n}}

which is the optimal form of R_n ≤ c·p_3n 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.

A further improvement of the upper bounds was done in late 2015 by Anitha Srinivasan and John W. Nicholson.^[7] They show that if

\alpha =1+{\frac {3}{\ln n+\ln \ln n-4}}

then $R_{n}<p_{\lfloor 2n\alpha \rfloor }$ for all $n>241$ , where $\lfloor \cdot \rfloor$ is the floor function. For large n, the bound is smaller and thus better than $p_{\lfloor 2nc\rfloor }$ for any fixed constant $c>1$ .

In 2016, Shichun Yang and Alain Togbe^[8] establish the estimates of the upper and lower bounds of Ramanujan primes $R_{n}$ when n is big: if $n>10^{300}$ and $R_{n}=p_{s}$ , then

\beta <s<\alpha ,

where

\alpha =2n{\Big (}1+{\frac {\ln 2}{\ln n}}-{\frac {\ln 2\ln \ln n-\ln ^{2}2-\ln 2-0.13}{\ln ^{2}n}}{\Big )},

\beta =2n{\Big (}1+{\frac {\ln 2}{\ln n}}-{\frac {\ln 2\ln \ln n-\ln ^{2}2-\ln 2+0.11}{\ln ^{2}n}}{\Big )}.

Generalized Ramanujan primes

Given a constant c between 0 and 1, the nth c-Ramanujan prime is defined as the smallest integer R_c,n with the property that for any integer x ≥ R_c,n there are at least n primes between cx and x, that is, $\pi (x)-\pi (cx)\geq n$ . In particular, when c = 1/2, the nth 1/2-Ramanujan prime is equal to the nth Ramanujan prime: R_0.5,n = R_n.

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

R_0.25,n = 2, 3, 5, 13, 17, ...

A193761,

R_0.75,n = 11, 29, 59, 67, 101, ...

A193880.

It is known^[9] that, for all n and c, the nth c-Ramanujan prime R_c,n exists and is indeed prime. Also, as n tends to infinity, R_c,n is asymptotic to p_{n/(1 − c)}

R_c,n ~ p_{n/(1 − c)} (n → ∞)

where p_{n/(1 − c)} is the $\lfloor$ n/(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. p_k is the kth prime and the nth Ramanujan prime.

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

Proof of Corollary:

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

An example of this corollary:

With n = 1000, R_n = p_k = 19403, and k = 2197, therefore i ≥ 2198 and i−n ≥ 1198. The smallest i − n prime is p_i−n = 9719, therefore 2p_i−n = 2 × 9719 = 19438. The 2198th prime, p_i, is between p_k = 19403 and 2p_i−n = 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 p_k. The values of $\pi (R_{n})\,$ are in the A179196.

The Ramanujan Prime Corollary is due to John Nicholson.

Srinivasan's Lemma ^[10] states that p_k−n < p_k/2 if R_n = p_k and n > 1. Proof: By the minimality of R_n, the interval (p_k/2,p_k] contains exactly n primes and hence p_k−n < p_k/2.

References

↑ Ramanujan, S. (1919), "A proof of Bertrand's postulate", Journal of the Indian Mathematical Society, 11: 181–182
↑ Jonathan Sondow. "Ramanujan Prime". MathWorld.
↑ Sondow, J. (2009), "Ramanujan primes and Bertrand's postulate", Amer. Math. Monthly, 116 (7): 630–635, arXiv:0907.5232, doi:10.4169/193009709x458609
↑ Laishram, S. (2010), "On a conjecture on Ramanujan primes" (PDF), International Journal of Number Theory, 6 (8): 1869–1873, doi:10.1142/s1793042110003848 .
↑ 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
↑ Axler, Christian (2014). "On generalized Ramanujan primes". The Ramanujan Journal. 39 (2016): 1. arXiv:1401.7179. doi:10.1007/s11139-015-9693-9.
↑ Srinivasan, Anitha; Nicholson, John (2015). "An Improved Upper Bound For Ramanujan Primes" (PDF). Integers. 15.
↑ Shichun, Yang; Alain, Togbe (2016). "On the estimates of the upper and lower bounds of Ramanujan primes". The Ramanujan Journal. 40 (2): 245–255. doi:10.1007/s11139-015-9706-8.
↑ Amersi, N.; Beckwith, O.; Miller, S.J.; Ronan, R.; Sondow, J. (2011), Generalized Ramanujan primes, arXiv:1108.0475
↑ Srinivasan, Anitha (2014), "An upper bound for Ramanujan primes" (PDF), Integers, 14

Prime number classes

By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))

By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin

By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient

Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin

Patterns	Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)

By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known

Complex numbers	Eisenstein prime Gaussian prime

Composite numbers	Pseudoprime Almost prime Semiprime Interprime

Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap

First 50 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229

List of prime numbers

This article is issued from Wikipedia - version of the 10/24/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.