Ramanujan prime
Encyclopedia
In mathematics
, a Ramanujan prime is a prime number
that satisfies a result proven by Srinivasa Ramanujan
relating to the prime-counting function.
. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:
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, of which the numbers 2, 11, 17, 29, 41 are the first examples. In other words:
Another way to put this is:
Since Rn is the smallest such number, it must be a prime:
and, hence, 
must increase by obtaining another prime at x = Rn. Since 
can increase by at most 1,
hold. If n > 1, then also
where pn is the nth prime number.
As n tends to infinity, Rn is asymptotic
to the 2nth prime, i.e.,
All these results were proved by Sondow (2009), except for the upper bound Rn < p3n which was conjectured by him and proved by Laishram (2010). The bound was improved by Sondow, Nicholson, and Noe (2011) to
which is optimal since it is an equality for n = 5.
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,
≥ 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
It is known 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)
where pn/(1-c) is the
n/(1-c)
th prime and 
is the floor function.
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 [pi-n,pk], one can see that the average prime gap is about ln(pk) using the following Rn / (2*n) ~ 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 > ... > pi-n > pi / 2, and so 2 pi-n > pi.
An example of this corollary:
With n = 1000, R_n = pk = 19403, and k = 2197, therefore i ≥ 2198 and i-n ≥ 1198.
The smallest i-n prime is pi-n = 9719, therefore 2 * pi-n = 2 * 9719 = 19438. The 2198th prime, pi, is between pk = 19403 and 2 * pi-n = 19438 and is 19417.
The left side of the Ramanujan Prime Corollary is the ; the right side is the .
The values of
are in the .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Ramanujan prime is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
that satisfies a result proven by Srinivasa Ramanujan
Srinivasa Ramanujan
Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...
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 ChebyshevPafnuty Chebyshev
Pafnuty Lvovich Chebyshev was a Russian mathematician. His name can be alternatively transliterated as Chebychev, Chebysheff, Chebyshov, Tschebyshev, Tchebycheff, or Tschebyscheff .-Early years:One of nine children, Chebyshev was born in the village of Okatovo in the district of Borovsk,...
. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:
-
≥ 1, 2, 3, 4, 5, ... for all x ≥ 2, 11, 17, 29, 41, ... respectively,
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, of which the numbers 2, 11, 17, 29, 41 are the first examples. In other words:
- The nth Ramanujan prime is the integer Rn that is the smallest to satisfy the condition
≥ n, for all x ≥ Rn.
Another way to put this is:
- Ramanujan primes are the integers Rn that are the smallest to guarantee there are at least n primes between x and x/2 for all x ≥ Rn.
Since Rn is the smallest such number, it must be a prime:
and, hence, must increase by obtaining another prime at x = Rn. Since can increase by at most 1,-
Rn
Rn
.
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
Asymptotic analysis
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
to the 2nth prime, i.e.,
- Rn ~ p2n (n → ∞).
All these results were proved by Sondow (2009), except for the upper bound Rn < p3n which was conjectured by him and proved by Laishram (2010). The bound was improved by Sondow, Nicholson, and Noe (2011) to
which is optimal since it is an equality for n = 5.
Generalized Ramanujan primes
Given a constant c between 0 and 1, the nth c-Ramanujan prime is defined as thesmallest 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,
≥ 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, ... ,
- R0.75,n = 11, 29, 59, 67, 101, ... .
It is known 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
n/(1-c)th prime and is the floor function.Ramanujan prime corollary
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 [pi-n,pk], one can see that the average prime gap is about ln(pk) using the following Rn / (2*n) ~ 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 > ... > pi-n > pi / 2, and so 2 pi-n > pi.
An example of this corollary:
With n = 1000, R_n = pk = 19403, and k = 2197, therefore i ≥ 2198 and i-n ≥ 1198.
The smallest i-n prime is pi-n = 9719, therefore 2 * pi-n = 2 * 9719 = 19438. The 2198th prime, pi, is between pk = 19403 and 2 * pi-n = 19438 and is 19417.
The left side of the Ramanujan Prime Corollary is the ; the right side is the .
The values of
are in the .