- Srinivasa Ramanujan
- 2 (angka)
- Daftar bilangan prima
- Faktorial
- Terence Tao
- Daftar angka
- Daftar topik teori bilangan
- Daftar tetapan matematis
- Fungsi phi Euler
- Fungsi zeta Riemann
- Ramanujan prime
- List of prime numbers
- Srinivasa Ramanujan
- 1729 (number)
- 2
- Prime-counting function
- Bertrand's postulate
- Ramanujan (disambiguation)
- List of integer sequences
- Ramanujan graph
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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
In 1919, Ramanujan published a new proof of Bertrand's postulate which, as he notes, was first proved 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
,
…
respectively
{\displaystyle \pi (x)-\pi \left({\frac {x}{2}}\right)\geq 1,2,3,4,5,\ldots {\text{ for all }}x\geq 2,11,17,29,41,\ldots {\text{ respectively}}}
OEIS: A104272
where
π
(
x
)
{\displaystyle \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
π
(
x
)
−
π
(
x
/
2
)
≥
n
,
{\displaystyle \pi (x)-\pi (x/2)\geq n,}
for all x ≥ Rn. 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 x ≥ Rn.
The first five Ramanujan primes are thus 2, 11, 17, 29, and 41.
Note that the integer Rn is necessarily a prime number:
π
(
x
)
−
π
(
x
/
2
)
{\displaystyle \pi (x)-\pi (x/2)}
and, hence,
π
(
x
)
{\displaystyle \pi (x)}
must increase by obtaining another prime at x = Rn. Since
π
(
x
)
−
π
(
x
/
2
)
{\displaystyle \pi (x)-\pi (x/2)}
can increase by at most 1,
π
(
R
n
)
−
π
(
R
n
2
)
=
n
.
{\displaystyle \pi (R_{n})-\pi \left({\frac {R_{n}}{2}}\right)=n.}
Bounds and an asymptotic formula
For all
n
≥
1
{\displaystyle n\geq 1}
, the bounds
2
n
ln
2
n
<
R
n
<
4
n
ln
4
n
{\displaystyle 2n\ln 2n
hold. If
n
>
1
{\displaystyle n>1}
, then also
p
2
n
<
R
n
<
p
3
n
{\displaystyle p_{2n}
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), 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
R
n
≤
41
47
p
3
n
{\displaystyle R_{n}\leq {\frac {41}{47}}\ p_{3n}}
which is the optimal form of Rn ≤ c·p3n since it is an equality for n = 5.