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.
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:
- [math]\displaystyle{ \pi(x) - \pi\left( \frac x 2 \right) \ge 1,2,3,4,5,\ldots \text{ for all } x \ge 2, 11, 17, 29, 41, \ldots \text{ respectively} }[/math] OEIS: A104272
where [math]\displaystyle{ \pi(x) }[/math] 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 [math]\displaystyle{ \pi(x) - \pi(x/2) \ge n, }[/math] for all x ≥ Rn.[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 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: [math]\displaystyle{ \pi(x) - \pi(x/2) }[/math] and, hence, [math]\displaystyle{ \pi(x) }[/math] must increase by obtaining another prime at x = Rn. Since [math]\displaystyle{ \pi(x) - \pi(x/2) }[/math] can increase by at most 1,
- [math]\displaystyle{ \pi(R_n) - \pi\left( \frac{R_n} 2 \right) = n. }[/math]
Bounds and an asymptotic formula
For all [math]\displaystyle{ n \geq 1 }[/math], the bounds
- [math]\displaystyle{ 2n\ln2n \lt R_n \lt 4n\ln4n }[/math]
hold. If [math]\displaystyle{ n \gt 1 }[/math], then also
- [math]\displaystyle{ p_{2n} \lt R_n \lt p_{3n} }[/math]
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
- [math]\displaystyle{ R_n \le \frac{41}{47} \ p_{3n} }[/math]
which is the optimal form of Rn ≤ c·p3n since it is an equality for n = 5.
References
- ↑ Ramanujan, S. (1919), "A proof of Bertrand's postulate", Journal of the Indian Mathematical Society 11: 181–182, http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper24/page1.htm
- ↑ Jonathan Sondow. "Ramanujan Prime". http://mathworld.wolfram.com/RamanujanPrime.html.
- ↑ Sondow, J. (2009), "Ramanujan primes and Bertrand's postulate", Amer. Math. Monthly 116 (7): 630–635, doi:10.4169/193009709x458609
- ↑ Laishram, S. (2010), "On a conjecture on Ramanujan primes", International Journal of Number Theory 6 (8): 1869–1873, doi:10.1142/s1793042110003848, http://www.isid.ac.in/~shanta/PAPERS/RamanujanPrimes-IJNT.pdf.
- ↑ Sondow, J.; Nicholson, J.; Noe, T.D. (2011), "Ramanujan primes: bounds, runs, twins, and gaps", Journal of Integer Sequences 14: 11.6.2, Bibcode: 2011arXiv1105.2249S, http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Noe/noe12.pdf
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