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Ramanujan Prime
The
th Ramanujan prime is the smallest number
such that
for all
, where
is theprime counting function. In other words, there are at least
primes between
and
whenever
. The smallest such number
must beprime, since the function
can increase only at a prime.
Equivalently,
 |
Using simple properties of the gamma function, Ramanujan (1919) gave a new proof ofBertrand's postulate. Then he proved the generalization that
, 2, 3, 4, 5, ... if
, 11, 17, 29, 41, ... (OEISA104272), respectively. These are the first few Ramanujan primes.
The case
for all
isBertrand's postulate.
See also
Bertrand's Postulate,Prime Counting FunctionThis entry contributed byJonathan Sondow (author's link)
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References
Ramanujan, S.Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc., pp. 208-209, 2000.Ramanujan, S. "A Proof of Bertrand's Postulate."J. Indian Math. Soc.11, 181-182, 1919.Sloane, N. J. A. SequenceA104272 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Ramanujan PrimeCite this as:
Sondow, Jonathan. "Ramanujan Prime." FromMathWorld--A Wolfram Resource, created byEric W. Weisstein.https://mathworld.wolfram.com/RamanujanPrime.html