Ramanujan Prime

The nth Ramanujan prime is the smallest number R_n such that pi(x)-pi(x/2)>=n for all x>=R_n, where pi(x) is the prime counting function. In other words, there are at least n primes between x/2 and x whenever x>=R_n. The smallest such number R_n must be prime, since the function pi(x)-pi(x/2) can increase only at a prime.



Using simple properties of the gamma function, Ramanujan (1919) gave a new proof of Bertrand's postulate. Then he proved the generalization that pi(x)-pi(x/2)>=1, 2, 3, 4, 5, ... if x>=2, 11, 17, 29, 41, ... (OEIS A104272), respectively. These are the first few Ramanujan primes.

The case pi(x)-pi(x/2)>=1 for all x>=2 is Bertrand's postulate.

See also

Bertrand's Postulate, Prime Counting Function

This entry contributed by Jonathan Sondow (author's link)

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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. Sequence A104272 in "The On-Line Encyclopedia of Integer Sequences."

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Ramanujan Prime

Cite this as:

Sondow, Jonathan. "Ramanujan Prime." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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