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# Ramanujan Prime

The th Ramanujan prime is the smallest number such that for all , where is the prime counting function. In other words, there are at least primes between and whenever . The smallest such number must be prime, since the function can increase only at a prime.

Equivalently,

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

The case for all is Bertrand's postulate.

Bertrand's Postulate, Prime Counting Function

This entry contributed by Jonathan 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. Sequence A104272 in "The On-Line Encyclopedia of Integer Sequences."

Ramanujan Prime

## Cite this as:

Sondow, Jonathan. "Ramanujan Prime." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/RamanujanPrime.html