Ramanujan Prime
المؤلف:
Ramanujan, S.
المصدر:
Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc
الجزء والصفحة:
...
8-10-2020
894
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,
![R_n=1+max_(k)<span style=]() {k:pi(k)-pi(1/2k)=n-1}. " src="https://mathworld.wolfram.com/images/equations/RamanujanPrime/NumberedEquation1.gif" style="height:29px; width:236px" /> |
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.
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."
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