Read More
Date: 23-1-2021
579
Date: 26-5-2020
1300
Date: 12-1-2021
1095
|
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.
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."
|
|
دور النظارات المطلية في حماية العين
|
|
|
|
|
العلماء يفسرون أخيرا السبب وراء ارتفاع جبل إيفرست القياسي
|
|
|
|
|
اختتام المراسم التأبينية التي أهدي ثوابها إلى أرواح شهداء المق*ا*و*مة
|
|
|