Ramanujan prime

Ramanujan prime

In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.

Origins and definition

In 1919, Ramanujan published a new proof [S. Ramanujan, "A proof of Bertrand's postulate". "Journal of the Indian Mathematical Society" 11 (1919), 181–182. [http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper24/page1.htm] ] of Bertrand's postulate which, as he says, was first proved by Chebyshev. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:

: pi(x) - pi(x/2) ≥ 1, 2, 3, 4, 5, ... for all "x" ≥ 2, 11, 17, 29, 41, ... ]

Another way to put this is:

:Ramanujan primes are the integers "Rn" that are the smallest to guarantee there would be "n" primes between "x" and "x/2" for all "x" ≥ "Rn".

Since "Rn" is the smallest such number, it must be a prime: pi(x) - pi(x/2) must increase by obtaining another prime.

References


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