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- Title
ON A CONJECTURE ON RAMANUJAN PRIMES.
- Authors
LAISHRAM, SHANTA
- Abstract
For n ≥ 1, the nth Ramanujan prime is defined to be the smallest positive integer Rn with the property that if x ≥ Rn, then $\pi(x)-\pi(\frac{x}{2})\ge n$ where π(ν) is the number of primes not exceeding ν for any ν > 0 and ν ∈ ℝ. In this paper, we prove a conjecture of Sondow on upper bound for Ramanujan primes. An explicit bound of Ramanujan primes is also given. The proof uses explicit bounds of prime π and θ functions due to Dusart.
- Subjects
LOGICAL prediction; PRIME numbers; MATHEMATICAL functions; NUMBER theory; MATHEMATICAL analysis; NATURAL numbers
- Publication
International Journal of Number Theory, 2010, Vol 6, Issue 8, p1869
- ISSN
1793-0421
- Publication type
Article
- DOI
10.1142/S1793042110003848