The $k^{th}$ prime is greater than $k(\ln k + \ln\ln k-1)$ for $k\geq 2$
نویسندگان
چکیده
منابع مشابه
N ov 2 00 3 Asymptotically exact heuristics for prime divisors of the sequence { a k + b k } ∞ k = 1
Let Na,b(x) count the number of primes p ≤ x with p dividing ak + bk for some k ≥ 1. It is known that Na,b(x) ∼ c(a, b)x/ log x for some rational number c(a, b) that depends in a rather intricate way on a and b. A simple heuristic formula for Na,b(x) is proposed and it is proved that it is asymptotically exact, i.e. has the same asymptotic behaviour as Na,b(x). Connections with Ramanujan sums a...
متن کاملAsymptotically Exact Heuristics for Prime Divisors of the Sequence { a k + b k } ∞ k = 1
Let Na,b(x) count the number of primes p ≤ x with p dividing ak + bk for some k ≥ 1. It is known that Na,b(x) ∼ c(a, b)x/ log x for some rational number c(a, b) that depends in a rather intricate way on a and b. A simple heuristic formula for Na,b(x) is proposed and it is proved that it is asymptotically exact, i.e., has the same asymptotic behavior as Na,b(x). Connections with Ramanujan sums a...
متن کاملThe Threshold for Random k - SAT is 2 k ( ln 2 + o ( 1 ) )
Let Fk(n, m) be a random k-SAT formula with n variables and m clauses selected uniformly andindependently among all 2(nk)possible k-clauses. It is well-known that if r ≥ 2 ln 2 then Fk(n, rn) isunsatisfiable with probability 1 − o(1). We prove that there exists a sequence tk = O(k) such that ifr ≤ 2 ln 2− tk, then Fk(n, rn) is satisfiable with probability 1− o(1)...
متن کاملThe k-variable property is stronger than H-dimension k
We study the notion of H-dimension and the formally stronger kvariable property, as considered by Gabbay and Immerman & Kozen. We exhibit a class of flows of time that has H-dimension 3, and admits a finite expressively complete set of one-dimensional connectives, but does not have the k-variable property for any finite k. Published in Journal of Philosophical Logic 26 (1997) 81–101. Publisher’...
متن کاملThe Threshold for Random k - SAT is 2 k log 2 − O ( k )
Let Fk(n, m) be a random k-CNF formula formed by selecting uniformly and independently m out of all possible k-clauses on n variables. It is well-known that if r ≥ 2 log 2, then Fk(n, rn) is unsatisfiable with probability that tends to 1 as n → ∞. We prove that if r ≤ 2 log 2 − tk, where tk = O(k), then Fk(n, rn) is satisfiable with probability that tends to 1 as n → ∞. Our technique, in fact, ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1999
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-99-01037-6