منابع مشابه
Sieving Very Thin Sets of Primes, and Pratt Trees with Missing Primes
Suppose P is a set of primes, such that for every p ∈ P , every prime factor of p − 1 is also in P . We apply a new sieve method to show that either P contains all of the primes or the counting function of P is O(x) for some c > 0, where c depends only on the smallest prime not in P . Our proof makes use of results connected with Artin’s primitive root conjecture.
متن کاملCatalan Numbers, Primes and Twin Primes
with C0 = 1. Their appearances occur in a dazzling variety of combinatorial settings where they are used to enumerate all manner of geometric and algebraic objects (see Richard Stanley’s collection [28, Chap. 6]; an online Addendum is continuously updated). Quite a lot is known about the divisibility of the Catalan numbers; see [2, 10]. They are obviously closely related to the middle binomial ...
متن کاملSmall Gaps between Primes or Almost Primes
Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that lim inf n→∞ (pn+1 − pn) log pn = 0. We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of exactly two distinct primes. We prove that lim inf n→∞ (qn+1 − qn) ≤ 26. If an appropriate generalization of ...
متن کاملRamanujan and Labos Primes, Their Generalizations, and Classifications of Primes
We study the parallel properties of the Ramanujan primes and a symmetric counterpart, the Labos primes. Further, we study all primes with these properties (generalized Ramanujan and Labos primes) and construct two kinds of sieves for them. Finally, we give a further natural generalization of these constructions and pose some conjectures and open problems.
متن کاملIrregularities in the Distribution of Primes and Twin Primes
The maxima and minima of sL(x)) — n(x), iR(x)) — n(x), and sL2(x)) — n2(x) in various intervals up to x = 8 x 10 are tabulated. Here n(x) and n2(x) are respectively the number of primes and twin primes not exceeding x, L(x) is the logarithmic integral, R(x) is Riemann's approximation to ir(x), and L2(x) is the Hardy-Littlewood approximation to ti"2(;c). The computation of the sum of inverses of...
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: Bulletin of the Australian Mathematical Society
سال: 2009
ISSN: 0004-9727,1755-1633
DOI: 10.1017/s0004972710000067