Irregularities in the Distribution of Primes and Twin Primes
نویسندگان
چکیده
منابع مشابه
Irregularities 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 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 کاملSome Remarks on the Distribution of twin Primes
The computer data up to 244 ≈ 1.76×1013 on the gaps between consecutive twins is presented. The simple derivation of the heuristic formula describing computer results contained in the recent papers by P.F.Kelly and T.Pilling [5], [6] is provided and compared with the “experimental” values.
متن کاملLargest known twin primes and Sophie Germain primes
The numbers 242206083 · 238880 ± 1 are twin primes. The number p = 2375063906985 · 219380 − 1 is a Sophie Germain prime, i.e. p and 2p + 1 are both primes. For p = 4610194180515 · 25056 − 1, the numbers p, p + 2 and 2p + 1 are all primes. In the first days of October, 1995, Harvey Dubner [4] found the largest known twin primes with 5129 decimal digits. (Our earlier twin prime record was 6970538...
متن کاملA Property of Twin Primes
We determine the product of the invertible quadratic residues in Zn. This is a variation on Gauss’ generalization of Wilson’s Theorem. From this we deduce that for twin primes p, p + 2, the product of the invertible quadratic residues in Zp(p+2) is ±(p+1), where the sign depends on the residue class of p modulo 4. We examine necessary and sufficient conditions for consecutive odd natural number...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1975
ISSN: 0025-5718
DOI: 10.2307/2005460