Vol'fson V.L.
In the statements 1, 2 of the article show that the density of
a strictly increasing sequence of integers with the addition of
integer sequences that have no members and have one member
in finite interval of the natural numbers, is the ultimate value of
a probability measure on the interval. There was obtained the
probability of the event that a large natural number is prime.
From this follows that the event based on the fact that a large
natural number is not divisible by distinct primes are dependent
events with constant 0,5eϒ, where ϒ – the Euler constant. The
author’s conjecture is - the events, which based on the fact that
large integers: x, x+2n1, ..., x+2n1+...+2nk-1 are not divided into
different primes are equally dependent, if each of the numbers
x, x+2n1, ..., x+2n1+...+2nk-1 are primes or not, consequently, are
dependent events with constant 0,5eϒ. From this conjecture
implies the validity of Hardy-Littlewoods’s conjecture. The author’s
conjecture can be extended in case when the event that large
integers: g1(x), g2(x), ..., gk(x) are not divided into different primes
are equally dependent, if each of these numbers is prime or not,
consequently, aredependenteventswithconstant0,5eϒ (g1, g2,..., gk –
sequence of irreducible polynomials with integer coefficients,
each of which takes an infinite number of prime values). This
generalized conjecture is true, if the Bateman-Horn’s conjecture
is true. Author introduces the density of one sequence, as a share
of the other sequence, which in general is not a natural numbers.
It is called the conditional density. In the statements 3, 4 show
that the conditional density in certain cases is the conditional
probability. There was obtained a new probabilistic model, free
from the assumption of independence of events that even and
odd numbers following each other are primes.
Key words: density of the sequence, probability, conditional
probability, probability model, conjecture, primes, Hardy-Littlewoods,
Bateman–Horn, Dickson, Schinzel, Kramer, Granville,
Legendre.
Contacts: E-mail: znakvicvolf@mail.ru
Pp. 87-98. |
