  This paper demonstrates that from the Cramer's, Hardy-Littlewood's and
Bateman-Horn's conjectures (suggest that the probability of a large positive
integer being $x$ a prime - $\frac {1} {\ln(x)}$) it follows that the events
consisting in a positive integer $x$ being not divisible by different primes
are dependent with the ratio $0,5e^{\gamma}$ ($\gamma$ - Euler's constant). In
establishing Hardy-Littlewood's and Bateman-Horn's conjectures, their authors
followed the first suggestion by another one assuming the independence of the
above-mentioned events, which on the basis of the first suggestion and Merten's
theorem is not exactly. This paper demonstrates why these suggestions do not
lead to an erroneous result using the Hardy-Littlewood's conjecture for twin
primes as an example. The author provides generalized conjectures, which if
taken together with the first suggestion, make Hardy-Littlewood's prime $k$ -
tuples and Bateman-Horn's conjectures true.
