Contradictions in some primes conjectures

TL;DR

This paper examines the dependencies in prime conjectures, showing that previous assumptions of independence are flawed, and proposes generalized conjectures that reconcile these issues, especially in the context of twin primes.

Contribution

It reveals the dependency structure in prime conjectures and introduces generalized conjectures that support the validity of Hardy-Littlewood and Bateman-Horn conjectures.

Findings

Dependencies among prime events are quantified with ratio 0.5e^{γ}.

Previous independence assumptions are invalid, affecting conjecture validity.

Generalized conjectures reconcile dependencies with existing prime conjectures.

Abstract

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.