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The paper substantiates the conjecture of the asymptotic behavior of thelargest distance between consecutive primes: $sup_{p_i \leq x}(p_{i+1}-p_i)\sim 2e^{-\gamma} \log^2(x)$, where $\gamma$ is the Euler constant. TheHardy-Littlewood conjecture about the number of prime tuples is investigatedand the rationale for this conjecture is given, taking into account the factthat a large natural number is not divisible by primes. It also substantiateswhy the accuracy of this conjecture is not affected by another assumption aboutthe probability of a natural number being prime, although such a probabilitydoes not exist. The paper also considers the distribution of prime tuples usinga mathematical model based on the Hardy-Littlewood conjecture.

Asymptotic of the greatest distance between adjacent primes and the Hardy-Littlewood conjecture