arXiv subjects
Publications and source records attributed to Boris B. Benyaminov.
We analyze the inequality $\sqrt{P_{k+1}}-\sqrt{P_{k}}<1,\ k\in\mathbb{N}$, discuss the existence of primes on arbitrary intervals $(r,s),\ r<s,\ r,s\in\mathbb{R}$, and finally address the issue of primes between squares of naturals.
We investigate logarithmic and square-root types of bounds for the general difference of two primes, $P_{k+q}-P_k$, $k, q\in\mathbb{N}$.
We derive properties of $π(x)$ reminiscent of those of the logarithm and absolute value functions. Two of these properties are similar to the relations defining the linearity of a function. Several applications of these properties of $π(x)$ are also examined.
We arrive at some new relations for the prime number $P_n$, based on the logarithmic and absolute-value properties of the function $π(x)$.
A hypothesis is put forward regarding the function $π_2(x)$ which describes the distribution of twin primes in the set of natural numbers. The function $π_2(x)$ is tested by evaluation and an empirical $π_2^{\ast}(x)$ is arrived at, which is shown to be highly accurate. Several other questions are also addressed.