Ramanujan Primes and Bertrand's Postulate

From a Consequence of Bertrand's Postulate to Hamilton Cycles

A consequence of Bertrand's postulate, proved by L. Greenfield and S. Greenfield in 1998, assures that the set of integers $\{1,2,\cdots, 2n\}$ can be partitioned into pairs so that the sum of each pair is a prime number for any positive integer $n$. Cutting through it from the angle of Graph Theory, this paper provides new insights into the problem. We conjecture a stronger statement that the set of integers $\{1,2,\cdots, 2n\}$ can be rearranged into a cycle so that the sum of any two adjacent integers is a prime number. Our main result is that this conjecture is true for infinitely many cas…

Bertrand's postulate and subgroup growth

In this article we investigate the L^1-norm of certain functions on groups called divisibility functions. Using these functions, their connection to residual finiteness, and integration theory on profinite groups, we define the residual average of a finitely generated group. One of the main results in this article is the finiteness of residual averages on finitely generated linear groups. Whether or not the residual average is finite depends on growth rates of indices of finite index subgroups. Our results on index growth rates are analogous to results on gaps between primes, and provide a var…

伯特兰公设向任意多个素数之和的推广

Generalizations of Bertrand's Postulate to Sums of Any Number of Primes

本文将伯特兰公设推广为：任意多个素数之和在足够大时严格超过另一组较少素数之和，并仅用素数定理证明该结果，同时给出寻找最小阈值N的方法及数值结果。

Derived Ramanujan Primes : R'_{n}

We study the Ramanujan-prime-counting function along the lines of Ramanujan's original work on Bertrand's Postulate. We show that the number of Ramanujan primes between x and 2x tends to infinity with x. This analysis leads us to define a new sequence of prime numbers, which we call derived Ramanujan primes. For n greater than or equal to 1 we define the nth derived Ramanujan prime as the smallest positive integer with the property that if x is greater than or equal to the nth derived Ramanujan prime, then the number of Ramanujan primes in the interval from x to x/2 is greater than or equal to…

另一个受埃罗什启发的不等式

Another inequality inspired by Erdős

本文通过算术方法研究并解决了一个受埃罗什经典证明启发的正整数不等式，旨在寻找伯特兰公式的纯算术证明。

受Erdős启发的一个不等式

One inequality inspired by Erdős

受Erdős证明Bertrand假设的启发，本文求解了一个正整数不等式，为证明任意正整数n≥2存在素数p满足n<p<2n提供了纯算术证明思路。

Generalized Ramanujan Primes

In 1845, Bertrand conjectured that for all integers $x\ge2$, there exists at least one prime in $(x/2, x]$. This was proved by Chebyshev in 1860, and then generalized by Ramanujan in 1919. He showed that for any $n\ge1$, there is a (smallest) prime $R_n$ such that $π(x)- π(x/2) \ge n$ for all $x \ge R_n$. In 2009 Sondow called $R_n$ the $n$th Ramanujan prime and proved the asymptotic behavior $R_n \sim p_{2n}$ (where $p_m$ is the $m$th prime). In the present paper, we generalize the interval of interest by introducing a parameter $c \in (0,1)$ and defining the $n$th $c$-Ramanujan prime as the …

On The Prime Numbers In Intervals

Bertrand's postulate establishes that for all positive integers $n>1$ there exists a prime number between $n$ and $2n$. We consider a generalization of this theorem as: for integers $n\geq k\geq 2$ is there a prime number between $kn$ and $(k+1)n$? We use elementary methods of binomial coefficients and the Chebyshev functions to establish the cases for $2\leq k\leq 8$. We then move to an analytic number theory approach to show that there is a prime number in the interval $(kn, (k+1)n)$ for at least $n\geq k$ and $2\leq k\leq 519$. We then consider Legendre's conjecture on the existence of a pr…