A new formula for the nth prime

In this paper, we propose a new primality test, and then we employ this test to find a formula for {\pi} that computes the number of primes within any interval. We finally propose a new formula that computes the nth prime number as well as the next prime for any given number

A new set of formulas for primes is presented. These formulas are more efficient and grow much slower than the two known formulas of Mills and Wright. 3 new formulas are explained.

In this article we gave a recurrence to obtain the n-th prime number as function of the (n-1)-th prime number.

We present a simple, closed formula which gives all the primes in order. It is a simple product of integer floor and ceiling functions.

We present in this work a heuristic expression for the density of prime numbers. Our expression leads to results which possesses approximately the same precision of the Riemann's function in the domain that goes from 2 to 1010 at least. Instead of using a constant as was done by Legendre and others in the formula of Gauss, we try to adjust the data through a function. This function has the remarkable property: its points of discontinuity are the prime numbers.

In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.

Using inequalities of Rosser and Schoenfeld, we prove formulas for pi(n) and the n-th prime that involve only the elementary operations +,-,/ on integers, together with the floor function. Pascal Sebah has pointed out that the formula for pi(n) operates in O(n^(3/2)) time. Similar formulas were proven using Bertrand's Postulate by Stephen Regimbal, An explicit formula for the k-th prime number, Mathematics Magazine, 48 (1975), 230-23

I develop a function that, for any integer $n \geq 2$, takes a value of 1 if $n$ is prime, 0 if $n$ is composite. I also discuss two applications: First, the characteristic function provides a new expression for the prime counting function. Second, the components of the characteristic function point to a new expression that gives the number of distinct prime factors for any integer greater than one.

Counting the number of prime numbers up to a certain natural number and describing the asymptotic behavior of such a counting function has been studied by famous mathematicians like Gauss, Legendre, Dirichlet, and Euler. The prime number theorem determines that such asymptotic behavior is similar to the asymptotic behavior of the number divided by its natural logarithm. In this paper, we take advantage of a multiplicative representation of a number and the properties of the l...

This is an expanded account of three lectures on the distribution of prime numbers given at the Montreal NATO school on equidistribution.