**M. Vassilev-Missana**

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 7, 2001, Number 1, Pages 15–20

**Full paper (PDF, 191 Kb)**

## Details

### Authors and affiliations

**M. Vassilev-Missana**

*5, V. Hugo Str., Sofia-1124, Bulgaria*

### Abstract

Let *C* = {*C _{n}*}

_{n≥1}be an arbitrary increasing sequence of natural numbers. By

*π*(

_{C}*n*) we denote the number of the terms of

*C*being not greater than

*n*(we agree that

*π*(0) = 0). In the first part of the paper we propose six different formulae for

_{C}*C*(

_{n}*n*= 1, 2, …), which depend on the numbers

*π*(

_{C}*k*) (

*k*= 0, 1, 2, …). Using these formulae, in the second part of the paper we obtain three different explicit formulae for the

*n*-th prime

*p*, which are the first main result of the present research. In the third part of the paper, using the formulae from the first part, we propose six explicit formulae for the

_{n}*n*-th term of the sequence of twin primes: 3, 5, 7, 11, 13, 17, 19,… – the second main result of the paper. The last three of them are main ones for the twin primes.

### References

- Davenport, H. Multiplicative Number Theory. Markham Publ. Co., Chicago, 1967.
- Mitrinovic, D., M. Popadic. Inequalities in Number Theory. Nis, Univ. of Nis, 1978.
- Ribenboim, P. The New Book of Prime Number Records. Springer, New York, 1995.
- Vassilev-Missana, M. Some new formulae for the twin primes counting function
*π*2(*n*).*Notes on Number Theory and Discrete Mathematics*, Vol. 7, 2001, No. 1,

10-14. - Atanassov, K. A new formula for the
*n*-th prime number. Comptes Rendus de

l’Academie Bulgare des Sciences, Vol. 54, No. 7, 5-6.

## Related papers

- Vassilev-Missana, M. (2001). Some new formulae for the twin primes counting function
*π*2(*n*).*Notes on Number Theory and Discrete Mathematics*, 7(1), 10-14. - Vassilev-Missana, M. V. (2003). On 15-th Smarandache’s problem.
*Notes on Number Theory and Discrete Mathematics*, 9(2), 42-45. - Atanassov, K. T. (2021). Formulas for the
*n*-th prime number.*Notes on Number Theory and Discrete Mathematics*, 27(4), 129-139.

## Cite this paper

Vassilev-Missana, M. (2001). Three formulae for *n*-th prime and six for *n*-th term of twin primes. *Notes on Number Theory and Discrete Mathematics*, 7(1), 15-20.