M. Vassilev-Missana
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 7, 2001, Number 1, Pages 15–20
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M. Vassilev-Missana
5, V. Hugo Str., Sofia-1124, Bulgaria
Abstract
Let C = {Cn}n≥1 be an arbitrary increasing sequence of natural numbers. By πC(n) we denote the number of the terms of C being not greater than n (we agree that πC(0) = 0). In the first part of the paper we propose six different formulae for Cn (n = 1, 2, …), which depend on the numbers πC(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 pn, 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-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 π2(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 π2(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.
