On arithmetic terms expressing the prime-counting function and the n-th prime
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911089556455424 |
|---|---|
| author | Prunescu, Mihai Shunia, Joseph M. |
| author_facet | Prunescu, Mihai Shunia, Joseph M. |
| contents | We present the first fixed-length elementary closed-form expressions for the prime-counting function, $π(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of elementary arithmetic operations from the set: addition, subtraction, multiplication, integer division, and exponentiation. Mazzanti proved that every Kalmar function can be represented as an arithmetic term. We develop an arithmetic term representing the prime omega function, $ω(n)$, which counts the number of distinct prime divisors of a positive integer $n$. From this term, we find immediately an arithmetic term for the prime-counting function, $π(n)$. Combining these results with a new arithmetic term for binomial coefficients and novel prime-related exponential Diophantine equations, we manage to develop an arithmetic term for the $n$-th prime number, $p(n)$, thereby providing a constructive solution to the fundamental question: Is there an order to the primes? |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14594 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On arithmetic terms expressing the prime-counting function and the n-th prime Prunescu, Mihai Shunia, Joseph M. Number Theory 11A41 (primary), 11A25, 03D20 (secondary) We present the first fixed-length elementary closed-form expressions for the prime-counting function, $π(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of elementary arithmetic operations from the set: addition, subtraction, multiplication, integer division, and exponentiation. Mazzanti proved that every Kalmar function can be represented as an arithmetic term. We develop an arithmetic term representing the prime omega function, $ω(n)$, which counts the number of distinct prime divisors of a positive integer $n$. From this term, we find immediately an arithmetic term for the prime-counting function, $π(n)$. Combining these results with a new arithmetic term for binomial coefficients and novel prime-related exponential Diophantine equations, we manage to develop an arithmetic term for the $n$-th prime number, $p(n)$, thereby providing a constructive solution to the fundamental question: Is there an order to the primes? |
| title | On arithmetic terms expressing the prime-counting function and the n-th prime |
| topic | Number Theory 11A41 (primary), 11A25, 03D20 (secondary) |
| url | https://arxiv.org/abs/2412.14594 |