On weakly prime-additive numbers with length $4k+3$

Jin-Hui Fang; Fang-Gang Xue

doi:10.5802/crmath.555

Research article - Number theory

On weakly prime-additive numbers with length

4 k + 3

Jin-Hui Fang ¹ ; Fang-Gang Xue ²

¹ School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China
² Nanjing University of Information Science & Technology, Nanjing 210044, P.R. China

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 275-278.

Abstract

If a positive integer $n$ has at least two distinct prime divisors and can be written as $n = p_{1}^{α_{1}} + \dots + p_{t}^{α_{t}}$ , where $p_{1} < \dots < p_{t}$ are prime divisors of $n$ and $α_{1}, \dots, α_{t}$ are positive integers, then we define such $n$ as weakly prime-additive. Obviously, $t \geq 3$ . Following Erdős and Hegyvári’s work, Fang and Chen [J. Number Theorey 182(2018), 258-270] obtained the following result: for any positive integer $m$ , there exist infinitely many weakly prime-additive numbers $n$ with $m ∣ n$ and $n = p_{1}^{α_{1}} + \dots + p_{5}^{α_{5}}$ , where $p_{1}, \dots, p_{5}$ are distinct prime divisors of $n$ and $α_{1}, \dots, α_{5}$ are positive integers. In this paper, we prove the existence of such $n$ with general length $t$ , where $t \equiv 3 (mod 4)$ and $t > 3$ . The main result is summarized as follows: for any positive integers $m, t$ with $t \equiv 3 (mod 4)$ and $t > 3$ , there exist infinitely many weakly prime-additive numbers $n$ with $m ∣ n$ and $n = p_{1}^{α_{1}} + \dots + p_{t}^{α_{t}}$ , where $p_{1}, \dots, p_{t}$ are distinct prime divisors of $n$ and $α_{1}, \dots, α_{t}$ are positive integers.

Received: 2023-07-28
Revised: 2023-08-14
Accepted: 2023-08-17
Published online: 2024-05-02

DOI: 10.5802/crmath.555

Classification: 11A07, 11A41
Keywords: weakly prime-additive numbers, Dirichlet’s theorem, the Chinese remainder theorem

Author's affiliations:

Jin-Hui Fang ¹; Fang-Gang Xue ²

¹ School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China
² Nanjing University of Information Science & Technology, Nanjing 210044, P.R. China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2024__362_G3_275_0,
     author = {Jin-Hui Fang and Fang-Gang Xue},
     title = {On weakly prime-additive numbers with length $4k+3$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {275--278},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.555},
     language = {en},
}

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Jin-Hui Fang; Fang-Gang Xue. On weakly prime-additive numbers with length $4k+3$. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 275-278. doi : 10.5802/crmath.555. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.555/

References
Cited by

[1] Jean-Marie De Koninck; Florian Luca Integers representable as the sum of powers of their prime factors, Funct. Approximatio, Comment. Math., Volume 33 (2005), pp. 57-72 | MR | Zbl

[2] Pál Erdős; Norbert Hegyvári On prime-additive numbers, Stud. Sci. Math. Hung., Volume 27 (1992) no. 1-2, pp. 207-212 | MR | Zbl

[3] Jin-Hui Fang Note on the weakly prime-additive numbers, J. Nanjing Norm. Univ., Nat. Sci. Ed., Volume 41 (2018) no. 4, pp. 26-28 | MR | Zbl

[4] Jin-Hui Fang A note on weakly prime-additive numbers, Int. J. Number Theory, Volume 18 (2022) no. 1, pp. 175-178 | DOI | MR | Zbl

[5] Jin-Hui Fang; Yong-Gao Chen On the shortest weakly prime-additive numbers, J. Number Theory, Volume 182 (2018), pp. 258-270 | DOI | MR | Zbl

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