Research article - Number theory
On weakly prime-additive numbers with length $4k+3$
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 275-278.

If a positive integer $n$ has at least two distinct prime divisors and can be written as $n={p}_{1}^{{\alpha }_{1}}+\cdots +{p}_{t}^{{\alpha }_{t}}$, where ${p}_{1}<\cdots <{p}_{t}$ are prime divisors of $n$ and ${\alpha }_{1},\cdots ,{\alpha }_{t}$ are positive integers, then we define such $n$ as weakly prime-additive. Obviously, $t\ge 3$. 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 $m\mid n$ and $n={p}_{1}^{{\alpha }_{1}}+\cdots +{p}_{5}^{{\alpha }_{5}}$, where ${p}_{1},\cdots ,{p}_{5}$ are distinct prime divisors of $n$ and ${\alpha }_{1},\cdots ,{\alpha }_{5}$ are positive integers. In this paper, we prove the existence of such $n$ with general length $t$, where $t\equiv 3\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}4\right)$ and $t>3$. The main result is summarized as follows: for any positive integers $m,t$ with $t\equiv 3\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}4\right)$ and $t>3$, there exist infinitely many weakly prime-additive numbers $n$ with $m\mid n$ and $n={p}_{1}^{{\alpha }_{1}}+\cdots +{p}_{t}^{{\alpha }_{t}}$, where ${p}_{1},\cdots ,{p}_{t}$ are distinct prime divisors of $n$ and ${\alpha }_{1},\cdots ,{\alpha }_{t}$ are positive integers.

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.555
Classification: 11A07, 11A41
Keywords: weakly prime-additive numbers, Dirichlet’s theorem, the Chinese remainder theorem

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
@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},
}
TY  - JOUR
AU  - Jin-Hui Fang
AU  - Fang-Gang Xue
TI  - On weakly prime-additive numbers with length $4k+3$
JO  - Comptes Rendus. Mathématique
PY  - 2024
SP  - 275
EP  - 278
VL  - 362
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.555
LA  - en
ID  - CRMATH_2024__362_G3_275_0
ER  - 
%0 Journal Article
%A Jin-Hui Fang
%A Fang-Gang Xue
%T On weakly prime-additive numbers with length $4k+3$
%J Comptes Rendus. Mathématique
%D 2024
%P 275-278
%V 362
%R 10.5802/crmath.555
%G en
%F CRMATH_2024__362_G3_275_0
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/

[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

Cited by Sources: