Comptes Rendus
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On weakly prime-additive numbers with length 4k+3
Jin-Hui Fang1  ; Fang-Gang Xue2
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.

If a positive integer n has at least two distinct prime divisors and can be written as n=p 1 α 1 +⋯+p t α t , where p 1 <⋯<p t are prime divisors of n and α 1 ,⋯,α t are positive integers, then we define such n as weakly prime-additive. Obviously, t≥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 +⋯+p 5 α 5 , where p 1 ,⋯,p 5 are distinct prime divisors of n and α 1 ,⋯,α 5 are positive integers. In this paper, we prove the existence of such n with general length t, where t≡3(mod4) and t>3. The main result is summarized as follows: for any positive integers m,t with t≡3(mod4) and t>3, there exist infinitely many weakly prime-additive numbers n with m∣n and n=p 1 α 1 +⋯+p t α t , where p 1 ,⋯,p t are distinct prime divisors of n and α 1 ,⋯,α t are positive integers.

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DOI : 10.5802/crmath.555
Classification : 11A07, 11A41
Mots clés : 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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     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/

[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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