no. G3
Article de recherche - Théorie des nombres
On weakly prime-additive numbers with length 4k+3
Fang, Jin-Hui1  ; Xue, Fang-Gang2
1School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China
2Nanjing University of Information Science & Technology, Nanjing 210044, P.R. China
Comptes Rendus. Mathématique, Tome 362 (2024) no. G3, 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
Keywords: weakly prime-additive numbers, Dirichlet’s theorem, the Chinese remainder theorem

Fang, Jin-Hui  1   ; Xue, Fang-Gang  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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     title = {On weakly prime-additive numbers with length $4k+3$},
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     pages = {275--278},
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Fang, Jin-Hui; Xue, Fang-Gang. On weakly prime-additive numbers with length $4k+3$. Comptes Rendus. Mathématique, Tome 362 (2024) no. G3, pp. 275-278. doi: 10.5802/crmath.555

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

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

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