On the number of prime factors of the composite numbers resulting after a change of digits of primes
Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 689-696.

Dans cette note, nous prouvons que pour tout entier fixé K2, pour tout ϵ>0 et pour tout x suffisamment grand, il existe au moins x 1-ϵ nombres premiers x<p(1+K -1 )x tels que tous les nombres entiers de la forme pj±a h k avec 2aK,0<|k|K,1jK,0hKlogx sont des nombres composés ayant au moins (loglogx) 1-ϵ facteurs premiers distincts.

In this note, we prove that for any fixed integer K2, for all ϵ>0 and for all sufficiently large x, there exist at least x 1-ϵ primes x<p(1+K -1 )x, such that all of the integers pj±a h k,2aK,0<|k|K,1jK,0hKlogx are composite having at least (loglogx) 1-ϵ distinct prime factors.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.1103
Classification : 11A41,  11P32
Mots clés : primes, digit, composite numbers
@article{JTNB_2019__31_3_689_0,
     author = {Benli, K\"ubra},
     title = {On the number of prime factors of the composite numbers resulting after a change of digits of primes},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {689--696},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {3},
     year = {2019},
     doi = {10.5802/jtnb.1103},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.1103/}
}
TY  - JOUR
AU  - Benli, Kübra
TI  - On the number of prime factors of the composite numbers resulting after a change of digits of primes
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2019
DA  - 2019///
SP  - 689
EP  - 696
VL  - 31
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - http://archive.numdam.org/articles/10.5802/jtnb.1103/
UR  - https://doi.org/10.5802/jtnb.1103
DO  - 10.5802/jtnb.1103
LA  - en
ID  - JTNB_2019__31_3_689_0
ER  - 
Benli, Kübra. On the number of prime factors of the composite numbers resulting after a change of digits of primes. Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 689-696. doi : 10.5802/jtnb.1103. http://archive.numdam.org/articles/10.5802/jtnb.1103/

[1] Birkhoff, George David; Vandiver, Harry S. On the Integral Divisors of a n -b n , Ann. Math., Volume 5 (1904) no. 4, pp. 173-180 | Article | MR 1503541 | Zbl 35.0205.01

[2] Bombieri, Enrico Le grand crible dans la théorie analytique des nombres, Astérisque, 18, Société Mathématique de France, 1987 | Zbl 0618.10042

[3] Erdős, Paul Solution to problem 1029: Erdős and the computer, Math. Mag., Volume 52 (1979), pp. 180-181

[4] Linnik, U. V. On the least prime in an arithmetic progression. I. The basic theorem, Mat. Sb., N. Ser., Volume 15 (1944) no. 57, pp. 139-178 | MR 12111 | Zbl 0063.03584

[5] Pan, Hao On the number of distinct prime factors of nj+a h k, Monatsh. Math., Volume 175 (2014) no. 2, pp. 293-305 | Article | MR 3260872 | Zbl 1311.11097

[6] Tao, Terence A remark on primality testing and decimal expansions, J. Aust. Math. Soc., Volume 91 (2011) no. 3, pp. 405-413 | MR 2900615 | Zbl 1251.11089

Cité par Sources :