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, Volume 31 (2019) no. 3, pp. 689-696.

In this note, we prove that for any fixed integer $K\ge 2$, for all $ϵ>0$ and for all sufficiently large $x$, there exist at least ${x}^{1-ϵ}$ primes $x, such that all of the integers $pj±{a}^{h}k,\phantom{\rule{0.166667em}{0ex}}2\le a\le K,\phantom{\rule{0.166667em}{0ex}}0<|k|\le K,\phantom{\rule{0.166667em}{0ex}}1\le j\le K,\phantom{\rule{0.166667em}{0ex}}0\le h\le Klogx$ are composite having at least ${\left(loglogx\right)}^{1-ϵ}$ distinct prime factors.

Dans cette note, nous prouvons que pour tout entier fixé $K\ge 2$, pour tout $ϵ>0$ et pour tout $x$ suffisamment grand, il existe au moins ${x}^{1-ϵ}$ nombres premiers $x tels que tous les nombres entiers de la forme $pj±{a}^{h}k$ avec $2\le a\le K,\phantom{\rule{0.166667em}{0ex}}0<|k|\le K,\phantom{\rule{0.166667em}{0ex}}1\le j\le K,\phantom{\rule{0.166667em}{0ex}}0\le h\le Klogx$ sont des nombres composés ayant au moins ${\left(loglogx\right)}^{1-ϵ}$ facteurs premiers distincts.

Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1103
Classification: 11A41, 11P32
Keywords: primes, digit, composite numbers
Benli, Kübra 1

1 Department of Mathematics University of Georgia Athens GA 30602, USA
@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://www.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
SP  - 689
EP  - 696
VL  - 31
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1103/
DO  - 10.5802/jtnb.1103
LA  - en
ID  - JTNB_2019__31_3_689_0
ER  - 
%0 Journal Article
%A Benli, Kübra
%T On the number of prime factors of the composite numbers resulting after a change of digits of primes
%J Journal de théorie des nombres de Bordeaux
%D 2019
%P 689-696
%V 31
%N 3
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.1103/
%R 10.5802/jtnb.1103
%G en
%F JTNB_2019__31_3_689_0
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, Volume 31 (2019) no. 3, pp. 689-696. doi : 10.5802/jtnb.1103. http://www.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 | DOI | MR | Zbl

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

[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 | Zbl

[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 | DOI | MR | Zbl

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

Cited by Sources: