On the number of prime factors of the composite numbers resulting after a change of digits of primes

Benli, Kübra

doi:10.5802/jtnb.1103

Benli, Kübra ¹

¹ Department of Mathematics University of Georgia Athens GA 30602, USA

Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 689-696.

Résumé
Abstract

Dans cette note, nous prouvons que pour tout entier fixé $K \geq 2$ , pour tout $ϵ > 0$ et pour tout $x$ suffisamment grand, il existe au moins $x^{1 - ϵ}$ nombres premiers $x < p \leq (1 + K^{- 1}) x$ tels que tous les nombres entiers de la forme $p j \pm a^{h} k$ avec $2 \leq a \leq K, 0 < | k | \leq K, 1 \leq j \leq K, 0 \leq h \leq K log x$ sont des nombres composés ayant au moins ${(log log x)}^{1 - ϵ}$ facteurs premiers distincts.

In this note, we prove that for any fixed integer $K \geq 2$ , for all $ϵ > 0$ and for all sufficiently large $x$ , there exist at least $x^{1 - ϵ}$ primes $x < p \leq (1 + K^{- 1}) x$ , such that all of the integers $p j \pm a^{h} k, 2 \leq a \leq K, 0 < | k | \leq K, 1 \leq j \leq K, 0 \leq h \leq K log x$ are composite having at least ${(log log x)}^{1 - ϵ}$ distinct prime factors.

Reçu le : 2019-01-11
Révisé le : 2019-05-26
Accepté le : 2019-07-15
Publié le : 2020-05-06

DOI : 10.5802/jtnb.1103

Classification : 11A41, 11P32
Mots-clés : primes, digit, composite numbers

Affiliations des auteurs :

Benli, Kübra ¹

¹ Department of Mathematics University of Georgia Athens GA 30602, USA

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     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 = {https://www.numdam.org/articles/10.5802/jtnb.1103/}
}

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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. https://www.numdam.org/articles/10.5802/jtnb.1103/

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