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 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.

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.

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

1 Department of Mathematics University of Georgia Athens GA 30602, USA
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Kübra Benli. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1103/

[1] George David Birkhoff; Harry S. Vandiver 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] Enrico Bombieri Le grand crible dans la théorie analytique des nombres, Astérisque, 18, Société Mathématique de France, 1987 | Zbl: 0618.10042

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

[4] U. V. Linnik 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] Hao Pan 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] Terence Tao 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

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