On shifted primes with large prime factors and their products

On shifted primes with large prime factors and their products

We estimate from below the lower density of the set of prime numbers p such that p - 1 has a prime factor of size at least [p.sup.c], where 1/4 [less than or equal to] c [less than or equal to] 1/2. We also establish upper and lower bounds on the counting function of the set of positive integers n [...

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Veröffentlicht in:Bulletin of the Belgian Mathematical Society, Simon Stevin Simon Stevin, 2015-03, Vol.22 (1), p.39-47
Hauptverfasser: Luca, Florian, Menares, Ricardo, Pizarro-Madariaga, Amalia
Format: Artikel
Sprache:eng
Schlagworte:
11N36
11N37
Functions (Mathematics)
Mathematical research
Prime numbers
Shifted primes
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Zusammenfassung:We estimate from below the lower density of the set of prime numbers p such that p - 1 has a prime factor of size at least [p.sup.c], where 1/4 [less than or equal to] c [less than or equal to] 1/2. We also establish upper and lower bounds on the counting function of the set of positive integers n [less than or equal to] x with exactly k prime factors, counted with or without multiplicity, such that the largest prime factor of gcd(p - 1 : p | n) exceeds [n.sup.1/2k]. Key words and phrases : Shifted primes.
ISSN:1370-1444
2034-1970
DOI:10.36045/bbms/1426856856