Smooth values of shifted primes in arithmetic progressions

Banks, William D.; Harcharras, Asma; Shparlinski, Igor E.

doi:10.1307/mmj/1100623415

Michigan Math. J.

2004

DOI: 10.1307/mmj/1100623415

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Smooth values of shifted primes in arithmetic progressions

William D. Banks¹,

Asma Harcharras²,

Igor E. Shparlinski³

Abstract: We study the problem of bounding the number of primes p ≤ x in an arithmetic progression for which the largest prime factor of p − h does not exceed y.

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Cited by 2 publications

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Autour des plus grands facteurs premiers d’entiers consÉcutifs voisins d’un entier criblé

Wang

2018

The Quarterly Journal of Mathematics

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Denote by P+(n) (resp. P−(n)) the largest (resp. the smallest) prime factor of the integer n. In this paper, we prove that there exists a positive proportion of integers n having no small prime factor such that P+(n)<P+(n+2). Especially, we prove that the pattern P+(P3)<P+(P3+2) is realized by a positive proportion of P3 with P−(P3)>x1/3−δ,0<δ≤112, where P3 denote an integer having at most three prime factors taken with multiplicity. We also prove that the pattern P+(p−1)<P+(p+1) holds for a positive proportion of primes under the Elliott-Halberstam conjecture.

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Autour des plus grands facteurs premiers d’entiers consÉcutifs voisins d’un entier criblé

Wang

2018

The Quarterly Journal of Mathematics

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Distributional properties of the largest prime factor

Banks¹,

Harman²,

Shparlinski³

2005

Michigan Math. J.

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Smooth values of shifted primes in arithmetic progressions

Abstract: We study the problem of bounding the number of primes p ≤ x in an arithmetic progression for which the largest prime factor of p − h does not exceed y.

Cited by 2 publications

References 16 publications

Autour des plus grands facteurs premiers d’entiers consÉcutifs voisins d’un entier criblé

Autour des plus grands facteurs premiers d’entiers consÉcutifs voisins d’un entier criblé

Distributional properties of the largest prime factor

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