Product of three primes in large arithmetic progressions-Reference-Cited by-同舟云学术

Product of three primes in large arithmetic progressions

Published:2022-10-15 Issue:04 Volume:19 Page:843-857
ISSN:1793-0421
Container-title:International Journal of Number Theory
language:en
Short-container-title:Int. J. Number Theory

Author:

Balasubramanian Ramachandran¹²,Ramaré Olivier³,Srivastav Priyamvad⁴

Affiliation:

1. Institute of Mathematical Sciences, Taramani, Chennai, India 600113, India

2. Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai, India 400094, India

3. CNRS/Institut de Mathématiques de Marseille, Aix Marseille Université, U.M.R. 7373, Site Sud, Campus de Luminy, Case 907, 13288 MARSEILLE Cedex 9, France

4. Mathematisches Institut, Bunsenstrafie 3-5 37073, Gottingen, Germany

Abstract

For any [Formula: see text], there exists [Formula: see text] such for any [Formula: see text] and any invertible residue class [Formula: see text] modulo [Formula: see text], there exists a natural number that is congruent to [Formula: see text] modulo [Formula: see text] and that is the product of exactly three primes, all of which are below [Formula: see text]. If we restrict our attention to odd moduli [Formula: see text] that do not have prime factors congruent to 1 mod 4, we can find such primes below [Formula: see text]. If we further restrict our set of moduli to prime [Formula: see text] that are such that [Formula: see text], we can find such primes below [Formula: see text]. Finally, for any [Formula: see text], there exists [Formula: see text] such that when [Formula: see text], there exists a natural number that is congruent to [Formula: see text] modulo [Formula: see text] and that is the product of exactly four primes, all of which are below [Formula: see text].

Funder

Indo-French Centre for the Promotion of Advanced Research — CEFIPRA

FWF-ANR project Arithrand

Publisher

World Scientific Pub Co Pte Ltd

Subject

Algebra and Number Theory

Link

https://www.worldscientific.com/doi/pdf/10.1142/S1793042123500422

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