1. Products of primes in arithmetic progressions: a footnote in parity breaking
- Author
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Olivier Ramaré, Aled Walker, Institut de Mathématiques de Marseille (I2M), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), Mathematical Institute, University of Oxford, University of Oxford [Oxford], Indo-French Centre for the Promotion of Advanced Research - CEFIPRA 5401-1, EPSRC, EP/M50659X/1, Walker, Aled [0000-0002-9879-988X], Apollo - University of Cambridge Repository, Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), University of Oxford, and I2m, Aigle
- Subjects
Arithmetic progressions ,Modulo ,Cardinality ,0102 computer and information sciences ,01 natural sciences ,law.invention ,Combinatorics ,law ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Linnik's theorem ,Mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,010102 general mathematics ,Prime numbers ,Prime number ,Mathematical inequalities ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,math.NT ,Invertible matrix ,010201 computation theory & mathematics ,Mathematical theorems ,Linnik's Theorem ,Parity (mathematics) ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
We prove that, if $x$ and $q\leqslant x^{1/16}$ are two parameters, then for any invertible residue class $a$ modulo $q$ there exists a product of exactly three primes, each one below $x^{1/3}$, that is congruent to $a$ modulo $q$., 6 pages, to submitted to Journal de Th\'{e}orie des Nombres de Bordeaux. Small corrections to previous version
- Published
- 2019
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