On short products of primes in arithmetic progressions
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Abstract:
We give several families of reasonably small integers $k, \ell \ge 1$ and real positive $\alpha , \beta \le 1$, such that the products $p_1\ldots p_k s$, where $p_1, \ldots , p_k \le m^\alpha$ are primes and $s \le m^\beta$ is a product of at most $\ell$ primes, represent all reduced residue classes modulo $m$. This is a relaxed version of the still open question of P. Erdős, A. M. Odlyzko and A. Sárközy (1987), that corresponds to $k = \ell =1$ (that is, to products of two primes). In particular, we improve recent results of A. Walker (2016).References
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Additional Information
- Igor E. Shparlinski
- Affiliation: Department of Pure Mathematics, University of New South Wales, 2052 NSW, Australia
- MR Author ID: 192194
- Email: igor.shparlinski@unsw.edu.au
- Received by editor(s): October 5, 2017
- Received by editor(s) in revised form: June 13, 2018
- Published electronically: November 16, 2018
- Additional Notes: This work was partially supported by the Australian Research Council Grant DP170100786.
- Communicated by: Matthew A. Papanikolas
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 977-986
- MSC (2010): Primary 11N25; Secondary 11B25, 11L07, 11N36
- DOI: https://doi.org/10.1090/proc/14289
- MathSciNet review: 3896048