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Journal Article

#### On short products of primes in arithmetic progressions

##### External Resource

https://doi.org/10.1090/proc/14289

(Publisher version)

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##### Fulltext (public)

arXiv:1705.06087.pdf

(Preprint), 183KB

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##### Citation

Shparlinski, I. E. (2019). On short products of primes in arithmetic progressions.* Proceedings of the American Mathematical Society,* *147*(3),
977-986. doi:10.1090/proc/14289.

Cite as: https://hdl.handle.net/21.11116/0000-0004-F813-F

##### 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. Erdos, A. M. Odlyzko and A. Sarkozy (1987), that corresponds to $k = \ell =1$ (that is, to

products of two primes). In particular, we improve recent results of A. Walker (2016).

products of two primes). In particular, we improve recent results of A. Walker (2016).