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On products of primes and almost primes in arithmetic progressions

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Abstract

We show that for any integers a and m with m ≥ 1 and gcd(a,m) = 1, there is a solution to the congruence pr ≡ a (modm) where p is prime, r is a product of at most k = 17 prime factors and p, r ≤ m. This is a relaxed version of the still open question, studied by P. Erdős, A. M. Odlyzko and A. Sárközy, that corresponds to k = 1 (that is, to products of two primes).

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Authors and Affiliations

  1. Department of Computing, Macquarie University, Sydney, NSW, 2109, Australia

    Igor E. Shparlinski

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  1. Igor E. Shparlinski
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Correspondence to Igor E. Shparlinski.

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Communicated by András Sárközy

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Shparlinski, I.E. On products of primes and almost primes in arithmetic progressions. Period Math Hung 67, 55–61 (2013). https://doi.org/10.1007/s10998-013-2736-3

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  • DOI: https://doi.org/10.1007/s10998-013-2736-3

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