TY - JOUR

T1 - On products of primes and almost primes in arithmetic progressions

AU - Shparlinski, Igor E.

PY - 2013/9

Y1 - 2013/9

N2 - 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. Erdo{double acute}s, A. M. Odlyzko and A. Sárközy, that corresponds to k = 1 (that is, to products of two primes).

AB - 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. Erdo{double acute}s, A. M. Odlyzko and A. Sárközy, that corresponds to k = 1 (that is, to products of two primes).

KW - primes

KW - almost primes

KW - sieve method

KW - exponential sums

UR - http://www.scopus.com/inward/record.url?scp=84881617013&partnerID=8YFLogxK

U2 - 10.1007/s10998-013-2736-3

DO - 10.1007/s10998-013-2736-3

M3 - Article

AN - SCOPUS:84881617013

VL - 67

SP - 55

EP - 61

JO - Periodica Mathematica Hungarica

JF - Periodica Mathematica Hungarica

SN - 0031-5303

IS - 1

ER -