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On the least almost-prime in arithmetic progression

Jinjiang Li, Min Zhang, Yingchun Cai (2023)

Czechoslovak Mathematical Journal

Let 𝒫 r denote an almost-prime with at most r prime factors, counted according to multiplicity. Suppose that a and q are positive integers satisfying ( a , q ) = 1 . Denote by 𝒫 2 ( a , q ) the least almost-prime 𝒫 2 which satisfies 𝒫 2 a ( mod q ) . It is proved that for sufficiently large q , there holds 𝒫 2 ( a , q ) q 1 . 8345 . This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range 1 . 845 in place of 1 . 8345 .

On the least almost-prime in arithmetic progressions

Liuying Wu (2024)

Czechoslovak Mathematical Journal

Let 𝒫 2 denote a positive integer with at most 2 prime factors, counted according to multiplicity. For integers a , q such that ( a , q ) = 1 , let 𝒫 2 ( q , a ) denote the least 𝒫 2 in the arithmetic progression { n q + a } n = 1 . It is proved that for sufficiently large q , we have 𝒫 2 ( q , a ) q 1 . 825 . This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained 𝒫 2 ( q , a ) q 1 . 8345 .

Currently displaying 21 – 40 of 46