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On the least almost-prime in arithmetic progression

Jinjiang LiMin ZhangYingchun 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 .

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