On the least almost‐prime in an arithmetic progression

Key Points

• To demonstrate the existence of integers with at most two prime factors in arithmetic progressions of sufficiently large numbers.
• Utilized the weighted sieve of Greaves–Halberstam–Richert
• Applied bilinear remainder terms in the analysis
• Employed Selberg's sieve for the proof.
• Identified at least one integer with two or fewer prime factors in each appropriate arithmetic progression.
• Confirmed that these integers are asymptotically less than specific bounds.