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We will establish a lower bound for the number of primes $p_r$, up to $X$, such that both $p_{r+1} - p_{r} < \\epsilon \\log p_r$ and $p_{r} \\equiv p_{r+1} \\equiv a \\bmod q$ simultaneously hold. As a lower bound for the number of primes satisfying the latter condition, the bound we obtain improves upon a bound obtained by D. 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