Abstract Let 0 < α, σ < 1 be arbitrary fixed constants, let $${{q}_{1}} < {{q}_{2}} < \ldots < {{q}_{n}} < {{q}_{{n + 1}}}$$ < … be the set of primes satisfying the condition $$\{ q_{n}^{\alpha }\} < \sigma $$ and indexed in ascending order, and let $$m \geqslant 1$$ be any fixed integer. Using an analogue of the Bombieri–Vinogradov theorem for the above set of primes, upper bounds are obtained for the constants c ( m ) such that the inequality $${{q}_{{n + m}}} - {{q}_{n}} \leqslant c\left( m \right)$$ holds for infinitely many n .

中文翻译：

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特殊形式素数之间的有界间隙

摘要 设 0 < α, σ < 1 为任意固定常数，令 $${{q}_{1}} < {{q}_{2}} < \ldots < {{q}_{n}} < {{q}_{{n + 1}}}$$ < ... 是满足条件 $$\{ q_{n}^{\alpha }\} < \sigma $$ 并按升序索引的素数集，并令 $$m \geqslant 1$$ 为任意固定整数。使用 Bombieri-Vinogradov 定理对上述素数集的模拟，可以得到常数 c ( m ) 的上限，使得不等式 $${{q}_{{n + m}}}​​ - {{q }_{n}} \leqslant c\left( m \right)$$ 对无穷多个 n 成立。