Let k ≥ 2 and s be positive integers. Let θ ∈ (0, 1) be a real number. In this paper, we establish that if s > k(k + 1) and θ > 0.55, then every sufficiently large natural number n, subject to certain congruence conditions, can be written as

, where p_i (1 ≤ i ≤ s) are primes in the interval \(({({n \over s})^{{1 \over k}}} - {n^{{\theta \over k}}},{({n \over s})^{{1 \over k}}} + {n^{{\theta \over k}}}]\). The second result of this paper is to show that if \(s > {{k(k + 1)} \over 2}\) and θ > 0.55, then almost all integers n, subject to certain congruence conditions, have the above representation.

设k ≥ 2 且s为正整数。令θ ∈ (0, 1) 为实数。在本文中，我们确定如果s > k ( k + 1) 且θ > 0.55，则每个足够大的自然数n在满足一定同余条件的情况下可以写为

，其中p _i (1 ≤ i ≤ s ) 是区间\(({({n \over s})^{{1 \over k}}} - {n^{{\theta \over k} }},{({n \over s})^{{1 \over k}}} + {n^{{\theta \over k}}}]\) 。本文的第二个结果是表明如果\(s > {{k(k + 1)} \over 2}\)且θ > 0.55，则几乎所有整数n在某些同余条件下都具有上述表示。