1934年,Romanoff证明了能表成2的方幂与一个素数之和形式的正整数在正整数集合中有正的比例．最近,本文作者证明了对充分大的x,能表成2的方幂与一个素数之和形式的正整数在不超过x的正整数中至少有0．0868x个．本文证明了:设 x≥5,则在不超过x的正整数中,能表成2的方幂与一个素数之和的数的个数不少于 0．005x,即给出了Romanoff定理的定量形式．

In 1934, Romanoff proved that there are positive proportion natural numbers which can be expressed as a sum of a prime and a power of 2. Recently, the authors proved that for all sufficiently larger x, the proportion is large than 0.0868. In this paper,a quantitative version of Romanoff's Theorem is given. The following result is proved: For all x≥5, the proportion of natural numbers which can be expressed as a sum of a prime and a power of 2 is larger than 0.005.