Elsholtz Christian, Luca Florian, Planitzer Stefan
1Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24, 8010 Graz, Austria.
2School of Mathematics, Wits University, Private Bag X3, Wits, Johannesburg, 2050 South Africa.
Ramanujan J. 2018;47(2):267-289. doi: 10.1007/s11139-017-9972-8. Epub 2018 Feb 8.
Romanov proved that the proportion of positive integers which can be represented as a sum of a prime and a power of 2 is positive. We establish similar results for integers of the form and where and , are primes. In the opposite direction, Erdős constructed a full arithmetic progression of odd integers none of which is the sum of a prime and a power of two. While we also exhibit in both cases full arithmetic progressions which do not contain any integers of the two forms, respectively, we prove a much better result for the proportion of integers not of these forms: (1) The proportion of positive integers not of the form is larger than . (2) The proportion of positive integers not of the form is at least .
罗曼诺夫证明了能够表示为一个质数与2的幂之和的正整数的比例是正的。我们针对形如(p + 2^k)（其中(p)和(q)是质数）的整数建立了类似的结果。在相反的方向上，埃尔德什构造了一个全是奇数的等差数列，其中没有一个数是质数与2的幂之和。虽然在这两种情况下我们也分别展示了不包含这两种形式中任何整数的全等差数列，但我们证明了对于不是这些形式的整数的比例有一个更好得多的结果：（1）不是(p + 2^k)形式的正整数的比例大于(\frac{1}{2})。（2）不是(p + q^k)形式的正整数的比例至少为(\frac{1}{4})。
