In this paper we give an algorithm for polignac numbers. That algorithm is also the base of the proof of the de Polignac's conjecture. The examples of the application present algorithm for twin, cousin, and sexy p...In this paper we give an algorithm for polignac numbers. That algorithm is also the base of the proof of the de Polignac's conjecture. The examples of the application present algorithm for twin, cousin, and sexy primes are included.展开更多
Let A be any subset of positive integers,and P the set of all positive primes.Two of our results are:(a) the number of positive integers which are less than x and can be represented as 2k + p(resp.p-2k) with k ∈ A an...Let A be any subset of positive integers,and P the set of all positive primes.Two of our results are:(a) the number of positive integers which are less than x and can be represented as 2k + p(resp.p-2k) with k ∈ A and p ∈ P is more than 0.03A(log x/log 2)π(x) for all sufficiently large x;(b) the number of positive integers which are less than x and can be represented as 2q + p with p,q ∈ P is(1 + o(1))π(log x/log 2)π(x).Four related open problems and one conjecture are posed.展开更多
文摘In this paper we give an algorithm for polignac numbers. That algorithm is also the base of the proof of the de Polignac's conjecture. The examples of the application present algorithm for twin, cousin, and sexy primes are included.
基金supported by National Natural Science Foundation of China (Grant No.10771103)
文摘Let A be any subset of positive integers,and P the set of all positive primes.Two of our results are:(a) the number of positive integers which are less than x and can be represented as 2k + p(resp.p-2k) with k ∈ A and p ∈ P is more than 0.03A(log x/log 2)π(x) for all sufficiently large x;(b) the number of positive integers which are less than x and can be represented as 2q + p with p,q ∈ P is(1 + o(1))π(log x/log 2)π(x).Four related open problems and one conjecture are posed.
