In this paper, we study the binary Goldbach problem in the set of the Piatetski-Shapiro primes. We obtain that for all most all large even integer n, the equationn = p1+p2, pi∈Pγi, i = 1, 2has solutions, where 0 &l...In this paper, we study the binary Goldbach problem in the set of the Piatetski-Shapiro primes. We obtain that for all most all large even integer n, the equationn = p1+p2, pi∈Pγi, i = 1, 2has solutions, where 0 < γ1,γ2 ≤ 1 are fixed real numbers, such that 73(1 - γ2) < 9, 73(1 - γ1) + 43(1 - γ2) < 9.展开更多
In 2012,Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form {n√zn},where {zn} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {zn+1/zn}is increasing(r...In 2012,Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form {n√zn},where {zn} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {zn+1/zn}is increasing(resp.,decreasing),then the sequence {n√zn} is strictly increasing(resp.,decreasing) subject to a certain initial condition. We also give some sufficient conditions when {zn+1/zn} is increasing,which is equivalent to the log-convexity of {zn}. As consequences,a series of conjectures of Zhi-Wei Sun are verified in a unified approach.展开更多
基金Supported by the Foundation of Shandong Provincial Education Department(03F06)
文摘In this paper, we study the binary Goldbach problem in the set of the Piatetski-Shapiro primes. We obtain that for all most all large even integer n, the equationn = p1+p2, pi∈Pγi, i = 1, 2has solutions, where 0 < γ1,γ2 ≤ 1 are fixed real numbers, such that 73(1 - γ2) < 9, 73(1 - γ1) + 43(1 - γ2) < 9.
基金supported by National Natural Science Foundation of China(Grant Nos.11071030,11201191 and 11371078)Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20110041110039)+1 种基金National Science Foundation of Jiangsu Higher Education Institutions(GrantNo.12KJB110005)the Priority Academic Program Development of Jiangsu Higher Education Institutions(Grant No.11XLR30)
文摘In 2012,Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form {n√zn},where {zn} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {zn+1/zn}is increasing(resp.,decreasing),then the sequence {n√zn} is strictly increasing(resp.,decreasing) subject to a certain initial condition. We also give some sufficient conditions when {zn+1/zn} is increasing,which is equivalent to the log-convexity of {zn}. As consequences,a series of conjectures of Zhi-Wei Sun are verified in a unified approach.
