Let ,4 = {1 ≤ a1 〈 a2 〈 ...} be a sequence of integers. ,4 is called a sum-free sequence if no ai is the sum of two or more distinct earlier terms. Let A be the supremum of reciprocal sums of sum-free sequences. In...Let ,4 = {1 ≤ a1 〈 a2 〈 ...} be a sequence of integers. ,4 is called a sum-free sequence if no ai is the sum of two or more distinct earlier terms. Let A be the supremum of reciprocal sums of sum-free sequences. In 1962, ErdSs proved that A 〈 103. A sum-free sequence must satisfy an ≥ （k ~ 1）（n - ak） for all k, n ≥ 1. A sequence satisfying this inequality is called a x-sequence. In 1977, Levine and O＇Sullivan proved that a x-sequence A with a large reciprocal sum must have al = 1, a2 = 2, and a3 = 4. This can be used to prove that λ 〈 4. In this paper, it is proved that a x-sequence A with a large reciprocal sum must have its initial 16 terms： 1, 2, 4, 6, 9, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, and 50. This together with some new techniques can be used to prove that λ 〈 3.0752. Three conjectures are posed.展开更多
This paper studies the property of the recursive sequences in the 3x ＋ 1 conjecture. The authors introduce the concept of μ function, with which the 3x ＋ 1 conjecture can be transformed into two other conjectures：...This paper studies the property of the recursive sequences in the 3x ＋ 1 conjecture. The authors introduce the concept of μ function, with which the 3x ＋ 1 conjecture can be transformed into two other conjectures： one is eventually periodic conjecture of the μ function and the other is periodic point conjecture. The authors prove that the 3x ＋ 1 conjecture is equivalent to the two conjectures above. In 2007, J. L. Simons proved the non-existence of nontrivial 2-cycle for the T function. In this paper, the authors prove that the μ function has nol-periodic points for 2 ≤ 1 ≤12. In 2005, J. L. Simons and B. M. M de Weger proved that there is no nontrivial/-cycle for the T function for 1 ≤68, and in this paper, the authors prove that there is no nontrivial l-cycle for the μ function for 2 ≤ 1≤ 102.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
For any given coprime integers p and q greater than 1, in 1959, B proved that all sufficiently large integers can be expressed as a sum of pairwise terms of the form p^aq^b. As Davenport observed, Birch＇s proof can b...For any given coprime integers p and q greater than 1, in 1959, B proved that all sufficiently large integers can be expressed as a sum of pairwise terms of the form p^aq^b. As Davenport observed, Birch＇s proof can be modified that the exponent b can be bounded in terms of p and q. In 2000, N. Hegyvari effective version of this bound. The author improves this bound.展开更多
基金supported by National Natural Science Foundation of China(Grant No.11071121)
文摘Let ,4 = {1 ≤ a1 〈 a2 〈 ...} be a sequence of integers. ,4 is called a sum-free sequence if no ai is the sum of two or more distinct earlier terms. Let A be the supremum of reciprocal sums of sum-free sequences. In 1962, ErdSs proved that A 〈 103. A sum-free sequence must satisfy an ≥ （k ~ 1）（n - ak） for all k, n ≥ 1. A sequence satisfying this inequality is called a x-sequence. In 1977, Levine and O＇Sullivan proved that a x-sequence A with a large reciprocal sum must have al = 1, a2 = 2, and a3 = 4. This can be used to prove that λ 〈 4. In this paper, it is proved that a x-sequence A with a large reciprocal sum must have its initial 16 terms： 1, 2, 4, 6, 9, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, and 50. This together with some new techniques can be used to prove that λ 〈 3.0752. Three conjectures are posed.
基金supported by Natural Science Foundation of China under Grant Nos.60833008 and 60902024
文摘This paper studies the property of the recursive sequences in the 3x ＋ 1 conjecture. The authors introduce the concept of μ function, with which the 3x ＋ 1 conjecture can be transformed into two other conjectures： one is eventually periodic conjecture of the μ function and the other is periodic point conjecture. The authors prove that the 3x ＋ 1 conjecture is equivalent to the two conjectures above. In 2007, J. L. Simons proved the non-existence of nontrivial 2-cycle for the T function. In this paper, the authors prove that the μ function has nol-periodic points for 2 ≤ 1 ≤12. In 2005, J. L. Simons and B. M. M de Weger proved that there is no nontrivial/-cycle for the T function for 1 ≤68, and in this paper, the authors prove that there is no nontrivial l-cycle for the μ function for 2 ≤ 1≤ 102.
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.
基金Project supported by the National Natural Science Foundation of China (Nos. 10771103, 11071121)
文摘For any given coprime integers p and q greater than 1, in 1959, B proved that all sufficiently large integers can be expressed as a sum of pairwise terms of the form p^aq^b. As Davenport observed, Birch＇s proof can be modified that the exponent b can be bounded in terms of p and q. In 2000, N. Hegyvari effective version of this bound. The author improves this bound.
