This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An...This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.展开更多
In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> for every positive integer n. We also prove three related conjectures. The m...In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> for every positive integer n. We also prove three related conjectures. The method that we use is to analyze binomial coefficients. It is developed by the author from the method of analyzing binomial central coefficients, that was used by Paul Erdős in his proof of Bertrand’s postulate - Chebyshev’s theorem.展开更多
This is a series of studies on Wu＇s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy＇s function f（z） （z = x ＋ iy） and its integral ...This is a series of studies on Wu＇s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy＇s function f（z） （z = x ＋ iy） and its integral J[f（z）]≡（2πi）-∮cf（t）（t-z）-1dt taken along the unit circle as contour C,inside which（the open domain D＋） f（z） is regular but has singularities distributed in open domain Doutside C. Resolution is given to the inverse problem that the singularities of f（z） can be determined in analytical form in terms of the values f（t） of f（z） numerically prescribed on C（｜t｜ = 1） ,as so enunciated by Wu＇s conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle,for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction,and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics,engineering and technology in analogy with resolving the inverse problem presented here.展开更多
Let k be a positive integer.A graph G is k-weight choosable if,for any assignment L(e)of k real numbers to each e∈E(G),there is a mapping f:E(G)→R such that f(uv)∈L(uv)and∑e∈∂(u)^f(e)≠∑e∈∂(u)^f(e)for each uv∈...Let k be a positive integer.A graph G is k-weight choosable if,for any assignment L(e)of k real numbers to each e∈E(G),there is a mapping f:E(G)→R such that f(uv)∈L(uv)and∑e∈∂(u)^f(e)≠∑e∈∂(u)^f(e)for each uv∈E(G),where?(v)is the set of edges incident with v.As a strengthening of the famous 1-2-3-conjecture,Bartnicki,Grytczuk and Niwcyk[Weight choosability of graphs.J.Graph Theory,60,242–256(2009)]conjecture that every graph without isolated edge is 3-weight choosable.This conjecture is wildly open and it is even unknown whether there is a constant k such that every graph without isolated edge is k-weight choosable.In this paper,we show that every connected graph of maximum degree 4 is 4-weight choosable.展开更多
文摘This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.
文摘In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> for every positive integer n. We also prove three related conjectures. The method that we use is to analyze binomial coefficients. It is developed by the author from the method of analyzing binomial central coefficients, that was used by Paul Erdős in his proof of Bertrand’s postulate - Chebyshev’s theorem.
文摘This is a series of studies on Wu＇s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy＇s function f（z） （z = x ＋ iy） and its integral J[f（z）]≡（2πi）-∮cf（t）（t-z）-1dt taken along the unit circle as contour C,inside which（the open domain D＋） f（z） is regular but has singularities distributed in open domain Doutside C. Resolution is given to the inverse problem that the singularities of f（z） can be determined in analytical form in terms of the values f（t） of f（z） numerically prescribed on C（｜t｜ = 1） ,as so enunciated by Wu＇s conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle,for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction,and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics,engineering and technology in analogy with resolving the inverse problem presented here.
基金Supported by National Natural Science Foundation of China(Grant Nos.11871397 and 11971205)the Natural Science Basic Research Plan in Shaanxi Province of China(Grant No.2020JM-083)the Fundamental Research Funds for the Central Universities(Grant No.3102019ghjd003)。
文摘Let k be a positive integer.A graph G is k-weight choosable if,for any assignment L(e)of k real numbers to each e∈E(G),there is a mapping f:E(G)→R such that f(uv)∈L(uv)and∑e∈∂(u)^f(e)≠∑e∈∂(u)^f(e)for each uv∈E(G),where?(v)is the set of edges incident with v.As a strengthening of the famous 1-2-3-conjecture,Bartnicki,Grytczuk and Niwcyk[Weight choosability of graphs.J.Graph Theory,60,242–256(2009)]conjecture that every graph without isolated edge is 3-weight choosable.This conjecture is wildly open and it is even unknown whether there is a constant k such that every graph without isolated edge is k-weight choosable.In this paper,we show that every connected graph of maximum degree 4 is 4-weight choosable.
