This article offers a simple but rigorous proof of Brouwer’s fixed point theorem using Sperner’s Lemma.The general method I have used so far in the proof is mainly to convert the n-dimensional shapes to the correspo...This article offers a simple but rigorous proof of Brouwer’s fixed point theorem using Sperner’s Lemma.The general method I have used so far in the proof is mainly to convert the n-dimensional shapes to the corresponding case under the Sperner’s Labeling and apply the Sperner’s Lemma to solve the question.展开更多
The author presents a new approach which is used to solve an important Diophantine problem. An elementary argument is used to furnish another fully transparent proof of Fermat’s Last Theorem. This was first stated by...The author presents a new approach which is used to solve an important Diophantine problem. An elementary argument is used to furnish another fully transparent proof of Fermat’s Last Theorem. This was first stated by Pierre de Fermat in the seventeenth century. It is widely regarded that no elementary proof of this theorem exists. The author provides evidence to dispel this belief.展开更多
Using the same method that we used in [1] to prove Fermat’s Last Theorem in a simpler and truly marvellous way, we demonstrate that Beal’s Conjecture yields—in the simplest imaginable manner, to our effort to prove...Using the same method that we used in [1] to prove Fermat’s Last Theorem in a simpler and truly marvellous way, we demonstrate that Beal’s Conjecture yields—in the simplest imaginable manner, to our effort to prove it.展开更多
This article presents a brief and new solution to the problem known as the “Fermat’s Last Theorem”. It is achieved without the use of abstract algebra elements or elements from other fields of modern mathematics of...This article presents a brief and new solution to the problem known as the “Fermat’s Last Theorem”. It is achieved without the use of abstract algebra elements or elements from other fields of modern mathematics of the twentieth century. For this reason it can be easily understood by any mathematician or by anyone who knows basic mathematics. The important thing is that the above “theorem” is generalized. Thus, this generalization is essentially a new theorem in the field of number theory.展开更多
If the concept of proof （including arithmetic proof） is syntactically restricted to closed sentences （or their Godel numbers）, then the standard accounts of Godel＇s Incompleteness Theorems （and Lob＇s Theorem） ...If the concept of proof （including arithmetic proof） is syntactically restricted to closed sentences （or their Godel numbers）, then the standard accounts of Godel＇s Incompleteness Theorems （and Lob＇s Theorem） are blocked. In these standard accounts （Godel＇s own paper and the exposition in Boolos＇ Computability and Logic are treated as exemplars）, it is assumed that certain formulas （notably so called ＂Godel sentences＂） containing the Godel number of an open sentence and an arithmetic proof predicate are closed sentences. Ordinary usage of the term ＂provable＂ （and indeed ＂unprovable＂） favors their restriction to closed sentences which unlike so-called open sentences can be true or false. In this paper the restricted form of provability is called strong provability or unprovability. If this concept of proof is adopted, then there is no obvious alternative path to establishing those theorems.展开更多
Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly...Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes, which implicitly proves Goldbach’s Conjecture for 2n as well.展开更多
Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying <i>ζ</i> and using analysis method likely are two incor-rect guides. Actually, a unique hope may study...Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying <i>ζ</i> and using analysis method likely are two incor-rect guides. Actually, a unique hope may study Riemann function <img alt="" src="Edit_8fcdfff5-6b95-42a4-8f47-2cabe2723dfc.bmp" />, <img alt="" src="Edit_6ce3a4bd-4c68-49e5-aabe-dec3e904e282.bmp" />, <img alt="" src="Edit_29ea252e-a81e-4b21-a41c-09209c780bb2.bmp" /> by geometric analysis, which has the symmetry: v=0 if <i>β</i>=0, and basic expression <img alt="" src="Edit_bc7a883f-312d-44fd-bcdd-00f25c92f80a.bmp" />. We show that |u| is single peak in each root-interval <img alt="" src="Edit_d7ca54c7-4866-4419-a4bd-cbb808b365af.bmp" /> of <i>u</i> for fixed <em>β</em> ∈(0,1/2]. Using the slope u<sub>t</sub>, we prove that <i>v</i> has opposite signs at two end-points of I<sub>j</sub>. There surely exists an inner point such that , so {|u|,|v|/<em>β</em>} form a local peak-valley structure, and have positive lower bound <img alt="" src="Edit_bac1a5f6-673e-49b6-892c-5adff0141376.bmp" /> in I<sub>j</sub>. Because each <i>t</i> must lie in some I<sub>j</sub>, then ||<em>ξ</em>|| > 0 is valid for any <i>t</i> (<i>i.e.</i> RH is true). Using the positivity <img alt="" src="Edit_83c3d2cf-aa7e-4aba-89f5-0eb44659918a.bmp" /> of Lagarias (1999), we show the strict monotone <img alt="" src="Edit_87eb4e9e-bc7b-43e3-b316-5dcf0efaf0d5.bmp" /> for <i>β</i> > <i>β</i><sub>0</sub> ≥ 0 , and the peak-valley structure is equiva-lent to RH, which may be the geometric model expected by Bombieri (2000). This research follows Liuhui’s methodology: “Computing can detect the un-known and method”.</i>展开更多
This is a lesson integrated with multiple approaches in geometry classroom to deepen middle school students’understanding of geometry and spatial sense in the topic of sums of interior angles in polygons.In three act...This is a lesson integrated with multiple approaches in geometry classroom to deepen middle school students’understanding of geometry and spatial sense in the topic of sums of interior angles in polygons.In three activities,teachers lead students to explore the pattern of interior angles throughout folding paper Origami,constructing animated polygons in Geometer’s Sketchpad,and establishing proof with Parallel Line Theorem.The lesson plan is developed with detailed procedures and prompting questions.The goal of the lesson is to identify the pattern of interior angles in polygons and to analyze the relationship among polygons in the setting of 25 to 30 middle school students.展开更多
The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact...The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes. The present paper contains explicit additional and complementary details of the proof, insisting on the existence and the number of Goldbach’s representations of even positive integers as sums of pairs of primes.展开更多
基金by Dr Kemp from National Mathematics and Science College.
文摘This article offers a simple but rigorous proof of Brouwer’s fixed point theorem using Sperner’s Lemma.The general method I have used so far in the proof is mainly to convert the n-dimensional shapes to the corresponding case under the Sperner’s Labeling and apply the Sperner’s Lemma to solve the question.
文摘The author presents a new approach which is used to solve an important Diophantine problem. An elementary argument is used to furnish another fully transparent proof of Fermat’s Last Theorem. This was first stated by Pierre de Fermat in the seventeenth century. It is widely regarded that no elementary proof of this theorem exists. The author provides evidence to dispel this belief.
文摘Using the same method that we used in [1] to prove Fermat’s Last Theorem in a simpler and truly marvellous way, we demonstrate that Beal’s Conjecture yields—in the simplest imaginable manner, to our effort to prove it.
文摘This article presents a brief and new solution to the problem known as the “Fermat’s Last Theorem”. It is achieved without the use of abstract algebra elements or elements from other fields of modern mathematics of the twentieth century. For this reason it can be easily understood by any mathematician or by anyone who knows basic mathematics. The important thing is that the above “theorem” is generalized. Thus, this generalization is essentially a new theorem in the field of number theory.
文摘If the concept of proof （including arithmetic proof） is syntactically restricted to closed sentences （or their Godel numbers）, then the standard accounts of Godel＇s Incompleteness Theorems （and Lob＇s Theorem） are blocked. In these standard accounts （Godel＇s own paper and the exposition in Boolos＇ Computability and Logic are treated as exemplars）, it is assumed that certain formulas （notably so called ＂Godel sentences＂） containing the Godel number of an open sentence and an arithmetic proof predicate are closed sentences. Ordinary usage of the term ＂provable＂ （and indeed ＂unprovable＂） favors their restriction to closed sentences which unlike so-called open sentences can be true or false. In this paper the restricted form of provability is called strong provability or unprovability. If this concept of proof is adopted, then there is no obvious alternative path to establishing those theorems.
文摘Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes, which implicitly proves Goldbach’s Conjecture for 2n as well.
文摘Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying <i>ζ</i> and using analysis method likely are two incor-rect guides. Actually, a unique hope may study Riemann function <img alt="" src="Edit_8fcdfff5-6b95-42a4-8f47-2cabe2723dfc.bmp" />, <img alt="" src="Edit_6ce3a4bd-4c68-49e5-aabe-dec3e904e282.bmp" />, <img alt="" src="Edit_29ea252e-a81e-4b21-a41c-09209c780bb2.bmp" /> by geometric analysis, which has the symmetry: v=0 if <i>β</i>=0, and basic expression <img alt="" src="Edit_bc7a883f-312d-44fd-bcdd-00f25c92f80a.bmp" />. We show that |u| is single peak in each root-interval <img alt="" src="Edit_d7ca54c7-4866-4419-a4bd-cbb808b365af.bmp" /> of <i>u</i> for fixed <em>β</em> ∈(0,1/2]. Using the slope u<sub>t</sub>, we prove that <i>v</i> has opposite signs at two end-points of I<sub>j</sub>. There surely exists an inner point such that , so {|u|,|v|/<em>β</em>} form a local peak-valley structure, and have positive lower bound <img alt="" src="Edit_bac1a5f6-673e-49b6-892c-5adff0141376.bmp" /> in I<sub>j</sub>. Because each <i>t</i> must lie in some I<sub>j</sub>, then ||<em>ξ</em>|| > 0 is valid for any <i>t</i> (<i>i.e.</i> RH is true). Using the positivity <img alt="" src="Edit_83c3d2cf-aa7e-4aba-89f5-0eb44659918a.bmp" /> of Lagarias (1999), we show the strict monotone <img alt="" src="Edit_87eb4e9e-bc7b-43e3-b316-5dcf0efaf0d5.bmp" /> for <i>β</i> > <i>β</i><sub>0</sub> ≥ 0 , and the peak-valley structure is equiva-lent to RH, which may be the geometric model expected by Bombieri (2000). This research follows Liuhui’s methodology: “Computing can detect the un-known and method”.</i>
文摘This is a lesson integrated with multiple approaches in geometry classroom to deepen middle school students’understanding of geometry and spatial sense in the topic of sums of interior angles in polygons.In three activities,teachers lead students to explore the pattern of interior angles throughout folding paper Origami,constructing animated polygons in Geometer’s Sketchpad,and establishing proof with Parallel Line Theorem.The lesson plan is developed with detailed procedures and prompting questions.The goal of the lesson is to identify the pattern of interior angles in polygons and to analyze the relationship among polygons in the setting of 25 to 30 middle school students.
文摘The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes. The present paper contains explicit additional and complementary details of the proof, insisting on the existence and the number of Goldbach’s representations of even positive integers as sums of pairs of primes.
