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Natural Numbers and the Strong Goldbach Conjecture
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作者 Ramon Carbó-Dorca 《Journal of Applied Mathematics and Physics》 2024年第9期3208-3236,共29页
This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the incl... This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the inclusion of the zero element as the source of a strong Goldbach conjecture reformulation. A prime Boolean vector is defined, pinpointing the positions of prime numbers within the odd number sequence. The natural unit primality is discussed in this context and transformed into a source of quantum-like indetermination. This approach allows for rephrasing the strong Goldbach conjecture, framed within a Boolean scalar product between the prime Boolean vector and its reverse. Throughout the discussion, other intriguing topics emerge and are thoroughly analyzed. A final description of two empirical algorithms is provided to prove the strong Goldbach conjecture. 展开更多
关键词 Natural Numbers Prime Numbers Vector Description of Natural Numbers Prime Boolean Vectors Primality of the Natural Unit strong Goldbach’s conjecture Vector Reversal Pairing conjecture Natural Matrix squeezing
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The Proofs of Legendre’s Conjecture and Three Related Conjectures
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作者 Wing K. Yu 《Journal of Applied Mathematics and Physics》 2023年第5期1319-1336,共18页
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
关键词 Legendre’s conjecture Bertrand’s Postulate - Chebyshev’s Theorem Oppermann’s conjecture Brocard’s conjecture Andrica’s conjecture
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Inequalities relating to L_p-version of Petty's conjectured projection inequality 被引量:1
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作者 王卫东 冷岗松 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2007年第2期269-276,共8页
Petty's conjectured projection inequality is a famous open problem in the theory of convex bodies. In this paper, it is shown that an inequality relating to Lp-version of the Petty's conjectured projection inequalit... Petty's conjectured projection inequality is a famous open problem in the theory of convex bodies. In this paper, it is shown that an inequality relating to Lp-version of the Petty's conjectured projection inequality is developed by using the notions of the Lp-mixed volume and the Lp-dual mixed volume, the relation of the Lp-projection body and the geometric body Г-pK, the Bourgain-Milman inequality and the Lp-Bnsemann-Petty inequality. In addition, for each origin-symmetric convex body, by applying the Jensen inequality and the monotonicity of the geometric body Г-pK, the reverses of Lp-version of the Petty's conjectured projection inequality and the Lp-Petty projection inequality are given, respectively. 展开更多
关键词 Lp-version Petty projection inequality Petty's conjectured projectioninequality Lp-projection body REVERsE
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A NEW RESULT ON ERDS-SóS CONJECTURE
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作者 王敏 赵艳青 李国君 《数学物理学报(A辑)》 CSCD 北大核心 1997年第S1期125-131,共7页
Erdosa and Sós conjectured in 1963 that if G is a graph o ofof ordeq >1/2p(k - 1), then G contains every tree of size k. It is shown in this paper that the conjecture is true if the complement G of G contains ... Erdosa and Sós conjectured in 1963 that if G is a graph o ofof ordeq >1/2p(k - 1), then G contains every tree of size k. It is shown in this paper that the conjecture is true if the complement G of G contains no a copy of K3 as an induced subgraph of G. 展开更多
关键词 packing IsOMORPHIC graphs. Erd■s-sós conjecture
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Some New Results on Terai's Conjecture
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作者 曹珍富 董晓蕾 《Journal of Harbin Institute of Technology(New Series)》 EI CAS 1998年第4期1-3,共3页
Terai presented the following conjecture: Ifa2 + b2 = c2 vith a, b, c ∈ N, gcd (a, b, c) = 1 and a even, then the diophantine equation x2 + bm = cn has the only peitive integral solution (x, m, n ) = (a,2, 2). In thi... Terai presented the following conjecture: Ifa2 + b2 = c2 vith a, b, c ∈ N, gcd (a, b, c) = 1 and a even, then the diophantine equation x2 + bm = cn has the only peitive integral solution (x, m, n ) = (a,2, 2). In this paper we prove that if c is a prime power, b 1 (mod 8) and b 1 (mod 16) if b2 + 1 = 2c, then Terai’s conjecture holds. 展开更多
关键词 Terai’s conjecture EXPONENTIAL DIOPHANTINE EQUATION imaginary QUADRATIC field
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A Simple and General Proof of Beal’s Conjecture (I)
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作者 Golden Gadzirayi Nyambuya 《Advances in Pure Mathematics》 2014年第9期518-521,共4页
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
关键词 Fermat’s LAsT THEOREM Beal’s conjecture PROOF
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On the Four Conjectures of G G Lorentz Type in L_ρ~2[-a,a]
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作者 王元恒 杨文善 《Chinese Quarterly Journal of Mathematics》 CSCD 2009年第3期343-348,共6页
In the space C[-1,1],G G Lorentz proposed four conjectures on the properties of the polynomials of the best approximation in 1977,1978 and 1980.The present paper transplants the four conjectures in the space Lρ2[-a,a... In the space C[-1,1],G G Lorentz proposed four conjectures on the properties of the polynomials of the best approximation in 1977,1978 and 1980.The present paper transplants the four conjectures in the space Lρ2[-a,a] and proves them being all right in only one theorem under the corresponding conditions,although each of the original conjectures is very difficulty. 展开更多
关键词 best approximation Lorentz's conjecture different norms property of polynomial
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Some notes on a conjecture of Zygmund
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作者 CHEN Jie-cheng ZHANG Chun-jie 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2008年第2期213-218,共6页
In this paper, some classes of differentiation basis are investigated and several positive answers to a conjecture of Zygmund on differentiation of integrals are presented.
关键词 Zygmund's conjecture differentiation of integrals maximal function.
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Scholz’s Third Conjecture: A Demonstration for Star Addition Chains
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作者 José Maclovio Sautto Vallejo Agustín Santiago Moreno +1 位作者 Carlos N. Bouza Herrera Verónica Campos Guzmán 《Applied Mathematics》 2013年第10期1-2,共2页
This paper presents a brief demonstration of Scholz’s third conjecture [1] for n numbers such that their minimum chain addition is star type [2]. The demonstration is based on the proposal of an algorithm that takes ... This paper presents a brief demonstration of Scholz’s third conjecture [1] for n numbers such that their minimum chain addition is star type [2]. The demonstration is based on the proposal of an algorithm that takes as input the star-adding chain of a number n, and returns a string in addition to x = 2n - 1??of length equal to l (n) + n - 1. As for any type addition chain star of a number n, this chain is minimal demonstrates the Scholz’s third Conjecture for such numbers. 展开更多
关键词 ADDITION CHAIN EXPONENTIATION short CHAIN scholz’s conjecture
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Scholz’s First Conjecture: A Brief Demonstration
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作者 José M. Sautto +3 位作者 Agustín Santiago Carlos N. Bouza Veró nica Campos 《Applied Mathematics》 2016年第1期70-76,共7页
This paper presents a brief demonstration of Schulz’s first conjecture, which sets the upper and lower limits on the length of the shortest chain of addition. Two methods of the upper limit are demonstrated;the secon... This paper presents a brief demonstration of Schulz’s first conjecture, which sets the upper and lower limits on the length of the shortest chain of addition. Two methods of the upper limit are demonstrated;the second one is based on the algorithm of one of the most popular methods for obtaining addition chains of a number, known as the binary method. 展开更多
关键词 Addition Chain EXPONENTIATION short Chain scholz’s conjecture Binary Method
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The Proof of Goldbach’s Conjecture on Prime Numbers
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作者 Silviu Guiasu 《Natural Science》 2019年第9期273-283,共11页
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
关键词 PROOF of Goldbach’s conjecture Hidden symmetry of PRIMEs Matching PRIMEs and Composite INTEGERs Arranging ODD INTEGERs in Frames The Existence and the Number of GOLDBACH solutions
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Remarks on Goldbach’s Conjecture on Prime Numbers
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作者 Silviu Guiasu 《Natural Science》 2019年第12期336-344,共9页
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
关键词 The Existence and the Number of GOLDBACH solutions The Proof of Goldbach’s conjecture Hidden symmetry of PRIMEs Matching PRIMEs and ODD Composite INTEGERs Arranging ODD INTEGERs in Frames
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Solutions to Beal’s Conjecture, Fermat’s Last Theorem and Riemann Hypothesis
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作者 A. C. Wimal Lalith de Alwis 《Advances in Pure Mathematics》 2016年第10期638-646,共9页
A Simple Mathematical Solutions to Beal’s Conjecture and Fermat’s Marginal Conjecture in his diary notes, Group Theoretical and Calculus Solutions to Fermat’s Last theorem & Integral Solution to Riemann Hypothe... A Simple Mathematical Solutions to Beal’s Conjecture and Fermat’s Marginal Conjecture in his diary notes, Group Theoretical and Calculus Solutions to Fermat’s Last theorem & Integral Solution to Riemann Hypothesis are discussed. 展开更多
关键词 Beal’s conjecture Fermat’s Last Theorem Riemann Hypothesis
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On resolution to Wu's conjecture on Cauchy function's exterior singularities
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作者 Theodore Yaotsu Wu 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2011年第3期309-317,共9页
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
关键词 Keywords Cauchy function. singularity distribution . Wu's conjecture - Resolution by induction
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Group-Theoretic Remarks on Goldbach’s Conjecture
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作者 Liguo He Gang Zhu 《Advances in Pure Mathematics》 2022年第11期624-637,共14页
The famous strongly binary Goldbach’s conjecture asserts that every even number 2n ≥ 8 can always be expressible as a sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Spe... The famous strongly binary Goldbach’s conjecture asserts that every even number 2n ≥ 8 can always be expressible as a sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we apply the element order prime graphs of alternating groups of degrees 2n and 2n &#8722;1 to characterize this conjecture, and present its six group-theoretic versions;and further prove that this conjecture is true for p +1 and p &#8722;1 whenever p ≥ 11 is a prime number. 展开更多
关键词 Alternating Group Element Order Prime Graph Goldbach’s conjecture CENTRALIZER
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Local Geometric Proof of Riemann Conjecture 被引量:3
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作者 Chuanmiao Chen 《Advances in Pure Mathematics》 2020年第10期589-610,共22页
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
关键词 Riemann conjecture Local Geometric Proof symmetry Peak-Valley struc-ture EQUIVALENCE Liuhui’s Methodology
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The Primes and Their Subsets as Continuum like the Integers
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作者 Pal Doroszlai Horacio Keller 《Advances in Pure Mathematics》 2023年第12期733-750,共18页
The present paper gives the proof of the set of primes as a continuum. It starts with the density of the primes, and shortly recapitulates the prime-number-formula and the complete-prime-number-formula. Reflecting the... The present paper gives the proof of the set of primes as a continuum. It starts with the density of the primes, and shortly recapitulates the prime-number-formula and the complete-prime-number-formula. Reflecting the series of the primes over any prime gives the double density of occupation of integer positions by the union of the series of multiples of the primes. The remaining free positions render it possible to prove Goldbach’s conjecture and the set of primes as a continuum. The theoretical evaluation is followed in annexes by numerical evaluation, demonstrating the theoretical results. The numerical evaluation results in different constants and relations, which represent inherent properties of the set of primes. 展开更多
关键词 Prime-Number-Formula Complete-Prime-Number-Formula Goldbach’s conjecture Primes as Continuum
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On Prime Numbers between kn and (k + 1) n
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作者 Wing K. Yu 《Journal of Applied Mathematics and Physics》 2023年第11期3712-3734,共23页
In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always e... In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always exists at least a prime number p such that kn p ≤ (k +1)n. The Bertrand-Chebyshev’s theorem is a special case of this theorem when k = 1. In the field of prime number distribution, just as the prime number theorem provides the approximate number of prime numbers relative to natural numbers, while the new theory indicates that prime numbers exist in the specific intervals between natural numbers, that is, the new theorem provides the approximate positions of prime numbers among natural numbers. 展开更多
关键词 Bertrand’s Postulate-Chebyshev’s Theorem The Prime Number Theorem Landau Problems Legendre’s conjecture Prime Number Distribution
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The Integral of the Inverse of the Primes
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作者 Pal Doroszlai Horacio Keller 《Advances in Pure Mathematics》 2023年第5期250-266,共17页
The present paper gives the proof of the set of primes as continuum and evaluates the analytical formula for the integral of the inverse of the primes over the distance. First it starts with the density of the primes,... The present paper gives the proof of the set of primes as continuum and evaluates the analytical formula for the integral of the inverse of the primes over the distance. First it starts with the density of the primes, shortly recapitulates the prime-number-formula and the complete-prime-number-formula, the proof of the set of primes as continuum. The theoretical evaluation is followed in annexes by numerical evaluation of the theoretical results and of different constants, which represent inherent properties of the set of primes. 展开更多
关键词 Density of Primes Prime-Number-Formula Complete-Prime-Number-Formula Goldbach’s conjecture Twin Primes K-Tuples of Primes Primes as Continuum
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Integral Formulas for Compact Spacelike Hypersurfaces in de Sitter Space and their Applications to Goddard's Conjecture 被引量:10
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作者 Li Haizhong (Department of Applied Mathematics,Tsinghua University,Beijing 100084,China)(Email:hli@math.tsinghua.edu.cn)Chen Weihuan (Department of Mathematics,Peking University,Beijing 100871,China)(Email:whchen@sxx0.math.pku,edu.cn) 《Acta Mathematica Sinica,English Series》 SCIE CSCD 1998年第2期285-288,共4页
We establish integral formulas of Minkowski’s type for compact spacelike hypersurfaces in de sitter space S<sub>1</sub><sup>n+1</sup>(1)and give their applications to the case of constant r-... We establish integral formulas of Minkowski’s type for compact spacelike hypersurfaces in de sitter space S<sub>1</sub><sup>n+1</sup>(1)and give their applications to the case of constant r-th mean curvature (r=1,2,…,n-1).When r=1 we recover Montiel’s result. 展开更多
关键词 spacelike hypersurfaces de sitter space goddard’s conjecture
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