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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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 −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 −1 whenever p ≥ 11 is a prime number.展开更多
文摘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.
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.
基金Project supported by the National Natural Science Foundation of China(No.10671117)the Key Science Research Foundation of Education Department of Hubei Province of China(No.2003A005).
文摘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.
文摘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.
文摘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.
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.
基金the National Natural Science Foundation of China(10571156)RSDP(20030335019)+1 种基金the Natural Science Foundation of Zhejiang Province(RC97017)supported partially by the National Natural Science Foundation of China(10601046).
文摘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.
文摘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.
文摘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.
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.
文摘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.
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.
文摘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 −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 −1 whenever p ≥ 11 is a prime number.
