The Proofs of Legendre’s Conjecture and Three Related Conjectures

In this paper, we prove Legendre’s conjecture: There is a prime number between n2 and (n +1)2 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.

Natural Numbers and the Strong Goldbach Conjecture

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.

Inequalities relating to L_p-version of Petty's conjectured projection inequality

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.

New Form of Legendre Polynomials Obtained by Virtue of Excited Squeezed State and IWOP Technique in Quantum Optics

Using the technique of integration within an ordered product of operators and the intermediate coordinatemomentum representation in quantum optics, as well as the excited squeezed state we derive a new form of Legendre polynomials.

A NEW RESULT ON ERDS-SóS CONJECTURE

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.

Some New Results on Terai's Conjecture

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.

A Simple and General Proof of Beal’s Conjecture (I)

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.

On the Four Conjectures of G G Lorentz Type in L_ρ~2[-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.

Some notes on a conjecture of Zygmund

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.

Scholz’s Third Conjecture: A Demonstration for Star Addition Chains

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.

Scholz’s First Conjecture: A Brief Demonstration

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.

The Proof of Goldbach’s Conjecture on Prime Numbers

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.

Remarks on Goldbach’s Conjecture on Prime Numbers

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.

Solutions to Beal’s Conjecture, Fermat’s Last Theorem and Riemann Hypothesis

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.

On resolution to Wu's conjecture on Cauchy function's exterior singularities

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.

Group-Theoretic Remarks on Goldbach’s Conjecture

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.

关于商高数的Jesmanowicz猜想

本研究主要利用简单同余、二次剩余、κ次剩余、四次剩余特征理论及因式分解法,对关于不定方程a^(x)+b^(y)=c^(z)的Jesmanowicz猜想的一类特殊情形进行证明,并得到如下结论:定理对于商高数组a=n^(2)-4,b=4n,c=n^(2)+4,2×n,当n+2含有素因子p■-1(mod 16)时,Jesmanowicz猜想成立.特别地,有推论对于上述商高数组,当n■-1(mod 16)时,Jesmanowicz猜想成立.

Local Geometric Proof of Riemann Conjecture

Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying ζ and using analysis method likely are two incor-rect guides. Actually, a unique hope may study Riemann function , , by geometric analysis, which has the symmetry: v=0 if β=0, and basic expression . We show that |u| is single peak in each root-interval of u for fixed β ∈(0,1/2]. Using the slope ut, we prove that v has opposite signs at two end-points of Ij. There surely exists an inner point such that , so {|u|,|v|/β} form a local peak-valley structure, and have positive lower bound in Ij. Because each t must lie in some Ij, then ||ξ|| > 0 is valid for any t (i.e. RH is true). Using the positivity of Lagarias (1999), we show the strict monotone for β > β0 ≥ 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”.

关于Legendre猜想的一个判定准则

Legendre猜想是数论中一个与素数间隙有关的猜想,它断言对于每个正整数n,在2n和2(n+1)之间都有一个素数2.为此,文章首先研究了夹在2n和(n+1)之间的奇数,刻画了小于一个给定平方数的奇素数的特点.然后证明了如下的判定准则:若nΣr=3,r∈odd(「(n+1)2-r2/2r■ -「n2-r2/2r■)<n,则Legendre猜想成立;反之,若nΣr=3,r∈odd(「(n+1)2-r2/2r■ -「n2-r2/2r■)≥n,则Legendre猜想不成立.

On Prime Numbers between kn and (k + 1) n

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.