Fermat and Pythagoras Divisors for a New Explicit Proof of Fermat’s Theorem:a4 + b4 = c4. Part I

In this paper we prove in a new way, the well known result, that Fermat’s equation a4 + b4 = c4, is not solvable in ℕ , when abc≠0 . To show this result, it suffices to prove that: ( F 0 ): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is not solvable in ℕ , (where a 1 , b 1 , c 1 ∈2ℕ+1 , pairwise primes, with necessarly 2≤s∈ℕ ). The key idea of our proof is to show that if (F0) holds, then there exist α 2 , β 2 , γ 2 ∈2ℕ+1 , such that ( F 1 ): α 2 4 + ( 2 s−1 β 2 ) 4 = γ 2 4 , holds too. From where, one conclude that it is not possible, because if we choose the quantity 2 ≤ s, as minimal in value among all the solutions of ( F 0 ) , then ( α 2 ,2 s−1 β 2 , γ 2 ) is also a solution of Fermat’s type, but with 2≤s−10) is solvable in ( a 1 ,2 s b 1 , c 1 ) , s ≥ 2 like above;afterwards, proceeding with “Pythagorician divisors”, we creat the notions of “Fermat’s b-absolute divisors”: ( d b , d ′ b ) which it uses hereafter. Then to conclude our proof, we establish the following main theorem: there is an equivalence between (i) and (ii): (i) (F0): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is solvable in ℕ , with 2≤s∈ℕ , ( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs. (ii) ∃( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs, for wich: ∃( b ′ 2 , b 2 , b ″ 2 )∈ ( 2ℕ+1 ) 3 coprime in pairs, and 2≤s∈ℕ , checking b 1 = b ′ 2 b 2 b ″ 2 , and such that for notations: S=s−λ( s−1 ) , with λ∈{ 0,1 } defined by c 1 − a 1 2 ≡λ( mod2 ) , d b =gcd( 2 s b 1 , c 1 − a 1 )= 2 S b 2 and d ′ b = 2 s−S b ′ 2 = 2 s B 2 d b , where ( 2 s B 2 ) 2 =gcd( b 1 2 , c 1 2 − a 1 2 ) , the following system is checked: { c 1 − a 1 = d b 4 2 2+λ = 2 2−λ ( 2 S−1 b 2 ) 4 c 1 + a 1 = 2 1+λ d ′ b 4 = 2 1+λ ( 2 s−S b ′ 2 ) 4 c 1 2 + a 1 2 =2 b ″ 2 4;and this system implies: ( b 1−λ,2 4 ) 2 + ( 2 4s−3 b λ,2 4 ) 2 = ( b ″ 2 2 ) 2;where: ( b 1−λ,2 , b λ,2 , b ″ 2 )={ ( b ′ 2 , b 2 , b ″ 2 ) if λ=0 ( b 2 , b ′ 2 , b ″ 2 ) if λ=1;From where, it is quite easy to conclude, following the method explained above, and which thus closes, part I, of this article. .

A New Proof for Congruent Number’s Problem via Pythagorician Divisors

Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2 t+1 ( 2 k Y ) 2 = T 2 , where k∈ℕ;and solutions ( X,Y,Z,T )=( D k , E k , f k , f ′ k )∈ ( 2ℕ+1 ) 4 , are given in pairwise prime numbers.2020-Mathematics Subject Classification 11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25 .

ON THE METHODS FOR FINDING ROOTS OF ALGEBRAIC EQUATIONS WITH WEIERSTRASS' CORRECTIONS

This is a study of the Durand-Kerner and Nourein methods for finding the roots of a given algebraic equation simultaneously. We consider the conditions under which the iterative methods fail. The numerical example is presented.

From Pythagoras Theorem to Fermat’s Last Theorem and the Relationship between the Equation of Degree <i>n</i>with One Unknown

The most interesting and famous problem that puzzled the mathematicians all around the world is much likely to be the Fermat’s Last Theorem. However, since the Theorem was proposed, people can’t find a way to solve the problem until Andrew Wiles proved the Fermat’s Last Theorem through a very difficult method called Modular elliptic curves in 1995. In this paper, I firstly constructed a geometric method to prove Fermat’s Last Theorem, and in this way we can easily get the conclusion below: If a and b are integer and?a = b, n ∈ Q and n > 1, the value of c satisfies the function an + bn = cn that can never be integer;if a, b and c are integer and a ≠ b, n is integer and n > 2, the function an + bn = cn cannot be established.

Spatial pattern of urban-rural integration in China and the impact of geography

Urban-rural integration (URI) is a global challenge that is highly related to inequalities, poverty, economic growth, and other Sustainable Development Goals (SDGs). Existing research has evaluated the extent of URI and explored its influencing factors, but urban-rural linkages are seldom incorporated in evaluation systems, and geographical factors are rarely recognized as the influencing factors. We construct a URI framework including regional economy, rural development, urban-rural linkage, and urban-rural gap. Based on a dataset consisting of 1,669 counties in China in 2020, we reveal the spatial pattern of URI and find a high correlation between the spatial pattern of URI and the relief degree of land surface (RDLS). Using structural equation modeling, we discover that topography has direct ( − 0.18, p < 0.001) and indirect ( − 0.17, p < 0.001) effects on URI. The indirect negative effects are mediated through the infrastructure, and the combination of localized advantages and modern technical conditions could mitigate the negative impact of topography. Finally, we identify 742 counties as lagging regions in URI, which can be clustered into eight types. Our findings could facilitate policy designing for those countries striving for integrated and sustainable development of urban and rural areas.

Evaluation and Driving Force Analysis of Cultivated Land Quality in Black Soil Region of Northeast China

Cultivated land is an important natural resource to ensure food,ecological and economic security.The cultivated land quality evaluation(CQE)is greatly significant for protecting and managing cultivated land.In this study,320 counties in the black soil region of Northeast China(BSRNC)represent the research units used to construct the CQE system measuring the soil properties(SP),cultivated land productivity(CLP),ecological environment(EE)and social economy(SE).The total of 19 factors were selected to calculate the integrated fertility index(IFI)and divided into grades.Simultaneously,we used the coupling coordination degree model to comprehensively analyze the spatial pattern of the cultivated land quality(CLQ)in the BSRNC,and use the structural equation model(SEM)to analyze the driving mechanism.The results show the following:1)The CLQ of 262 counties in the BSRNC is in a state of coupling and coordination,and the coupling and coordination degree presents a spatial distribution pattern of‘high in the southwest and northeast,low in the northwest and southeast’.The coordinated development degree of 271 counties is between 0.4 and 0.6,which is in a transitional state between coordination and disorder.2)The CLQ in the BSRNC is generally good,with an average grade of 3.High-quality cultivated land accounts for 58.45%of all counties,middle-and upper-quality cultivated land accounts for 27.05%,and poor-quality cultivated land accounts for 14.49%.3)The SEM analysis shows that the SP,CLP,EE,and SE all influence the CLQ.Among them,the SP has the largest driving force on the CLQ,while the SE has the smallest driving force on it.The results confirm that the main factors affecting the evaluation results are crop productivity level,normalized difference vegetation index,ratio vegetation index,difference vegetation index,and organic carbon content.When implementing protection measures in counties with a low CLQ,considering a balanced coordination of multiple systems and reasonably controlling the quality degradation are important.This study provides the current situation and driving factors of the CLQ in the BSRNC and will play an important role in black soil governance and utilization.

ON THE VERTEX PARTITION EQUATION OF ROOTED LOOPLESS PLANAR MAPS

This paper provides a functional equation astisfied by the generating function for enumerating rooted loopless planar maps with vertex partition. A kind of applications in enumerating, by providing explicit formulae, a type of rooted loopless planar maps with the maximum valency of vertices given are described. Meanwhile, the functional equation for enumerating rooted loopless planar maps (connected) with the edge number and the valency of root-vertex as the parameters is also derived directly.

Numerical Simulation about the Characteristics of the Store Released from the Internal Bay in Supersonic Flow

To understand the influence of the initial release conditions on the separation characteristics of the store and improve it under high Mach number(Ma=4)flight conditions,the overset grid method and the Realizable k−εturbulence model coupled with an equation with six degrees of freedom are used to simulate the store released from the internal bay.The motion trajectory and the attitude angle of the store separation under the conditions of different centroid,velocity,height and control measures are given by the calculated result.Through analysis,the position of the centroid will affect the separation of the store,which needs to be considered in the design.Increasing the launching height is conducive to the separation of the store.If the store has an initial velocity,it can leave the internal bay more quickly and reduce the probability of collision with the wall.Cylindrical rod and slanted aft wall control measures can improve the attitude of the store and make the store fall more smoothly.

The Existence of Periodic Solutions of a Class of <i>n</i>-Degree Polynomial Differential Equations*

This paper deals with a class of n-degree polynomial differential equations. By the fixed point theorem and mathematical analysis techniques, the existence of one (n is an odd number) or two (n is an even number) periodic solutions of the equation is obtained. These conclusions have certain application value for judging the existence of periodic solutions of polynomial differential equations with only one higher-order term.

ON THE METHOD OF SOLUTION FOR A KIND OFNONLINEAR SINGULAR INTEGRAL EQUATION

The solutions of the nonlinear singular integral equation , t 6 L, are considered, where L is a closed contour in the complex plane, b ≠ 0 is a constant and f(t) is a polynomial. It is an extension of the results obtained in [1] when f(t) is a constant. Certain special cases are illustrated.

Degree Splitting of Root Square Mean Graphs

Let be an injective function. For a vertex labeling f, the induced edge labeling is defined by, or;then, the edge labels are distinct and are from . Then f is called a root square mean labeling of G. In this paper, we prove root square mean labeling of some degree splitting graphs.

The Theory of Membership Degree of Γ-Conclusion in Several n-Valued Logic Systems

Based on the analysis of the properties of Γ-conclusion by means of deduction theorems, completeness theorems and the theory of truth degree of formulas, the present papers introduces the concept of the membership degree of formulas A is a consequence of Γ (or Γ-conclusion) in Lukasiewicz n-valued propositional logic systems, Godel n-valued propositional logic system and the R0 n-valued propositional logic systems. The condition and related calculations of formulas A being Γ-conclusion were discussed by extent method. At the same time, some properties of membership degree of formulas A is a Γ-conclusion were given. We provide its algorithm of the membership degree of formulas A is a Γ-conclusion by the constructions of theory root.

APPROXIMATE SOLUTION OF A p-th ROOT FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN (2,β)-BANACH SPACES

In this paper, using the Brzdek's fixed point theorem [9,Theorem 1] in non-Archimedean(2,β)-Banach spaces, we prove some stability and hyperstability results for an p-th root functional equation ■where p∈{1, …, 5}, a_1,…, a_k are fixed nonzero reals when p ∈ {1,3,5} and are fixed positive reals when p ∈{2,4}.

AN APPLICATION OF HARDY-BOEDEWADT'S THEOREM TO ITERATED FUNCTIONAL EQUATIONS

In this paper an iterated functional equation of polynomial type which does not possess the firt order iterative term g(x) is to be discussed. The difficulties resulted from loss of the first order term are overcome by utilization of Hardy-Boedewadt's theorem.

The Solution of Nonlinear Equations via the Method of Hurwitz-Radon Matrices

Image analysis and computer vision are interested in suitable methods to solve the nonlinear equations. Coordinate x??for f (x)?= 0?is crucial because each equation can be transformed into f (x)?= 0. A novel method of Hurwitz-Radon Matrices (MHR) can be used in approximation of a root of function in the plane. The paper contains a way of data approximation via MHR method to solve any equation. Proposed method is based on the family of Hurwitz-Radon (HR) matrices. The matrices are skew-symmetric and possess columns composed of orthogonal vectors. The operator of Hurwitz-Radon (OHR), built from these matrices, is described. Two-dimensional data are represented by discrete set of curve??f points. It is shown how to create the orthogonal OHR operator and how to use it in a process of data interpolation. MHR method is interpolating the curve point by point without using any formula or function.

Nonrecurrence Theorems of Periodic Solutions for Functional Differential Equations

nonrecurrence theorem on the existence of periodic solutions for functional differential equations is proved by employing the topological method, and some applications are given.

EXTRACTION OF CHARACTERISTIC ROOTS ON BESSEL-NEUMANN’S MIXED EQUATIONS

This paper discusses the problem of the extraction of characteristic roots on Bessel-Neumann's mixed equations. It gives the expressions and the evaluation of the minimum root. The advantage of the method has no use for the table of the multi-figure number Bessel function and it does not need computer but can calculate all the characteristic roots The precision of these roots is still high.

Solving Systems of Transcendental Equations Involving the Heun Functions

The Heun functions have wide application in modern physics and are expected to succeed the hypergeometrical functions in the physical problems of the 21st century. The numerical work with those functions, however, is complicated and requires filling the gaps in the theory of the Heun functions and also, creating new algorithms able to work with them efficiently. We propose a new algorithm for solving a system of two nonlinear transcendental equations with two complex variables based on the Müller algorithm. The new algorithm is particularly useful in systems featuring the Heun functions and for them, the new algorithm gives distinctly better results than Newton’s and Broyden’s methods. As an example for its application in physics, the new algorithm was used to find the quasi-normal modes (QNM) of Schwarzschild black hole described by the Regge-Wheeler equation. The numerical results obtained by our method are compared with the already published QNM frequencies and are found to coincide to a great extent with them. Also discussed are the QNM of the Kerr black hole, described by the Teukolsky Master equation.

EXISTENCE THEOREMS OF SOLUTIONS FOR NONLINEAR IMPULSIVE VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACES

In this paper, we apply the coincidence degree theory to study nonlinear second order impulsive periodic boundary value problems (PBVP), and show some sufficient conditions for the existence of solutions.

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-SEPARATED TYPE INTEGRAL BOUNDARY CONDITIONS

In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.