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. .

Algorithm for Visualization of Zero Divisor Graphs of the Ring ℤn Using MAPLE Coding

This research investigates the comparative efficacy of generating zero divisor graphs (ZDGs) of the ring of integers ℤn modulo n using MAPLE algorithm. Zero divisor graphs, pivotal in the study of ring theory, depict relationships between elements of a ring that multiply to zero. The paper explores the development and implementation of algorithms in MAPLE for constructing these ZDGs. The comparative study aims to discern the strengths, limitations, and computational efficiency of different MAPLE algorithms for creating zero divisor graphs offering insights for mathematicians, researchers, and computational enthusiasts involved in ring theory and mathematical computations.

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 Fermat Last Theorem: The New Efficient Expression of a Hypothetical Solution as a Function of Its Fermat Divisors

Denote by a non-trivial primitive solution of Fermat’s equation (p prime).We introduce, for the first time, what we call Fermat principal divisors of the triple defined as follows. , and . We show that it is possible to express a,b and c as function of the Fermat principal divisors. Denote by the set of possible non-trivial solutions of the Diophantine equation . And, let (p prime). We prove that, in the first case of Fermat’s theorem, one has . In the second case of Fermat’s theorem, we show that , ,. Furthermore, we have implemented a python program to calculate the Fermat divisors of Pythagoreans triples. The results of this program, confirm the model used. We now have an effective tool to directly process Diophantine equations and that of Fermat. .

Analysis of Distance-Based Topological Polynomials Associated with Zero-Divisor Graphs

Chemical compounds are modeled as graphs.The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges.The topological indices representing the molecular graph corresponds to the different chemical properties of compounds.Let a,b be are two positive integers,andΓ(Z_(a)×Z_(b))be the zero-divisor graph of the commutative ring Z_(a)×Z_(b).In this article some direct questions have been answered that can be utilized latterly in different applications.This study starts with simple computations,leading to a quite complex ring theoretic problems to prove certain properties.The theory of finite commutative rings is useful due to its different applications in the fields of advanced mechanics,communication theory,cryptography,combinatorics,algorithms analysis,and engineering.In this paper we determine the distance-based topological polynomials and indices of the zero-divisor graph of the commutative ring Z_(p^(2))×Z_(q)(for p,q as prime numbers)with the help of graphical structure analysis.The study outcomes help in understanding the fundamental relation between ring-theoretic and graph-theoretic properties of a zero-divisor graphΓ(G).

Zero-divisor Graphs for Direct Products of Rings

In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those graphs whose diameters ≥ 1 and discussed six cases. This paper further discusses the other nine cases and also gives a complete characterization for the possible diameters for left Artin rings.

On the Cozero-Divisor Graphs of Commutative Rings

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs and such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.

The Zero-divisor Graphs of Abelian Regular Rings

We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular, moreover, the zero-divisor graph of such a ring is studied.

APPROXIMATE COMMON DIVISORS OF POLYNOMIALS AND DEGREE REDUCTION FOR RATIONAL CURVES

Abstract This paper deals with how to perturb a given set of polynomials so as to include a common linear factor. An algorithm is derived for determining such a set of perturbation polynomials which are subject to certain constrains at the endpoints of a prescribed parametric interval and minimized in a certain sense. This result can be combined with subdivision technique to obtain a continuous piecewise approximation to a rational curve.

Some Conjectures on the Divisor Function

We propose the following conjecture on(σ)n the sum-of-divisors function:log(e'nlog logn-σ(n))/log(e'nlog log n)will increase strictly and converge to1 when n runs from the colossally abundant numbers to infinity This conjecture is a sufficient condition for the and converge to1 when n nuns from the colossally abundant numbers to infinity.This conjecture is a sufficient condition for the Ricemann hypothesis by Robin's theorem,and it is confirmed for n from10^(4 )up to 10^(103078) Further,we present two additional Riemann hypothesis by Robin's theorem,and it is confirmed forn from 10^(4) up to 10^(103078) Further,we present two additional conjectures that are related to Robin's theorem.

New Asymptotic Results on Fermat-Wiles Theorem

We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.

关于一道抽象代数题的推广

用初等的方法证明了:设G是有限群,p是G的最小素因子,若对任意n∣|G|都有|{x∈G∣x^(n)=1}|≤(p-1)n,则G是循环群.推广了教材[1]第1.4节的习题3.

On the Order of Magnitude of the Divisor Function

Let D be an increasing sequence of positive integers, and consider the divisor functions：d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1,where [d,δ]=1.c.m.（d,δ）. A probabilistic argument is introduced to evaluate the series ∑n=1^∞and（n,D） and ∑n=1^∞and2（n,D）.

Logarithmic vanishing theorems for effective q-ample divisors

Let X be a compact K?hler manifold and D be a simple normal crossing divisor. If D is the support of some effective q-ample divisor, we show H^i(X, ?_X^j (log D)) = 0, for i + j > n + q.

无小环大列重QC-LDPC短码的显式构造

针对列重较大的无4环且无6环的准循环(Quasi-Cyclic,QC)低密度奇偶校验(Low-Density Parity-Check,LDPC)码,本文提出了三种新的显式构造方法.新方法的指数矩阵由两个整数序列完全定义,其中第一个序列是从0开始且公差为1的等差序列,第二个序列是由符合最大公约数约束的整数组成的特殊序列.对于现有显式方法只能提供较大循环块尺寸的多种行重类型,新显式构造方法在这些行重类型下均获得了相当小的循环块尺寸,从而将最小循环块尺寸降低到大约只有原来的一半.与近期提出的基于搜索的对称结构法相比,新的显式构造方法具有类似或更优的译码性能、极低的描述复杂度且不需要计算机搜索.

多通道奇异频率信号的稳相合成研究

为了解决多通道奇异频率间相位差稳定的难题,理论上分析了锁相环稳相原理,提出了一种多通道奇异频率的稳相算法。该算法通过求解奇异频率间的最大公约数,联动输出频率的同时可满足稳相合成的条件。电路实物加入适当的环路阶型设计,当输出频率在S波段时,相位差稳定性≤4°,满足使用需求,同时很好地验证了该算法的可行性和灵活性。

The complete principal divisor lattices

In this paper,we introduce a concept of principal divisor lattice and describe the structure of its elements.We first give a necessary and sufficient condition for the existence of irredundant join irreducible decompositions in complete principal divisor distributive lattices,and prove that the complete lower continuous,principal divisor lattices have irredundant join irreducible decompositions.In the end,we show the descriptions of lattices that have unique(resp.replaceable) irredundant join irreducible decompositions in complete lower continuous principal divisor lattices.

Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function

Let Δ(x) and E(t) denote respectively the remainder terms in the Dirichlet divisor problem and the mean square formula for the Riemann zeta-function on the critical line.This article is a survey of recent developments on the research of these famous error terms in number theory.These include upper bounds,Ω-results,sign changes,moments and distribution,etc.A few open problems are also discussed.

围长为8的较大列重准循环低密度奇偶校验码的行重普适代数构造

适合于任意行重(即行重普适(RWU))的无小环准循环(QC)低密度奇偶校验(LDPC)短码,对于LDPC码的理论研究和工程应用具有重要意义。具有行重普适特性且消除4环6环的现有构造方法,只能针对列重为3和4的情况提供QC-LDPC短码。该文在最大公约数(GCD)框架的基础上,对于列重为5和6的情况,提出了3种具有行重普适特性且消除4环6环的构造方法。与现有的行重普适方法相比,新方法提供的码长从目前的与行重呈4次方关系锐减至与行重呈3次方关系,因而可以为QC-LDPC码的复合构造和高级优化等需要较大列重基础码的场合提供行重普适的无4环无6环短码。此外,与基于计算机搜索的对称结构QC-LDPC码相比,新码不仅无需搜索、描述复杂度更低,而且具有更好的译码性能。

Commutative Rings Whose Zero-divisor Graph Is a Proper Refinement of a Star Graph

A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G＊ be the subgraph of G induced on the vertex set V（G） ／ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G=Г（R）, which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G＊ has at least two connected components. We prove that the diameter of the induced graph G＊ is two if Z（R）2 ≠{0}, Z（R）3 = {0} and Gc is connected. We determine the structure of R which has two distinct nonadjacent vertices a, fl C Z（R）＊／{c} such that the ideal [N（a）N（β）]{0} is generated by only one element of Z（R）＊／{c}. We also completely determine the correspondence between commutative rings and finite complete graphs Kn with some end vertices adjacent to a single vertex of Kn.