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.

Small Modular Solutions to Fermat’s Last Theorem

The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.

Whole Perfect Vectors and Fermat’s Last Theorem

A naïve discussion of Fermat’s last theorem conundrum is described. The present theorem’s proof is grounded on the well-known properties of sums of powers of the sine and cosine functions, the Minkowski norm definition, and some vector-specific structures.

An Elementary Proof of Fermat’s Last Theorem for Epsilons

The author presents a new approach which is used to solve an important Diophantine problem. An elementary argument is used to furnish another fully transparent proof of Fermat’s Last Theorem. This was first stated by Pierre de Fermat in the seventeenth century. It is widely regarded that no elementary proof of this theorem exists. The author provides evidence to dispel this belief.

Very Original Proofs of Two Famous Problems: “Are There Any Odd Perfect Numbers?” (Unsolved until to Date) and “Fermat’s Last Theorem: A New Proof of Theorem (Less than One and a Half Pages) and Its Generalization”

This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. Note that the second solution is very short and does not exceed one and a half pages. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.

A Brief New Proof to Fermat’s Last Theorem and Its Generalization

This article presents a brief and new solution to the problem known as the “Fermat’s Last Theorem”. It is achieved without the use of abstract algebra elements or elements from other fields of modern mathematics of the twentieth century. For this reason it can be easily understood by any mathematician or by anyone who knows basic mathematics. The important thing is that the above “theorem” is generalized. Thus, this generalization is essentially a new theorem in the field of number theory.

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.

One More Assertion to Fermat’s Last Theorem

Around 1637, Fermat wrote his Last Theorem in the margin of his copy “It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers”. With n, x, y, z ∈ N (meaning that n, x, y, z are all positive numbers) and n > 2, the equation xn + yn = zn has no solutions. In this paper, I try to prove Fermat’s statement by reverse order, which means no two cubes forms cube, no two fourth power forms a fourth power, or in general no two like powers forms a single like power greater than the two. I used roots, powers and radicals to assert Fermat’s last theorem. Also I tried to generalize Fermat’s conjecture for negative integers, with the help of radical equivalents of Pythagorean triplets and Euler’s disproven conjecture.

Products of Odd Numbers or Prime Number Can Generate the Three Members’ Families of Fermat Last Theorem and the Theorem Is Valid for Summation of Squares of More Than Two Natural Numbers

Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that An + Bn = Cn, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A2 + B2 + C2 + D2 + so on =An2 where all are natural numbers.

On a Simpler, Much More General and Truly Marvellous Proof of Fermat’s Last Theorem (I)

English mathematics Professor, Sir Andrew John Wiles of the University of Cambridge finally and conclusively proved in 1995 Fermat’s Last Theorem which had for 358 years notoriously resisted all gallant and spirited efforts to prove it even by three of the greatest mathematicians of all time—such as Euler, Laplace and Gauss. Sir Professor Andrew Wiles’s proof employed very advanced mathematical tools and methods that were not at all available in the known World during Fermat’s days. Given that Fermat claimed to have had the “truly marvellous” proof, this fact that the proof only came after 358 years of repeated failures by many notable mathematicians and that the proof came from mathematical tools and methods which are far ahead of Fermat’s time, has led many to doubt that Fermat actually did possess the “truly marvellous” proof which he claimed to have had. In this short reading, via elementary arithmetic methods, we demonstrate conclusively that Fermat’s Last Theorem actually yields to our efforts to prove it.

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.

A COMMENT ON THE PROOF OF FERMAT'S LAST THEOREM

In this paper, some conmments on the proof of Fermat’s last theorem are proposed.The main resuilt is thai the proof proposed by Wong Chiahe is only part of proof for fermat’s last theorem. That is to sqy ,the proof is not all-full proof to Fermat’s last theorem.

THE PROOF OF FERMAT'S LAST THEOREM

i) Instead of x￣n+ y￣n = z￣n ,we use as the general equation of Fermat's Last Theorem (FLT),where a and b are two arbitrary natural numbers .By means of binomial expansion ,(0.1) an be written as Because a￣r-(-b)￣r always contains a +b as its factor ,(0.2) can be written as where φ_r =[a￣r-(-b)￣r]/ (a+b ) are integers for r=1 . 2, 3. ...n (ii) Lets be a factor of a+b and let (a +b) = se. We can use x= sy to transform (0.3 ) to the following (0.4)(iii ) Dividing (0.4) by s￣2 we have On the left side of (0.5) there is a polynomial of y with integer coefficient and on the right side there is a constant cφ/s .If cφ/s is not an integer ,then we cannot find an integer y to satisfy (0.5), and then FLT is true for this case. If cφ_n/s is an integer ,we may change a and c such the cφ_n/s≠an integer .

A Geometric Proof of Fermat’s Little Theorem

We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in using the idea of colorings applied to regular polygons to establish a number-theoretic result. A lemma traditionally, if ambiguously, attributed to Burnside provides a critical enumeration step.

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. .

On Fermat’s Last Theorem and Galaxies of Sequences of Positive Integers

We construct sequences of positive integers which are solutions of the equation x2+y2=z2. We introduce Mouanda’s choice functions which allow us to construct galaxies of sequences of positive integers. We give many examples of galaxies of numbers. We show that the equation x2n+y2n=z2n (n ≥2) has no integer solutions. We prove that the equation xn+yn=zn (n ≥3) has no solutions in N. We introduce the notion of the planetary representation of a galaxy of numbers which allow us to predict the structure, laws of the universe and life in every planet system of every galaxy of the universe. We show that every multiverse contains a finite number of universes.

A Journey into Fermat＇s Equation

As expounded in some recent mathematical conferences, this research on that amazing source of algebraic ideas known as Fermat＇s equation is aimed to prove how Fermat triples can be limited until the impossible existence through a criterion of incompatible parities related to unexplored properties of the binomial coefficients. In this paper, the authors use a technique based on the analysis of four numbers and their internal relations with three basic compulsory factors. It leads to the practical impossibility to find any triple of natural numbers candidate to satisfy Fermat＇s equation, because when the authors try to meet a condition between parity and range the authors are compelled to violate the other one, so that they are irreducibly alternative. In particular, there is a parity violation when the authors choose all the basic factors in the allowed range and the authors obtain exceeding values of one of the involved variables when the authors try to restore the parity. Since Fermat＇s last theorem would consequently be demonstrated, many readers could recall the never found elementary proof of FLT （Fermat＇s last theorem） claimed by Pierre de Fermat. The authors are not encouraging such an interpretation because this paper is intended as a journey into Fermat＇s equation and the reader＇s attitude should be towards the algebraic achievements here proposed, with their possible hidden flaws and future developments, rather than to legendary problems like Fermat＇s riddle.

A Note on Fermat Equation＇s Fascination

As former Fermatist, the author tried many times to prove Fermat＇s Last Theorem in an elementary way. Just few insights of the proposed schemes partially passed the peer-reviewing and they motivated the subsequent fruitful collaboration with Prof. Mario De Paz. Among the author＇s failures, there is an unpublished proof emblematic of the FLT＇s charming power for the suggestive circumstances it was formulated. As sometimes happens with similar erroneous attempts, containing out-of-context hints, it provides a germinal approach to power sums yet to be refined.

层次化的K.Fan’s Theorem

在L-拓扑空间中,对任意LF集,我们利用层次化的K.Fan’s Theorem定义了一种层次连通并研究了其基本性质.

基于S-R和分解定理的二维几何非线性问题的虚单元法求解

应变-旋转(Strain-Rotation,S-R)和分解定理为分析几何非线性问题提供了合理可靠的理论基础,但用有限元求解时会遇到大变形发生后的网格畸变问题。近年提出的虚单元法(Virtual element method,VEM)适用于一般的多边形网格,因此,该文尝试使用一阶虚单元求解基于S-R和分解定理的二维几何非线性问题,以克服网格畸变的影响。基于重新定义的多项式位移空间基函数,推演获得一阶虚单元分析线弹性力学问题时允许位移空间向多项式位移空间的投影表达式;按照虚单元法双线性格式的计算规则,分析处理基于更新拖带坐标法和势能率原理的增量变分方程;进而建立离散系统方程及其矩阵表达形式,并编制MATLAB求解程序;采用常规多边形网格和畸变网格,应用该文算法分析均布荷载下的悬臂梁和均匀内压下的厚壁圆筒变形。结果与已有文献和ANSYS软件的对比表明:该文算法在两种网格中均可有效执行且具备足够数值精度。总体该文算法为基于S-R和分解定理的二维几何非线性问题求解提供了一种鲁棒方法。