Proof Of Fermat Research Articles

The proof of Fermat’s last theorem based on the geometric principle

This paper provides another proof of Fermat's theorem. As in the previous work, a geometric approach is used, namely: instead of integers a, b, c, a triangle with side lengths a, b, c is considered. To preserve the completeness of the proof of the theorem in this work, the proof is repeated for the cases of right and obtuse triangles. In this case, the Fermat equation ap+bp=cp has no solutions for any natural number p>2 and arbitrary numbers a, b, c. When considering the case when the numbers a, b, c are sides of an acute triangle, it is proven that Fermat’s equation has no solutions for any natural number p>2 and non-zero integer numbers a, b, c. Numbers a=k, b=k+m, c=k+n, where k, m, n are natural numbers that satisfy the inequalities n>m, n<k+m, exhaust all possible variants of natural numbers a, b, c, which are the sides of the triangle. In an acute triangle, the following condition is additionally satisfied:
 To study the Fermat equation, an auxiliary function f(k,p)=kp+(k+m)p–(k+n)p, is introduced, which is a polynomial of natural degree p in the variable k. The equation f(k,p)=0 has a single positive root for any natural
 A recurrent formula connecting the functions f(k,p+1) and f(k,p) has been proven: f(k,p+1)=kf(k,p)-[n(k+n)p-m(k+m)p]. The proof of the main proposition 2 is based on considering all possible relationships between the assumed integer solution of the equation f(k,p+1)=0 and the number corresponding to this solution
 The proof was carried out using the mathematical apparatus of number theory, elements of higher algebra and the foundations of mathematical analysis. These studies are a continuation of the author’s works, in which some special cases of Fermat’s theorem were proved

Fermat's Two Page Claim of his Last Problem Verified

In the 1650s Fermat claimed a short two-page proof of his last theorem but could not fit it into the margin. This article vindicates Fermat’s claim that has never been verified. The proof of Fermat’s claim has held up for over 360 years. The solution to the problem given in this article shows that the problem can be solved in two pages and clarifies why the problem resisted solution by thousands of mathematicians both professional and amateur. The author solved the problem in 1998 but opted not to publish it until recent years. However, the author featured the proof publicly in his website challenging anyone to find a flaw in the proof. No objections ever occurred. The proof uses only theorems and proof techniques available to Fermat. It is a devious proof that can be understood by a gifted high school student because it only uses theorems taught in high school along with an interesting algorithm devised by the author for the purpose of addressing infinity issues obstructing the finding of a simple two-page solution. The algorithm invented by the author ended the quest to solve the theorem and allows the viewer of the proof to understand why this problem has resisted so many attempts. The author believes this solution is the only one possible. The author Dr. Joseph E. L. Brierly is skeptical about the many-page solution claimed by Sir Andrew Wiles some years ago. Even professional mathematicians avoid attempting to verify pages of subtle reasoning. There is a strong likelihood of errors in proofs of that magnitude. As everyone knows a computer operating system invariably has multiple bugs due to the large number of lines of coding. In essence, there is no reason for an impractical-to-verify solution taking pages of subtle reasoning when there exists an easily reviewed two-page solution given in this research article. There exist few important theorems in mathematics and physics that require more than 5 pages of reasoning.

Epistemic phase transitions in mathematical proofs

Mathematical proofs are both paradigms of certainty and some of the most explicitly-justified arguments that we have in the cultural record. Their very explicitness, however, leads to a paradox, because the probability of error grows exponentially as the argument expands. When a mathematician encounters a proof, how does she come to believe it? Here we show that, under a cognitively-plausible belief formation mechanism combining deductive and abductive reasoning, belief in mathematical arguments can undergo what we call an epistemic phase transition: a dramatic and rapidly-propagating jump from uncertainty to near-complete confidence at reasonable levels of claim-to-claim error rates. To show this, we analyze an unusual dataset of forty-eight machine-aided proofs from the formalized reasoning system Coq, including major theorems ranging from ancient to 21st Century mathematics, along with five hand-constructed cases including Euclid, Apollonius, Hernstein's Topics in Algebra, and Andrew Wiles's proof of Fermat's Last Theorem. Our results bear both on recent work in the history and philosophy of mathematics on how we understand proofs, and on a question, basic to cognitive science, of how we justify complex beliefs.

Proof Of Fermat’s Last Theorem By Choosing Two Unknowns in the Integer Solution Are Prime Exponents

In this paper we are revisits well known problem in number theory ‘ proof of Fermat’s last theorem ‘ with different perspective .Also we are presented for n greater than 2, Diophantine equations K(xn+yn)=zn and xn+yn=L zn are satisfied by some positive prime exponents of x,y,z with some sufficient values of K and L. But it is not possible to find positive integers x,y and z, which are satisfies above equations with exactly K=1 and L=1. Clearly it proves Fermat’s last theorem, which states that No positive integers of x, y, z are satisfies the equation xn+yn=zn for n greater than 2.

Retraction of: An elementary proof of Fermat’s Last Theorem for all even exponents

Retraction of: An elementary proof of Fermat’s Last Theorem for all even exponents

A modern proof of fermat's little theorem

A modern proof of fermat's little theorem

A Second New Proof of Fermat's Last Conjecture

This article introduces a second new proof of Fermat’s Last Conjecture (Nwogugu [2020] introduced the first new proof of Fermat’s Last Conjecture) which is studied in the context of Sub-Rings of Complex Numbers (Integers; Rationals).

Fermat’s Last Theorem Proved by Induction (and Accompanied by a Philosophical Comment)

A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of 𝑛 = 3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite descent. The infinite descent is linked to induction starting from 𝑛 = 3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat’s last theorem. As far as Fermat had been proved the theorem for 𝑛 = 4, one can suggest that the proof for 𝑛 ≥ 4 was accessible to him. An idea for an elementary arithmetical proof of Fermat’s last theorem (FLT) by induction is suggested. It would be accessible to Fermat unlike Wiles’s proof (1995).

Geometry of Figurate Numbers and Sums of Powers of Consecutive Natural Numbers

First, we give a geometric proof of Fermat’s fundamental formula for figurate numbers. Then we use geometrical reasoning to derive weighted identities with figurate numbers and observe some of their applications. Next, we utilize figurate numbers to provide a matrix formulation for the closed forms of the sums thus generating Bernoulli numbers. Finally, we present a formula—motivated by the inclusion-exclusion principle—for as a linear combination of figurate numbers.

Review of Modern Cryptography and Elliptic Curves, A Beginner's Guide by Thomas R. Shemanske

The equation y2 = x3 + ax2 + bx + c might seem a little innocuous at first. However, studying the sets of rational points (x; y) obeying this equation has proven to be one of the most far-reaching and fruitful areas of mathematics. For example, it led, aided and abetted by much of the most powerful mathematics of the past century, to Wiles' proof of Fermat's Last Theorem. And furthermore, these so-called "elliptic curves" (the terminology having little to do with ellipses) are actually useful. You can factor numbers with them! And send secret messages!

The y-Increment Puzzle, the Law-of-Large-Numbers, and a Second New Proof of Fermat’s Last Conjecture.

The y-Increment Puzzle is a new phenomenon that is introduced herein, and occurs in exponential equations of the type x = rs-qs; in positive integers where r = (n+y1), q=(n-y2), and y1 and y2 may or may not be equal. This class of functions has wide applications in computer science, applied math, physics and finance/economics. Given the y-Increment Puzzle, the Law-of-Large-Numbers (LLN) isn’t valid for all time series.

English

English

English

In this research, a proof of Beal’s conjecture is presented. A possible Pythagorean algebraic relationship between the terms of the conjecture problem will be proposed and used to arrive at the proof results. In the process of seeking the proof the solution of the congruent number problem through a family of cubic curves will be discussed. Key words: Proof of Beal’s conjecture, proof of ABC conjecture, algebraic proof of Fermat’s last theorem, the congruent number problem, rational points on the elliptic curve, Pythagorean triples

The Easiest Proof of Fermat’s Principle

This article presents the easiest direct proof of Fermat’s principle with the help of differential calculus for all types of surfaces.

A new proof of fermat’s last theorem

A new proof of fermat’s last theorem

A Proof of Fermat's Last Theorem using an Euler's Equation: A Corollary

A Proof of Fermat's Last Theorem using an Euler's Equation: A Corollary

Proof of de Smit’s conjecture: a freeness criterion

Let $A\rightarrow B$ be a morphism of Artin local rings with the same embedding dimension. We prove that any $A$-flat $B$-module is $B$-flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond’s criterion [The Taylor–Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379–391, Theorem 2.1]. We also prove that if there is a nonzero $A$-flat $B$-module, then $A\rightarrow B$ is flat and is a relative complete intersection. Then we explain how this result allows one to simplify Wiles’s proof of Fermat’s last theorem: we do not need the so-called ‘Taylor–Wiles systems’ any more.

A Proof of Fermat's Last Theorem using an Euler's Equation

A Proof of Fermat's Last Theorem using an Euler's Equation

Rational solutions of an elliptic curve

The study of elliptic curves is long standing. A fundamental problem for an elliptic curve is to find all rational solutions. The problem has played and continues to play a fundamental role in the development of many areas of number theory. The theory of elliptic curves was crucial in the proof of Fermat’s Last Theorem. In this paper, we introduce rational solutions of lines and conics first. A natural problem of rational solutions of plane cubic arises, which is the problem of rational solutions of an elliptic curve in most cases. There is a natural addition on the rational solutions of an elliptic curve. So the set of rational solutions of an elliptic curve forms an abelian group. In 1922, Mordell proved that the set of rational solutions of an elliptic curve was a finitely generated abelian group, which confirmed a conjecture of Poincare in 1901. By the fundamental structure theorem of finitely generated abelian group, the group of rational solutions of an elliptic curve is completely determined by its rank and torsion subgroup. The rank of the rational solutions of an elliptic curve is called the rank of the elliptic curve, which is an import invariant of an elliptic curve. Therefore, an elliptic curve having infinitely many rational solutions is equivalent to the rank of elliptic curve being positive. After many efforts, people are clear about the torsion subgroup of the rational solutions of an elliptic curve. However, people know a little about the rank of an elliptic curve. There are many conjectures of the rank of elliptic curves. One of them says most elliptic curves has rank 0 or 1. Birch and Swinnerton-Dyer reduce an elliptic curve over rational numbers to the elliptic curve over a finite field. The advantage of an elliptic curve over a finite field is that its solutions are finite. Then Birch and Swinnerton-Dyer define Hasse-Weil L -function L ( E , s ) regarding of solutions of the elliptic curve over the finite field. The Hasse-Weil L -function L ( E , s ) has good analytic properties and is somehow computable. Birch and Swinnerton-Dyer conjecture that the rank of an elliptic curve E and the order of s =1 of the Hasse-Weil L -function L ( E , s ) are equal. The BSD conjecture offers an approach to determine whether an elliptic curve has infinitely many rational solutions. However, the conjecture is extremely difficult, and it was chosen as one of the seven Millennuim Prize Problems listed by Clay Mathematics Institute, which has offered a $1,000,000 prize for the first correct proof. We will introduce current progress of the conjecture. In the end, we introduce Fermat’s Last Theorem and congruent numbers. These two problems are closely related to the theory of elliptic curves. Fermat’s Last Theorem was complete solved by Wiles in 1995. The problem of congruent numbers is not clear so far. We will introduce the recent important progress of the problem by Chinese mathematician Ye Tian. Besides, important and useful applications of elliptic curves have been found in many fields, such as cryptography, factoring large integers, and primality proving.

Fermat's Last theorem Algebraic Proof

In 1995, A, Wiles announced, using cyclic groups ( a subject area which was not available at the time of Fermat), a proof of Fermat's Last Theorem, which is stated as follows: If is an odd prime and x; y; z; are relatively prime positive integers, then z 6= x + y: In this note, an elegant proof of this result is given. It is proved, using elementary algebra, that if is an odd prime and x; y; z are positive integers satisfying z = x+y; then z; x; y are each divisible by :