scholarly journals ALGEBRAIC PROOFS FERMAT'S LAST THEOREM, BEAL'S CONJECTURE

2016 ◽  
Vol 12 (9) ◽  
pp. 6576-6577
Author(s):  
James E Joseph
Keyword(s):  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Algebraic Proofs ◽  
Positive Integers

In this paper, the following statememt of Fermat's Last Theorem is proved. If x; y; z are positive integers, _ is an odd prime and z_ = x_ + y_; then x; y; z are all even. Also, in this paper, is proved Beal's conjecture; the equation z_ = x_ + y_ has no solution in relatively prime positive integers x; y; z; with _; _; _ primes at least 3:

Fermat's Last Theorem: 1637—1988

1989 ◽  
Vol 82 (8) ◽  
pp. 637-640
Author(s):  
Charles Vanden Eynden
Keyword(s):  
West Germany ◽  
Max Planck ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
The World ◽  
Positive Integers

Around 1637 the French jurist and amateur mathematician Pierre de Fermat wrote in the margin of his copy of Diophantus's Arithmetic that he had a “truly marvelous” proof that the equation xn + yn = zn has no solution in positive integers if n > 2. Unfortunately the margin was too narrow to contain it. In 1988 the world thought that the Japanese mathematician Yoichi Miyaoka, working at the Max Planck Institute in Bonn, West Germany, might have discovered a proof of this theorem. Such a proof would be of considerable interest because no evidence has been found that Fermat ever wrote one down, and no one has been able to find one in the 350 years since. In fact Miyaoka's announcement turned out to be premature, and a few weeks later articles reported holes in his argument that could not be repaired.

scholarly journals The asymptotic Fermat’s Last Theorem for five-sixths of real quadratic fields

2015 ◽  
Vol 151 (8) ◽  
pp. 1395-1415 ◽  
Author(s):  
Nuno Freitas ◽  
Samir Siksek
Keyword(s):  
Analytic Number Theory ◽  
Real Field ◽  
Real Quadratic Fields ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Fermat Equation ◽  
Image Position ◽  
Totally Real ◽  
Positive Integers ◽  
Level Lowering

Let $K$ be a totally real field. By the asymptotic Fermat’s Last Theorem over$K$ we mean the statement that there is a constant $B_{K}$ such that for any prime exponent $p>B_{K}$, the only solutions to the Fermat equation $$\begin{eqnarray}a^{p}+b^{p}+c^{p}=0,\quad a,b,c\in K\end{eqnarray}$$ are the trivial ones satisfying $abc=0$. With the help of modularity, level lowering and image-of-inertia comparisons, we give an algorithmically testable criterion which, if satisfied by $K$, implies the asymptotic Fermat’s Last Theorem over $K$. Using techniques from analytic number theory, we show that our criterion is satisfied by $K=\mathbb{Q}(\sqrt{d})$ for a subset of $d\geqslant 2$ having density ${\textstyle \frac{5}{6}}$ among the squarefree positive integers. We can improve this density to $1$ if we assume a standard ‘Eichler–Shimura’ conjecture.

scholarly journals Fermat's Last Theorem: A Proof by Contradiction

2021 ◽  
Author(s):  
Benson Schaeffer
Keyword(s):  
Positive Integer ◽  
Diophantine Equations ◽  
Algebraic Proof ◽  
Fermat's Last Theorem ◽  
Binomial Expansion ◽  
Proof By Contradiction ◽  
Fermat’S Last Theorem ◽  
Integer Solutions ◽  
Positive Integers ◽  
Fermat's Equation

In this paper I offer an algebraic proof by contradiction of Fermat’s Last Theorem. Using an alternative to the standard binomial expansion, (a+b) n = a n + b Pn i=1 a n−i (a + b) i−1 , a and b nonzero integers, n a positive integer, I show that a simple rewrite of the Fermat’s equation stating the theorem, A p + B p = (A + B − D) p , A, B, D and p positive integers, D < A < B, p ≥ 3 and prime, entails the contradiction, A(B − D) X p−1 i=2 (−D) p−1−i "X i−1 j=1 A i−1−j (A + B − D) j−1 # + B(A − D) X p−1 i=2 (−D) p−1−i "X i−1 j=1 B i−1−j (A + B − D) j−1 # = 0, the sum of two positive integers equal to zero. This contradiction shows that the rewrite has no non-trivial positive integer solutions and proves Fermat’s Last Theorem. AMS 2020 subject classification: 11A99, 11D41 Diophantine equations, Fermat’s equation ∗The corresponding author. E-mail: [email protected] 1 1 Introduction To prove Fermat’s Last Theorem, it suffices to show that the equation A p + B p = C p (1In this paper I offer an algebraic proof by contradiction of Fermat’s Last Theorem. Using an alternative to the standard binomial expansion, (a+b) n = a n + b Pn i=1 a n−i (a + b) i−1 , a and b nonzero integers, n a positive integer, I show that a simple rewrite of the Fermat’s equation stating the theorem, A p + B p = (A + B − D) p , A, B, D and p positive integers, D < A < B, p ≥ 3 and prime, entails the contradiction, A(B − D) X p−1 i=2 (−D) p−1−i "X i−1 j=1 A i−1−j (A + B − D) j−1 # + B(A − D) X p−1 i=2 (−D) p−1−i "X i−1 j=1 B i−1−j (A + B − D) j−1 # = 0, the sum of two positive integers equal to zero. This contradiction shows that the rewrite has no non-trivial positive integer solutions and proves Fermat’s Last Theorem.

scholarly journals The Proof of the Insolubility in Natural Numbers for n>2, the Fermat's Last Theorem and Beal's Conjecture for Co-Prime Integers Arranged in a Pair A, B, D in the Equations An+Bn=Dn and An+By=Dz (Elementary Aspect)

2013 ◽  
Vol 5 ◽  
pp. 44-47
Author(s):  
K. Raja Rama Gandhi ◽  
Reuven Tint
Keyword(s):  
Natural Numbers ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Coprime Integers ◽  
Positive Integers

We give the corresponding identities for different solutions of the equations: aAx+bBx=cDx [1] and aAx+bBy=cDz [2]: As for coprime integers a, b, c, A, B, D and arbitrary positive integers x, y, z further, for not coprime integers, if A0x0+B0x0=D0xo [3] and A0x0+B0yo=D0z0 [4], where x0, y0, z0, A0, B0, D0 - are any solutions in positive integers.

5. More triangles and squares

2020 ◽  
pp. 79-96
Author(s):  
Robin Wilson
Keyword(s):  
Diophantine Equations ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Whole Number ◽  
Perfect Squares ◽  
Positive Integers ◽  
Lagrange's Theorem

‘More triangles and squares’ explores Diophantine equations, named after the mathematician Diophantus of Alexandria. These are equations requiring whole number solutions. Which numbers can be written as the sum of two perfect squares? Joseph-Louis Lagrange’s theorem guarantees that every number can be written as the sum of four squares, and Edward Waring correctly suggested that there are similar results for higher powers. In 1637, Fermat conjectured that no three positive integers, a, b, and c, can satisfy the equation an+bn=cn, if n is greater than 2. Known as ‘Fermat’s last theorem’, this conjecture was eventually proved by Andrew Wiles in 1995.

scholarly journals A Deeper Analysis on a Generalization of Fermat´s Last Theorem

2018 ◽  
Vol 10 (2) ◽  
pp. 1
Author(s):  
Leandro Torres Di Gregorio
Keyword(s):  
Prime Factor ◽  
Fermat's Last Theorem ◽  
Complementary Approach ◽  
Fermat’S Last Theorem ◽  
Positive Integers

In 1997, the following conjecture was considered by Mauldin as a generalization of Fermat's Last Theorem: “for X, Y, Z, n$_1$, n$_2$ and n$_3$ positive integers with n$_1$, n$_2$, n$_3$> 2, if $X^{n$_1$} + Y^{n$_2$}= Z^{n$_3$}$ then X, Y, Z must have a common prime factor”. The present work provides an investigation focusing in various aspects of this conjecture, exploring the problem´s specificities with graphic resources and offering a complementary approach to the arguments presented in our previous paper. In fact, we recently discovered the general form of the counterexamples of this conjecture, what is explored in detail in this article. We also analyzed the domain in which the conjecture is valid, defined the situations in which it could fail and previewed some characteristics of its exceptions, in an analytical way.

Kepler and Wiles: Models of Perseverance

2000 ◽  
Vol 93 (8) ◽  
pp. 680-681
Author(s):  
Paul G. Shotsberger
Keyword(s):  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Formal Presentation ◽  
Positive Integers

If you have seen the videotape The Proof or read the companion book by Simon Singh (1997) called Fermat's Enigma, you know that the story of Andrew Wiles's journey toward the proof of Fermat's last theorem is a remarkable tale of hope, disappointment, persistence, and ultimate triumph. Discovered by Pierre Fermat around 1637, the theorem is simple to state: “The conjecture that xn + yn = zn, where n > 2, has no solution with x, y, and z positive integers” (James and James 1992, p. 163). Yet the proof of this well-known theorem eluded generations of mathematicians. Wiles spent nearly a decade attempting to prove the theorem before he finally succeeded in 1994. The videotape and the book each offer an all-too-rare glimpse into the private struggles that preceded the more formal presentation of a finished product.

scholarly journals 2016 ALGEBRAIC PROOF FERMAT'S LAST THEOREM (2-18)

2016 ◽  
Vol 12 (1) ◽  
pp. 5825-5826
Author(s):  
JAMES E JOSEPH
Keyword(s):  
Subject Area ◽  
Algebraic Proof ◽  
Elementary Algebra ◽  
Cyclic Groups ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Positive Integers ◽  
Elegant Proof

In 1995, A, Wiles [2], [3], 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 fol-lows: If is an odd prime and x; y; z; are relatively prime positive integers, then z 6= x + y: In this note, a new elegant proof of this result is presented. 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; y; x; are each divisible by :

Fermat's Last Theorem

1986 ◽  
Vol 59 (2) ◽  
pp. 76 ◽  
Author(s):  
Jonathan P. Dowling
Keyword(s):  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem
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