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 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.

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 On a few Diophantine equations, in particular, Fermat's last theorem

2003 ◽  
Vol 2003 (71) ◽  
pp. 4473-4500
Author(s):  
C. Levesque
Keyword(s):  
Diophantine Equations ◽  
Number Fields ◽  
Linear Forms In Logarithms ◽  
Open Problems ◽  
Abc Conjecture ◽  
Linear Forms ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
The Subject ◽  
Integral Solutions

This is a survey on Diophantine equations, with the purpose being to give the flavour of some known results on the subject and to describe a few open problems. We will come across Fermat's last theorem and its proof by Andrew Wiles using the modularity of elliptic curves, and we will exhibit other Diophantine equations which were solvedà laWiles. We will exhibit many families of Thue equations, for which Baker's linear forms in logarithms and the knowledge of the unit groups of certain families of number fields prove useful for finding all the integral solutions. One of the most difficult conjecture in number theory, namely, theABC conjecture, will also be described. We will conclude by explaining in elementary terms the notion of modularity of an elliptic curve.

On a Few Diophantine Equations Related to Fermat’s Last Theorem

2002 ◽  
Vol 45 (2) ◽  
pp. 247-256 ◽  
Author(s):  
O. Kihel ◽  
C. Levesque
Keyword(s):  
Trivial Solution ◽  
Diophantine Equations ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem

AbstractWe combine the deep methods of Frey, Ribet, Serre and Wiles with some results of Darmon, Merel and Poonen to solve certain explicit diophantine equations. In particular, we prove that the area of a primitive Pythagorean triangle is never a perfect power, and that each of the equations X4−4Y4 = Zp, X4 + 4Yp = Z2 has no non-trivial solution. Proofs are short and rest heavily on results whose proofs required Wiles’ deep machinery.

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 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:

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.

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.

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