scholarly journals A plane trigonometric proof for the case n = 4 of Fermat’s Last Theorem

2021 ◽  
Vol 27 (4) ◽  
pp. 154-163
Author(s):  
Giri Prabhakar ◽  
Keyword(s):  
Geometric Construction ◽  
Real Numbers ◽  
Positive Real ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Acute Triangle ◽  
The Law ◽  
Law Of Cosines ◽  
Fermat Equation ◽  
Insight Into

We present a plane trigonometric proof for the case n = 4 of Fermat’s Last Theorem. We first show that every triplet of positive real numbers (a, b, c) satisfying a4 + b4 = c4 forms the sides of an acute triangle. The subsequent proof is founded upon the observation that the Pythagorean description of every such triangle expressed through the law of cosines must exactly equal the description of the triangle from the Fermat equation. On the basis of a geometric construction motivated by this observation, we derive a class of polynomials, the roots of which are the sides of these triangles. We show that the polynomials for a given triangle cannot all have rational roots. To the best of our knowledge, the approach offers new geometric and algebraic insight into the irrationality of the roots.

scholarly journals Asymptotic generalized Fermat’s last theorem over number fields

2019 ◽  
Vol 16 (05) ◽  
pp. 907-924
Author(s):  
Yasemin Kara ◽  
Ekin Ozman
Keyword(s):  
Number Fields ◽  
Real Field ◽  
Quadratic Number ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Imaginary Quadratic Number ◽  
Fermat Equation ◽  
Totally Real ◽  
Totally Real Fields ◽  
Generalized Fermat Equation

Recent work of Freitas and Siksek showed that an asymptotic version of Fermat’s Last Theorem (FLT) holds for many totally real fields. This result was extended by Deconinck to the generalized Fermat equation of the form [Formula: see text], where [Formula: see text] are odd integers belonging to a totally real field. Later Şengün and Siksek showed that the asymptotic FLT holds over number fields assuming two standard modularity conjectures. In this work, combining their techniques, we show that the generalized Fermat’s Last Theorem (GFLT) holds over number fields asymptotically assuming the standard conjectures. We also give three results which show the existence of families of number fields on which asymptotic versions of FLT or GFLT hold. In particular, we prove that the asymptotic GFLT holds for a set of imaginary quadratic number fields of density 5/6.

Ten Lessons From the Proof of Fermat's Last Theorem

1999 ◽  
Vol 92 (6) ◽  
pp. 530-531
Author(s):  
Jeremy Kahan
Keyword(s):  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Insight Into

The proof of Fermat's last theorem establishes the truth of an old mathematical conjecture in ways that are beyond the understanding of most students and their teachers. Yet the lessons from the proof process can help teachers and their students gain insight into constructing mathematics.

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 Sex Work in Slovenia: Assessing the Needs of Sex Workers

2021 ◽  
pp. 136078042110184
Author(s):  
Leja Markelj ◽  
Alisa Selan ◽  
Tjaša Dolinar ◽  
Matej Sande
Keyword(s):  
Needs Assessment ◽  
Sex Work ◽  
Sex Workers ◽  
Point Of View ◽  
Social Distress ◽  
Legal Advice ◽  
The Law ◽  
Client Relations ◽  
Insight Into

The research comprehensively identifies the needs and problems of sex workers in Slovenia from the point of view of three groups of actors in a decriminalized setting. The objective of the rapid needs assessment was to identify the needs of sex workers as perceived by themselves. In order to gain a deeper insight into this topic, we analyzed the functioning of the organizations working with the population, and examined the perspective of the clients. The results of the study show that no aid programmes have been developed for sex workers, even though organizations from various fields often come in contact with this population. Sex workers express the need to be informed about various topics (health, the law, legal advice) and emphasize client relations as the primary issue. The findings indicate the need for the development of a specialized aid programmes to address the fields of advocacy, reducing social distress and providing psychosocial assistance.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

Determinacy of Banach games

1985 ◽  
Vol 50 (1) ◽  
pp. 110-122
Author(s):  
Howard Becker
Keyword(s):  
Winning Strategy ◽  
Similar Type ◽  
Infinite Games ◽  
Real Numbers ◽  
Positive Real ◽  
Infinite Game ◽  
First Time

For any A ⊂ R, the Banach game B(A) is the following infinite game on reals: Players I and II alternately play positive real numbers a1; a2, a3, a4,… such that for n > 1, an < an−1. Player I wins iff ai exists and is in A.This type of game was introduced by Banach in 1935 in the Scottish Book [15], Problem 43. The (rather vague) problem which Banach posed was to characterize those sets A for which I (II) has a winning strategy in B(A). (There are three parts to Problem 43. In the first, Mazur defined a game G**(A) for every set A ⊂ R and conjectured that II has a winning strategy in G**(A) iff A is meager and I has a winning strategy in G**(A) iff A is comeager in some neighborhood; this conjecture was proved by Banach. Presumably Banach had this result in mind when he asked the question about B(A), and hoped for a similar type of characterization.) Incidentally, Problem 43 of the Scottish Book appears to be the first time infinite games of any sort were studied by mathematicians.This paper will not provide the reader with any answer to Banach's question. I know of no nontrivial way to characterize when player I (or II) wins, and I suspect there is none. This paper is concerned with a different (also rather vague) question: For which sets A is the Banach game B(A) determined? To say that B(A) is determined means, of course, that one of the players has a winning strategy for B(A).

scholarly journals On the Solutions of the System of Difference Equationsxn+1=max{A/xn,yn/xn},yn+1=max{A/yn,xn/yn}

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Dağistan Simsek ◽  
Bilal Demir ◽  
Cengiz Cinar
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Real Numbers ◽  
System Of Difference Equations ◽  
Positive Real

We study the behavior of the solutions of the following system of difference equationsxn+1=max⁡{A/xn,yn/xn},yn+1=max⁡{A/yn,xn/yn}where the constantAand the initial conditions are positive real numbers.

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

scholarly journals Boundedness of solutions and stability of certain second-order difference equation with quadratic term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Emin Bešo ◽  
Senada Kalabušić ◽  
Naida Mujić ◽  
Esmir Pilav
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Second Order ◽  
Quadratic Term ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

AbstractWe consider the second-order rational difference equation $$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$xn+1=γ+δxnxn−12, where γ, δ are positive real numbers and the initial conditions $x_{-1}$x−1 and $x_{0}$x0 are positive real numbers. Boundedness along with global attractivity and Neimark–Sacker bifurcation results are established. Furthermore, we give an asymptotic approximation of the invariant curve near the equilibrium point.

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