Related papers: Understanding Fermat's Last Theorem's Proofs
We give an explicit formulae for obtaining the translation symmetries in the cartesian product $X^N$, where $N$ is some positive integer and $X$ is some finite set. Moreover, we obtain some fundamental results from elementary number theory.
This paper's first aim is to prove a modernized Occam's razor beyond a reasonable doubt. To summarize the main argument in one sentence: If we consider all possible, intelligible, scientific models of ever-higher complexity, democratically,…
We present the proof of Diophantus' 20th problem (book VI of Diophantus' Arithmetica), which consists in wondering if there exist right triangles whose sides may be measured as integers and whose surface may be a square. This problem was…
Last three years have seen new developments in the theory of last passage percolation, which has variety applications to random permutations, random growth and random vicious walks. It turns out that a few class of models have determinant…
We study the polynomials $x^n + (1-x)^n + a^n, a \in\mathbb{Q}$, whose rational roots would yield counterexamples to Fermat's Last Theorem. We investigate their factorization over $\mathbb{Q}$. In the case $a \notin \{0, \pm 1\}$, we ask…
A significant amount of research has considered mathematical proofs, the students who learn them, and the instructors that teach them, from a variety of perspectives. This paper considers this topic from four main perspectives: students'…
In this article, I explain why I abandoned my original plan to write a novel about a mathematician proving Fermat's Last Theorem, letting my protagonist solve the entirely fictional Wild Number Problem instead. I enumerate the steps that…
While preparing the correspondence between Leonhard Euler and Christian Goldbach for publication, Martin Mattmueller asked whether the lemma given in the postscript of Euler's letter dated July 26, 1749, was enough for completing the proof…
The original proof of the Sharkovsky theorem is presented in full detail. The proof should be accessible to readers with basic Real Analysis background. Although nowadays there are several alternative proofs of this classical result, we…
We review the accomplishments of nuclear theory during the past five years and identify the future challenges and research opportunities. [Final, substantially revised version of the nuclear theory white paper]
This paper provides a proof of a LLT-like test for Fermat numbers, based on the properties of Lucas Sequences and on the method of Lehmer.
In this paper we study the logical foundations of automated inductive theorem proving. To that aim we first develop a theoretical model that is centered around the difficulty of finding induction axioms which are sufficient for proving a…
For Fatou's interpolation theorem of 1906 we suggest a new elementary proof.
We give a new proof of the fundamental theorem of algebra. It is entirely elementary, focused on using long division to its fullest extent. Further, the method quickly recovers a more general version of the theorem recently obtained by…
These notes were written from a series of lectures given in March 2010 at the Universidad Complutense of Madrid and then in Barcelona for the centennial anniversary of the Spanish Mathematical Society (RSME). Our aim is to give an…
We provide a new simple and transparent proof of the version of Kummer's test given in [Tong, J. (1994). Amer. Math. Monthly. 101(5): 450--452]. Our proof is based on an application of a Hardy--Littlewood Tauberian theorem.
Based on the recently proposed framework of general relativistic stochastic mechanics [{\em J. Stat. Phys.}, 190:193, 2023; {\em J. Stat. Phys.}, 190:181, 2023] and stochastic thermodynamics [{\em SciPost Physics Core} 7, 082, 2024] at the…
In fuzzy group theory many versions of the well-known Lagrange's theorem have been studied. The aim of this article is to investigate the converse of one of those results. This leads to an interesting characterization of finite cyclic…
The purpose of this paper is to present a generalization of Forelli's theorem. In particular, we prove an all dimensional version of the two-dimensional theorem of Chirka of 2005.
Not only motivated by the fact that the publication of the GAFT first appeared 60 years ago in print we reconstruct its history and so show that it is no exaggeration to claim that it has appeared already 75 years ago!