First Case of Fermat's Last Theorem

Finding the Fermat point via analysis

Let $P_1,P_2,P_3$ be three given points in $\mathbf{R}^2$, and $P$ be an arbitrary point in $\mathbf{R}^2$. The classical Fermat's problem to Torricelli asks for the location of the point $P$ such that $|PP_1|+|PP_2|+|PP_3|$ is a minimum. There exist several elegant geometrical solutions in the literature.In this note, we consider finding the Fermat point by using methods in advanced calculus. The main tools we use are the extreme value theorem,Fermat's theorem, and the intermediate value theorem.

Eisenstein's criterion, Fermat's last theorem, and a conjecture on powerful numbers

Given integers $\ell > m >0$, we define monic polynomials $X_n$, $Y_n$, and $Z_n$ with the property that $μ$ is a zero of $X_n$ if and only if the triple $(μ,μ+m,μ+\ell)$ satisfies $x^n + y^n = z^n$. It is shown that the irreducibility of these polynomials implies Fermat's last theorem. It is also shown, in a precise asymptotic sense, that for a vast majority of cases, these polynomials are irreducible via Eisenstein's criterion. We conclude by offering a conjecture on powerful numbers.

费马两平方定理的一个缺乏想象力的证明

An Unimaginative Proof of Fermat's Two Squares Theorem

本文给出费马两平方定理的一个简单直接证明，不使用复杂概念，仅靠仔细记账，易于初学者理解。

费马是如何发现他的定理的？

How did Fermat discover his theorem?

本文通过考察费马对完全数的研究及其书信线索，重构Mersenne数的素因数分解模式如何引导他发现费马小定理。

Observations on a certain theorem of Fermat and on others concerning prime numbers

E26 in the Enestrom index. Translated from the Latin original, "Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus" (1732). In this paper Euler gives a counterexample to Fermat's claim that all numbers of the form 2^{2^m}+1 are primes, by showing 2^{2^5}+1=4294967297 is divisible by 641. He also considers many cases in which we are guaranteed that a number is composite, but he notes clearly that it is not possible to have a full list of circumstances under which a number is composite. He then gives a theorem and several corollaries of it, but he says that he …

Theoremata arithmetica nova methodo demonstrata

Euler presents a third proof of the Fermat theorem, the one that lets us call it the Euler-Fermat theorem. This seems to be the proof that Euler likes best. He also proves that the smallest power x^n that, when divided by a numer N, prime to x, and that leaves a remainder of 1, is equal to the number of parts of N that are prime to n, that is to say, the number of distinct aliquot parts of N. The translation is presnted from Euler's Latin original into German.

索菲·热尔曼，非凡的数学家：比小说更离奇的故事

Sophie Germain, mathématicienne extraordinaire: A story stranger than fiction

本文回顾索菲·热尔曼在数论领域的贡献，揭示她对费马大定理的深入研究及与拉格朗日、勒让德和高斯的互动，重新评估其在数论中的重要性。

Fermat's Four Squares Theorem

It is easy to find a right-angled triangle with integer sides whose area is 6. There is no such triangle with area 5, but there is one with rational sides (a `\emph{Pythagorean triangle}'). For historical reasons, integers such as 6 or 5 that are (the squarefree part of) the area of some Pythagorean triangle are called `\emph{congruent numbers}'. These numbers actually are interesting for the following reason: Notice the sequence $\frac14$, $6\frac14$, $12\frac14$. It is an arithmetic progression with common difference 6, consisting of squares $(\frac12)^2$, $(\frac52)^2$, $(\frac72)^2$ of rat…