One curious proof of Fermat's little theorem

Jacob阶梯、Hardy-Littlewood积分(1918)及Euler Gamma函数的新渐近函数方程，连同Fermat-Wiles定理的第十等价式

Jacob's ladders, Hardy-Littlewood integral (1918) and new asymptotic functional equations for Euler's Gamma function together with the tenth equivalent of the Fermat-Wiles theorem

本文基于Hardy-Littlewood积分的几乎线性增量构造了新$Γ$泛函，生成了Fermat-Wiles定理的$Γ$等价式及Euler $Γ$函数的新因式分解公式集。

Jacob's ladders，通过Raabe积分和Fermat-Wiles定理的第十三个等价形式对Hardy-Littlewood积分（1918）某些增量的几乎精确分解

Jacob's ladders, almost exact decomposition of certain increments of the Hardy-Littlewood integral (1918) by means of the Raabe's integral and the thirteenth equivalent of the Fermat-Wiles theorem

本文利用Jacob's ladders理论，通过Raabe积分获得了Fermat-Wiles定理的第十三个等价形式，并实现了Hardy-Littlewood积分某些增量的几乎精确分解。

雅可比梯、哈代-利特尔伍德积分（1918）的几乎线性增量及其与塞尔伯格公式（1946）和费马-怀尔斯定理的关系

Jacob's ladders, almost linear increments of the Hardy-Littlewood integral (1918) and their relations to the Selberg's formula (1946) and the Fermat-Wiles theorem

本文从哈代-利特尔伍德积分增量公式出发，建立了费马-怀尔斯定理的一个 ζ 等价形式，该形式仅用临界线上的黎曼 ζ 函数与雅可比梯表达。

Jacob梯子，我们的渐近公式（1981）以及Fermat-Wiles定理的下一个ζ等价物，连同Riemann ζ振荡器的分解与合成

Jacob's ladders, our asymptotic formulae (1981) and next $ζ$-equivalents of the Fermat-Wiles theorem together with decomposition and synthesis of the Riemann's $ζ$-oscillators

本文利用1981年提出的渐近公式（将Hardy-Littlewood指数改善33.3%），生成了Fermat-Wiles定理的新ζ等价物，并探讨了Riemann ζ振荡器的分解与合成。

An elementary proof of a congruence by Skula and Granville

Let $p\ge 5$ be a prime, and let $q_p(2):=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base 2. The following curious congruence was conjectured by L. Skula and proved by A. Granville $$ q_p(2)^2\equiv -\sum_{k=1}^{p-1}\frac{2^k}{k^2}\pmod{p}. $$ In this note we establish the above congruence by entirely elementary number theory arguments.

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。

On Fibonacci Polynomial Expressions for Sums of $m$th Powers, their implications for Faulhaber's Formula and some Theorems of Fermat

Denote by $Σn^m$ the sum of the $m$-th powers of the first $n$ positive integers $1^m+2^m+\ldots +n^m$. Similarly let $Σ^r n^m$ be the $r$-fold sum of the $m$-th powers of the first $n$ positive integers, defined such that $Σ^0 n^{m}=n^m$, and then recursively by $Σ^{r+1} n^{m}=Σ^{r} 1^{m}+Σ^{r} 2^{m}+\ldots + Σ^{r} n^{m}$. During the early 17th-century, polynomial expressions for the sums $Σ^r n^m$ and their factorisation and polynomial basis representation properties were examined by Johann Faulhaber, who published some remarkable theorems relating to these $r$-fold sums in his Academia Alge…

Theorems on residues obtained by the division of powers

This is an English translation of Euler's ``Theoremata circa residua ex divisione potestatum relicta'', Novi Commentarii academiae scientiarum Petropolitanae 7 (1761), 49-82. E262 in the Enestrom index. Euler gives many elementary results on power residues modulo a prime number p. He shows that the order of a subgroup generated by an element a in F_p^* must divide the order p-1 of F_p^* (i.e. a special case of Lagrange's theorem for cyclic groups). Euler also gives a proof of Fermat's little theorem, that a^{p-1} = 1 mod p for a relatively prime to p (i.e. not 0 mod p). He remarks that this pr…