Bjoern Grohmann
In this article an upper bound for the first consecutive zeros of the Fermat quotient is given in terms of the zeros of a Mirimanoff polynomial. This bound is obtained by investigating a relation between these polynomials and the factor system of Gauss sums of a certain number field.
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This paper explores some previously-unrecognized consequences of Lerch's 1905 formula for the Fermat quotient, with special attention to the sums which he introduced in this context. A generalization of his result is proved, and a new proof given of a sharpened result by Skula (2008). We also sharpen the criteria given by Emma Lehmer in 1938 for a Wieferich prime to be simultaneously a Mirimanoff prime.
This paper presents some refinements of the representation of the Fermat quotient of base 2 as an alternating series which was discovered by Eisenstein in 1850, including some evaluations that are believed to be new.
We improve an estimate of A.Granville (1987) on the number of vanishing Fermat quotients $q_p(\ell)$ modulo a prime $p$ when $\ell$ runs through primes $\ell \le N$. We use this bound to obtain an unconditional improvement of the conditional (under the Generalised Riemann Hypothesis) estimate of Y. Ihara (2006) on a certain sum, related to vanishing Fermat quotients. In turn this sum appears in the study of the index of certain subfields of of cyclotomic fields $\Q(\exp(2 πi/p^2))$.
We show that for any $\varepsilon > 0$ and a sufficiently large cube-free $q$, any reduced residue class modulo $q$ can be represented as a product of $14$ integers from the interval $[1, q^{1/4e^{1/2} + \varepsilon}]$. The length of the interval is at the lower limit of what is possible before the Burgess bound on the smallest quadratic nonresidue is improved. We also consider several variations of this result and give applications to Fermat quotients.
Wilson's theorem modulo higher prime powers I: Fermat and Wilson quotients
该研究利用Bell多项式与Newton恒等式，证明威尔逊定理及威尔逊商可由含费马商幂和项的任意更高素数幂模超同余式描述，还给出相关多元多项式的递推公式与更多性质，推进了数论中超同余问题的研究。
Irreducibility and locus of complex roots of polynomials related to Fermat's Last Theorem
研究有理数a下多项式x^n+(1-x)^n+a^n的不可约性，证明多个无穷族的不可约性，并分析复根位置，同时揭示a=±1时与Cauchy-Mirimanoff多项式及Nanninga多项式的因式分解关系。
For a prime $p$ and an integer $u$ with $\gcd(u,p)=1$, we define Fermat quotients by the conditions $$ q_p(u) \equiv \frac{u^{p-1} -1}{p} \pmod p, \qquad 0 \le q_p(u) \le p-1. $$ D. R. Heath-Brown has given a bound of exponential sums with $N$ consecutive Fermat quotients that is nontrivial for $N\ge p^{1/2+ε}$ for any fixed $ε>0$. We use a recent idea of M. Z. Garaev together with a form of the large sieve inequality due to S. Baier and L. Zhao, to show that on average over $p$ one can obtain a nontrivial estimate for much shorter sums starting with $N\ge p^ε$. We also obtain lower bounds on …
Polynomial identities and Fermat quotients
本文通过证明多项式恒等式，推导出费马商模 $p^2$ 的更简单同余式，推广 Eisenstein 经典同余式，并得到调和数相关和式及高阶二项和公式。
Fermat quotients and the Ankeny-Artin-Chowla conjecture
本文为 Ankeny-Artin-Chowla 关于实二次域基本单位数的结论提供了精简证明，推广了该猜想，并据此将模 p 的二次非剩余的费马商与调和数之和联系起来。
For a fixed integer a>1, we suggest that the probability of nullity of the p-Fermat quotient q(p,a) is much lower than 1/p for any arbitrary large prime number p. For this we use various heuristics, justified by means of numerical computations and analytical results, which may imply the finiteness of the q(p,a) equal to 0 and the existence of integers a such that q(p,a) is different from 0 for all p. However no proofs are obtained concerning these heuristics.
Some Congruences Involving Binomial Coefficient and Fermat Quotient
本文研究了涉及二项式系数和费马商的某些同余式，探讨了相关求和形式的性质。
The classifying element for quotients of Fermat curves
该研究针对射影直线上在0、1、∞三点分歧的循环伽罗瓦覆叠C，在温和分歧条件下，借助适配绝对伽罗瓦群作用研究的基，确定了其基本群下中心序列的分次李代数结构。
Pattern formation Statistics on Fermat Quotients
本文研究费马商矩阵中图案形成的统计特性，证明矩阵行具有随机性，而规则图案上的距离统计符合预期，揭示了其缺乏规律性与规则排列的对比。
Root numbers for twisted Fermat quotient curves
本文基于扭曲费马商曲线的Hecke特征，推导出根数的显式公式，扩展了Stoll(2002)和Shu(2021)的研究成果。
We obtain some theoretic and experimental results concerning various properties (the number of fixed points, image distribution, cycle lengths) of the dynamical system naturally associated with Fermat quotients acting on the set $\{0, ..., p-1\}$. We also consider pseudorandom properties of Fermat quotients such as joint distribution and linear complexity.
Given an odd prime p, we prove a congruence on the p-residue of the Fermat quotient q_p(a) in base a with 0<a<p, which arises from a generalization of the Voronoi congruences and from some other congruences on sums and weighted sums of divided Bernoulli numbers. As an application in the base 2 case, we retrieve a congruence for the first generalized harmonic number of order (p-1)/2, a generalization originally due to Z-H. Sun of a classical congruence known since long for the harmonic number of order (p-1)/2 as a special case of the Lerch formula. We find a sharpening of the Voronoi congruence…
We describe the trace representations of two families of binary sequences derived from Fermat quotients modulo an odd prime $p$ (one is the binary threshold sequences, the other is the Legendre-Fermat quotient sequences) via determining the defining pairs of all binary characteristic sequences of cosets, which coincide with the sets of pre-images modulo $p^2$ of each fixed value of Fermat quotients. From the defining pairs, we can obtain an earlier result of linear complexity for the binary threshold sequences and a new result of linear complexity for the Legendre-Fermat quotient sequences und…
We obtain the trace map image of the values of certain harmonic volumes for some quotients of Fermat curves. This provides the algorithm that the algebraic cycles called by the k-th Ceresa cycles are not algebraically equivalent to zero in the Jacobian varieties.
Let C denote the Fermat curve over Q of prime exponent l. The Jacobian Jac(C) of C splits over Q as the product of Jacobians Jac(C_k), 0< k < l-1, where C_k are curves obtained as quotients of C by certain subgroups of automorphisms of C. It is well known that Jac(C_k) is the power of an absolutely simple abelian variety B_k with complex multiplication. We call degenerate those pairs (l,k) for which B_k has degenerate CM type. For a non-degenerate pair (l,k), we compute the Sato-Tate group of Jac(C_k), prove the generalized Sato-Tate Conjecture for it, and give an explicit method to compute th…
Root numbers for twisted Fermat quotient curves II
本文研究二次费马商曲线根数，放宽了技术假设ℓ∤rst的限制，计算了当ℓ^{N-1}‖r且ℓ∤stδ时的根数。
We classify all geometric torsion points on the Fermat quotients $y^n = x^d + 1$ where $n, d \ge 2$ are coprime. In addition, we classify all geometric torsion points on the generic superelliptic curve $y^n = (x - a_1) \cdots (x - a_d)$, extending a result of Poonen and Stoll, who considered the hyperelliptic $n = 2$ case.
We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), Ernvall-Metsankyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2001), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsankyla residues to some full convolutions of the same kind. We also establish some congruences concerning other related weighed sums of powers of …
We give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, the central value of the $L$-function of the relevant motive is non-vanishing and the Ceresa cycle is torsion in the Griffiths group, consistent with the conjectures of Beilinson and Bloch. We speculate on a possible explanation for the existence of these torsion Ceresa classes, based on some computations with cyclic Fermat quotients.
Given a prime $\ell\geq 3$ and a positive integer $k \le \ell-2$, one can define a matrix $D_{k,\ell}$, the so-called Demjanenko matrix, whose rank is equal to the dimension of the Hodge group of the Jacobian ${\mathrm Jac}({\mathcal C}_{k,\ell})$ of a certain quotient of the Fermat curve of exponent $\ell$. For a fixed $\ell$, the existence of $k$ for which $D_{k,\ell}$ is singular (equivalently, for which the rank of the Hodge group of ${\mathrm Jac}({\mathcal C}_{k,\ell})$ is not maximal) has been extensively studied in the literature. We provide an asymptotic formula for the number of such…
The Fermat quotient $q_p(a):=(a^{p-1}-1)/p$, for prime $p\nmid a$, and the Wilson quotient $w_p:=((p-1)!+1)/p$ are integers. If $p\mid w_p,$ then $p$ is a Wilson prime. For odd $p,$ Lerch proved that $(\sum_{a=1}^{p-1} q_p(a) - w_p)/p$ is also an integer; we call it the Lerch quotient $\ell_p.$ If $p\mid\ell_p$ we say $p$ is a Lerch prime. A simple Bernoulli-number test for Lerch primes is proven. There are four Lerch primes 3, 103, 839, 2237 up to $3\times10^6$; we relate them to the known Wilson primes 5, 13, 563. Generalizations are suggested. Next, if $p$ is a non-Wilson prime, then $q_p(w…
Carmichael quotients for an integer $m\ge 2$ are introduced analogous to Fermat quotients, by using Carmichael function $λ(m)$. Various properties of these new quotients are investigated, such as basic arithmetic properties, sequences derived from Carmichael quotients, Carmichael-Wieferich numbers, and so on. Finally, we link Carmichael quotients to perfect nonlinear functions.
Chebyshev's bias for Fermat curves of prime degree
本文证明素数次 Fermat 曲线及其部分商的素数竞赛渐近公式与 DRH 相关部分等价，并据此在 DRH 下计算其二阶矩 L 函数在 s=1 处的零点阶数。
We give q-analogues of the following congruences by Z.-W. Sun: \sum_{k=1}^{p-1}\frac{D_k}{k} \equiv -\frac{2^{p-1}-1}{p} \pmod p,\\ \sum_{k=1}^{p-1}\frac{H_k}{k 2^k}\equiv 0 \pmod{p},\quad p\geqslant 5, where p is a prime, D_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k} are the Delannoy numbers, and H_n=\sum_{k=1}^n\frac{1}{k} are the harmonic numbers. We also prove that, for any positive integer m and prime p>m+1, \sum_{1\leqslant k_1\leqslant \cdots \leqslant k_m\leqslant p-1}\frac{1}{k_1\cdots k_m 2^{k_m}} \equiv\frac{1}{2}\sum_{k=1}^{p-1}\frac{(-1)^{k-1}}{k^m} \pmod p, which is a multiple g…
Wilson's theorem modulo higher prime powers II: Bernoulli numbers and polynomials
该研究将作者此前用费马商幂和超同余描述威尔逊定理与威尔逊商的成果，转化为幂和与伯努利数的同余式，给出更简洁的证明，还算出了模p⁴的威尔逊商等结果并可推广到更高素数幂。
We extend the results of the first author on nontrivial elements in the Shafarevich-Tate group of the jacobian of a quotient of a Fermat curve of prime degree, and use the methods of the second author to derive a result bounding the quadratic points on the Fermat curve of degree 19.
Fermat-Euler quotients arose from the study of the first case of Fermat's Last Theorem, and have numerous applications in number theory. Recently they were studied from the cryptographic aspects by constructing many pseudorandom binary sequences, whose linear complexities and trace representations were calculated. In this work, we further study their correlation measures by using the approach based on Dirichlet characters, Ramanujan sums and Gauss sums. Our results show that the $4$-order correlation measures of these sequences are very large. Therefore they may not be suggested for cryptograp…
Let $p>3$ be a prime, and let $q_p(2)=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base 2. Recently, Z. H. Sun proved that \sum_{k=1}^{p-1}\frac{1}{k\cdot 2^k}\equiv q_p(2)-\frac{p}{2}q_p(2)^2 \pmod{p^2} which is a generalization of a congruence due to W. Kohnen. In this note we give an elementary proof of the above congruence which is based on several combinatorial identities and congruences involving the Fermat quotient $q_p(2)$, harmonic or alternating harmonic sums.
A resolution of Kellner's conjectures on Wilson quotients
本研究针对凯尔纳关于威尔逊商的猜想，利用\(p\)进对数和指数函数给出同余式的\(p\)进证明，还通过完全贝尔多项式给出多项式\(\psi_\nu\)的显式生成函数，从而解决了凯尔纳的两个猜想。
For an odd prime $p$ and an integer $w\ge 1$, polynomial quotients $q_{p,w}(u)$ are defined by $$ q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~~ \mathrm{with}~~ 0 \le q_{p,w}(u) \le p-1, ~~u\ge 0, $$ which are generalizations of Fermat quotients $q_{p,p-1}(u)$. First, we estimate the number of elements $1\le u<N\le p$ for which $f(u)\equiv q_{p,w}(u) \bmod p$ for a given polynomial $f(x)$ over the finite field $\mathbb{F}_p$. In particular, for the case $f(x)=x$ we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest…
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…
Irrationality of finite logarithms in a congruence-class adèle ring
该文在$abc$猜想成立前提下，将有限对数无理性的已有结果扩展到素数受限为$p\equiv 1\bmod m$的算术级数情形，还证明其无法为二次无理数。
In the paper we prove a new upper bound for Heilbronn's exponential sum and obtain some applications of our result to distribution of Fermat quotients.
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.
We show that, modulo some odd prime p, the powers of two weighted sum of the first p-2 divided Bernoulli numbers equals the Agoh-Giuga quotient plus twice the number of permutations on p-2 letters with an even number of ascents and distinct from the identity. We provide a combinatorial characterization of Wieferich primes, as well as of primes p for which p^2 divides the Fermat quotient q_p(2).
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