Fermat Quotients Research Articles

Experiments with Ceresa classes of cyclic Fermat quotients

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 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.

Some congruences on the hyper-sums of powers of integers involving Fermat quotient and Bernoulli numbers

For a given prime p ≥ 5, let ℤ_p denote the set of rational p-integers (those rational numbers whose denominator is not divisible by p). In this paper, we establish some congruences modulo a prime power p5 on the hyper-sums of powers of integers in terms of Fermat quotient, Wolstenholme quotient, Bernoulli and Euler numbers.

Statistical distribution of Fermat quotients

Let qpb=bp−1−1pmodp, for 0 ≤ b ≤ p2 − 1 and gcd(b,p) = 1, be the Fermat quotients arranged in the p × (p − 1) Fermat quotient matrix FQM(p). We study the elements of the matrix and prove two results: (i) Under the Generalized Riemann Hypothesis, ℤp is fully covered fast by the quotients qpq with q prime and q ≪ plog2p, as p → ∞ and (ii) The matrix FQM(p) passes the pair size correlation test, in which any pair (u,v) of entries in the matrix, which lie apart under any apriori chosen fixed geometric pattern, are in the limit, as p → ∞, with equal probability 1/2 in the size relation u ≤ v or u ≥ v.

Proof of Two Congruences Concerning Legendre Polynomials

The Legendre polynomials $$P_n(x)$$ are defined by $$\begin{aligned} P_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n+k\\ k\end{array}}\right) \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( \frac{x-1}{2}\right) ^k\quad (n=0,1,2,\ldots ). \end{aligned}$$ In this paper, we prove two congruences concerning Legendre polynomials. For any prime $$p>3$$ , by using the symbolic summation package Sigma, we show that $$\begin{aligned} \sum _{k=0}^{p-1}(2k+1)P_k(-5)^3\equiv p-\frac{10}{3}p^2q_p(2)\pmod {p^3}, \end{aligned}$$ where $$q_p(2)=(2^{p-1}-1)/p$$ is the Fermat quotient. This confirms a conjecture of Z.-W. Sun. Furthermore, we prove the following congruence which was conjectured by V.J.W. Guo $$\begin{aligned}&\sum _{k=0}^{p-1}(-1)^k(2k+1)P_k(2x+1)^4\\ \equiv&p\sum _{k=0}^{(p-1)/2}(-1)^k\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2(x^2+x)^k(2x+1)^{2k}\pmod {p^3},\end{aligned}$$ where p is an odd prime and x is an integer.

On the correlation measures of orders $ 3 $ and $ 4 $ of binary sequence of period $ p^2 $ derived from Fermat quotients

Let $ p $ be a prime and let $ n $ be an integer with $ (n, p) = 1 $. The Fermat quotient $ q_p(n) $ is defined as \begin{document}$ q_p(n)\equiv \frac{n^{p-1}-1}{p} \ (\bmod\ p), \quad 0\leq q_p(n)\leq p-1. $\end{document} We also define $ q_p(kp) = 0 $ for $ k\in \mathbb{Z} $. Chen, Ostafe and Winterhof constructed the binary sequence $ E_{p^2} = \left(e_0, e_1, \cdots, e_{p^2-1}\right)\in \{0, 1\}^{p^2} $ as$ \begin{equation*} \begin{split} e_{n} = \left\{\begin{array}{ll} 0, & \hbox{if }\ 0\leq \frac{q_p(n)}{p}<\frac{1}{2}, \\ 1, & \hbox{if }\ \frac{1}{2}\leq \frac{q_p(n)}{p}<1, \end{array} \right. \end{split} \end{equation*} $and studied the well-distribution measure and correlation measure of order $ 2 $ by using estimates for exponential sums of Fermat quotients. In this paper we further study the correlation measures of the sequence. Our results show that the correlation measure of order $ 3 $ is quite good, but the $ 4 $-order correlation measure of the sequence is very large.

Almost perfect sequence pairs derived from Fermat quotient

Sequence pairs with good correlation can be applied in many practical systems. In this Letter, the authors construct a class of p-ary sequence pairs of period 2 p 2 by interleaving two p-ary sequences derived from Fermat quotient, where p is an odd prime. In the authors' construction, sequence pairs have almost perfect correlation.

In brief

PAGE 589 Swarm reinforcement learning that extends standard Q-learning to multi-agent systems via the development of a real-time swarm algorithm has been presented by a large collaboration of researchers from Italy. The novel approach achieves a substantial speed increase over standard learning methods and promises enhanced learning performance in more complex environments. PAGE 573 Researchers from Belgium and China jointly present a multi-functional dual band frequency reconfigurable antenna with pattern diversity for use in wireless communication systems. The antenna consists of a composite right/left-handed transmission line inspired structure, with simulations and experiment validating the new concept. Optimal maze solution. PAGE 619 A team of researchers from Japan have produced a neural optical implant which is monolithically integrated with a micro-light-emitting diode (µLED), a micro-photodiode, and a polymer waveguide. This tool can suppress the influence of temperature while making good use of the advantages offered by the LED, with the µLED placed outside, but still able to optically stimulate, the brain. Metamaterial antenna prototype. PAGE 599 Sequence pairs with good correlation can be applied in many practical systems, with a pair of Chinese researchers having constructed a class of sequence pairs by interleaving two sequences derived from the Fermat quotient. The produced sequence pairs subsequently have almost perfect correlation. SEM image of substrate after deep RIE etching. PAGE 574 Researchers from China have presented a compact self-isolated MIMO antenna system based on a quarter-mode substrate integrated waveguide cavity. With the advantages of compact size, high isolation, and good radiation performance, the proposed antenna has many applications in wireless communications. A section of the cross-correlation proof employed. Half-mode electric field distribution.

On error linear complexity of new generalized cyclotomic binary sequences of period p2

We consider the k-error linear complexity of a new generalized cyclotomic binary sequence of period p2 for an odd prime p. The new sequences were introduced recently by Z. Xiao, X. Zeng, C. Li and T. Helleseth by defining a new kind of generalized cyclotomic classes modulo p2. They proved that the sequences had large linear complexity. In this work, we determine the values of the k-error linear complexity in terms of the theory of Fermat quotients. The results indicate that such sequences have good stability, that is, the linear complexity does not significantly decrease by changing a few terms.

Pseudo-random subsets constructed by using Fermat quotients

Pseudo-random subsets constructed by using Fermat quotients

On $k$-error linear complexity of pseudorandom binary sequences derived from Euler quotients

We investigate the $k$-error linear complexity of pseudorandom binary sequences of period $p^{\mathfrak{r}}$ derived from the Euler quotients modulo $p^{\mathfrak{r}-1}$, a power of an odd prime $p$ for $\mathfrak{r}\geq 2$. When $\mathfrak{r}=2$, this is just the case of polynomial quotients (including Fermat quotients) modulo $p$, which has been studied in an earlier work of Chen, Niu and Wu. In this work, we establish a recursive relation on the $k$-error linear complexity of the sequences for the case of $\mathfrak{r}\geq 3$. We also state the exact values of the $k$-error linear complexity for the case of $\mathfrak{r}=3$. From the results, we can find that the $k$-error linear complexity of the sequences (of period $p^{\mathfrak{r}}$) does not decrease dramatically for $k<p^{\mathfrak{r}-2}(p-1)^2/2$.

Approche p-adique de la conjecture de Greenberg pour les corps totalement réels

Let k be a totally real number field ant let k$\infty$ be its cyclotomic Zp-extension for a prime p\textgreater{}2. We give (Theorem 3.2) a sufficient condition of nullity of the Iwasawa invariants lambda, mu, when p totally splits in k, and we obtain important tables of quadratic fields and p for which we can conclude that lambda = mu=0.We show that the number of ambiguous p-classes of kn (nth stage in k$\infty$) is equal to the order of the torsion group T, of the Galois group of the maximal Abelian p-ramified pro-p-extension of k (Theorem 4.2), for all n \textgreater{}\textgreater{} e, where p^e is the exponent of U*/ adh(E) (in terms of local and global units of k). Then we establish analogs of Chevalley's formula using a family (Lambda\_i^n)\_{0$\le$i$\le$m\_n} of subgroups of k* containing E, in which any x is norm of an ideal of kn. This family is attached to the classical filtration of the p-class group of kn defining the algorithm of computation of its order in m\_n steps. From this, we prove (Theorem 6.1) that m\_n $\ge$ (lambda.n + mu.p^n + nu)/v\_p(T\_k) and that the condition m\_n = O(1) (i.e., lambda = mu=0) essentially depends on the P-adic valuations of the (x^(p-1)-1)/p, x in Lambda\_i^n, for P I p, so that Greenberg's conjecture is strongly related to "Fermat quotients" in k*. Heuristics and statistical analysis of these Fermat quotients (Sections 6, 7, 8) show that they follow natural probabilities, linked to T\_k whatever n, suggesting that lambda = mu=0 (Heuristics 7.1, 7.2, 7.3). This would imply that, for a proof of Greenberg's conjecture, some deep p-adic results (probably out of reach now), having some analogy with Leopoldt's conjecture, are necessary before referring to the sole algebraic Iwasawa theory.

On the additive energy of the Heilbronn subgroup

A new upper bound for the additive energy of the Heilbronn subgroup is found. Several applications to the distribution of Fermat quotients are obtained.

Arithmetic Euler top

The theory of differential equations has an arithmetic analogue in which derivatives of functions are replaced by Fermat quotients of numbers. Many classical differential equations (Riccati, Weierstrass, Painlevé, etc.) were previously shown to possess arithmetic analogues. The paper introduces and studies an arithmetic analogue of the Euler differential equations for the rigid body.

An extension of binary threshold sequences from Fermat quotients

We extend the construction of $p^2$-periodic binary thresholdsequences derived from Fermat quotients to the $d$-ary case where $d$is an odd prime divisor of $p-1$, and then by defining cyclotomic classesmodulo $p^{2}$, we present exact values of the linear complexityunder the condition of $d^{p-1}\not \equiv 1 \pmod {p^2}$. Also, weextend the results to the Euler quotients modulo $p^{r}$ with oddprime $p$ and $r \geq 2$. The linear complexity is very close to theperiod and is of desired value for cryptographic purpose. Theresults extend the linear complexity of the corresponding $d$-arysequences when $d$ is a primitive root modulo $p^2$ in earlier work.Finally, partial results for the linear complexity of the sequenceswhen $d^{p-1} \equiv 1 \pmod {p^2}$ is given.

Painlevé VI equations in p-adic time

Using the description of Painleve VI family of differential equations in terms of a universal elliptic curve, going back to R. Fuchs, we translate it into the realm of Buium’s p-adic Arithmetic Differential Equations, where the role of derivative is played by a version of Fermat quotient.

Lesθ-régulateurs locaux d'un nombre algébrique : Conjecturesp-adiques

AbstractLet K/ℚ be Galois and let η K×be such that Reg∞(η)=0 .We define the local θ–regulatorfor the ℚp–irreducible characters θ of G = Gal(Kℚ). Let Vθbe the θ-irreducible representation. A linear representationis associated withwhose nullity is equivalent to δ≥1. Eachyields Regθpmodulopin the factorizationof(normalizedp–adic regulator). From Probf≥ 1 is a residue degree) and the Borel–Cantelli heuristic, we conjecture that for p large enough, RegGp(η) is ap–adic unit(a single with f = δ=1); this obstruction may be led assuming the existence of a binomial probability law confirmed through numerical studies (groups C3, C5, D6) is conjecture would imply that for all p large enough, Fermat quotients, normalized p–adic regulators are p–adic units and that number fields are p-rational.We recall some deep cohomological results that may strengthen such conjectures.

Arithmetic differential equations on GLn, I: Differential cocycles

The theory of differential equations has an arithmetic analogue [8] in which derivatives are replaced by Fermat quotients. One can then ask what is the arithmetic analogue of a linear differential equation. The study of usual linear differential equations is the same as the study of the differential cocycle from GLn into its Lie algebra given by the logarithmic derivative [14] (equivalently by the Maurer–Cartan connection [17]). However we prove here that there are no such cocycles in the context of arithmetic differential equations. In sequels of this paper [10,11] we will remedy the situation by introducing arithmetic analogues of Lie algebras and a skew version of differential cocycles; this will lead to a theory of linear arithmetic differential equations.

Optimal Families of Perfect Polyphase Sequences From the Array Structure of Fermat-Quotient Sequences

We show that a $p$ -ary polyphase sequence of period $p^{2}$ from the Fermat quotients is perfect. That is, its periodic autocorrelation is zero for all non-trivial phase shifts. We call this Fermat-quotient sequence. We propose a collection of optimal families of perfect polyphase sequences using the Fermat-quotient sequences in the sense of the Sarwate bound. That is, the cross correlation of two members in a family is upper bounded by $p$ . To investigate some relation between Fermat-quotient sequences and Frank-Zadoff sequences and to construct optimal families including these sequences, we introduce generators of $p$ -ary polyphase sequences of period $p^{2}$ using their $p\times p$ array structures. We call an optimal generator to be the generator of some $p$ -ary polyphase sequences which are perfect and which gives an optimal family by the proposed construction. Finally, we propose an algebraic construction for optimal generators as another main result. A lot of optimal families of size $p-1$ can be constructed from these optimal generators, some of which are known to be from the Fermat-quotient sequences or from the Frank-Zadoff sequences, but some families are new for $p\geq 11$ . The relation between the Fermat-quotient sequences and the Frank-Zadoff sequences is determined as a by-product.

Estimates of trigonometric sums over subgroups and some of their applications

In this paper, we obtain new upper bounds for trigonometric sums over subgroups Γ ⊂ Z*p whose size belongs to [p28/95, p182/487]. Using an approach due to Malykhin, we refine estimates of such sums in \(\mathbb{Z}_{{p^r}}^*\) and apply them to the divisibility problem for Fermat quotients.

Arithmetic differential equations on $$GL_n$$ G L n , III Galois groups

Differential equations have arithmetic analogues (Buium in Arithmetic differential equations, Mathematical Surveys and Monographs, vol 118. American Mathematical Society, Providence 2005) in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations, and the present paper is concerned with the “linear” ones. The equations themselves were introduced in a previous paper (Buium and Dupuy, in Arithmetic differential equations on $$GL_{n}$$ , II: arithmetic Lie–Cartan theory, arXiv:1308.0744 ). In the present paper we deal with the solutions of these equations as well as with the Galois groups attached to the solutions.