We obtain a lower bound for the normalised height of a non-torsion hypersurface $V$ of a C.M. abelian variety $A$ which is a refinement of a precedent result. This lower bound is optimal in terms of the geometric degree of $V$, up to an absolute power of a ``log'' (independant of the dimension of $A$). We thus extend the results of F. Amoroso and S. David on the same problem on a multiplicative group $\mathbb{G}_m^n$. When $A$ is an elliptic curve and $V=\bar{P}$ is the set of conjugates of a non torsion $\bar{k}$-point, we reobtain the result of M. Laurent on the elliptic Lehmer's problem.
Lehmer's totient problem asks if there exists a composite number $d$ such that its totient divide $d-1$. In this article we generalize the Lehmer's totient problem in algebraic number fields. We introduce the notion of a Lehmer number. Lehmer numbers are defined to be the natural numbers which obey the Lehmer's problem in the ring of algebraic integers of a number field.
We consider a generalisation of the classical Lehmer problem about the distribution of modular inverses in arithmetic progression, introduced by E. Alkan, F. Stan and A. Zaharescu. Using bounds of sums of multiplicative characters instead of traditionally applied to this kind of problem Kloosterman sums, we improve their results in several directions.
We formulate Lehmer's Problem about the Mahler measure of polynomials for general compact abelian groups, introducing a Lehmer constant for each such group. We show that all nontrivial connected compact groups have the same Lehmer constant, and conjecture the value of the Lehmer constant for finite cyclic groups. We also show that if a group has infinitely many connected components then its Lehmer constant vanishes.
We find the minimal non-trivial integer variable group determinant for any dihedral group of order less than $3.79\times 10^{47}$. We think of this as the Lind-Lehmer problem for the dihedral group. We give a complete description of the determinants for some dihedral groups including $D_{2p}$ and $D_{4p}$.
We tackle the "relative" Lehmer problem on algebraic subvarieties of a multiplicative torus. Generalizing a theorem of F. Amoroso and U. Zannier, we give a lower bound for the normalized height of a non torsion hypersurface in terms of its obstruction index over $\Q^{ab}$, the maximal abelian extension of $\Q$. We prove up to $\eps$ the sharpest conjecture that can be formulated.
We consider a generalisation of the classical Lehmer problem about the parity distribution of an integer and its modular inverse. We use some known estimates of exponential sums to study a more general question of simultaneous distribution of the residues of any fixed number of negative and positive powers of integers in prescribed arithmetic progressions. In particular, we improve and generalise a recent result of Y. Yi and W. Zhang.
The Euler's totient function $ φ(n) $ counts the positive integers up to a given integer $ n$ that are relatively prime to $ n $. We solve a problem due to Lehmer that there is no composite number $ n $ such that $ φ(n)\mid n-1 $. We also note that there are infinitely many primes that are not Sophie Germain primes.
We draw connections between the various conjectures which are included in G. Rémond's generalized Lehmer problems. Specifically, we show that the degree one form of his conjecture for the multiplicative group is, in a sense, almost as strong as the strong form. Our insights into the conjectures apply a basic result on what may be called Diophantine approximation in the metric induced by the canonical height. We also present a previously unpublished partial result of Rémond on his conjectures.
We introduce vector space norms associated to the Mahler measure by using the L^p norm versions of the Weil height recently introduced by Allcock and Vaaler. In order to do this, we determine orthogonal decompositions of the space of algebraic numbers modulo torsion by Galois field and degree. We formulate L^p Lehmer conjectures involving lower bounds on these norms and prove that these new conjectures are equivalent to their classical counterparts, specifically, the classical Lehmer conjecture in the p = 1 case and the Schinzel-Zassenhaus conjecture in the p = infinity case.
A composite number $n$ is called a Lehmer number when $ϕ(n) | n - 1$, where $ϕ$ is the Euler totient function. Lehmer's totient problem asks if there exist any composite numbers $n$ such that $ϕ(n)| n-1$? No such numbers are known. In this paper we establish an Euler Totient Inequality and relate some new parameters to Lehmer numbers. As an application of what we have done, we show that for all prime numbers $p$ and for all odd square-free numbers $n$, at most one of $n$ or $pn$ is a Lehmer number. Finally we suggest some open problems for the future investigations on the Lehmer numbers.
Lehmer's totient problem consists of determining the set of positive integers $n$ such that $φ(n)|n-1$ where $φ$ is Euler's totient function. In this paper we introduce the concept of $k$-Lehmer number. A $k$-Lehmer number is a composite number such that $φ(n)|(n-1)^k$. The relation between $k$-Lehmer numbers and Carmichael numbers leads to a new characterization of Carmichael numbers and to some conjectures related to the distribution of Carmichael numbers which are also $k$-Lehmer numbers.
We solve Lehmer's problem for a class of polynomials arising from Hermitian matrices over the Eisenstein and Gaussian integers, that is, we show that all such polynomials have Mahler measure at least Lehmer's number τ_0 = 1.17628... .
Let $A/K$ be an abelian variety over a number field $K$. We prove in this article that a good lower bound (in terms of the degree $[K(P):K]$) for the Néron-Tate height of the points $P$ of infinite order modulo every strict abelian subvarieties of $A$ implies a good lower bound for the height of all the non-torsion points of $A$. In particular when $A$ is of C.M. type, a theorem of David and Hindry enables us to deduce, up to ``log'' factors, an optimal lower bound for the height of the non-torsion points of $A$. In the C.M. type case, this improves the previous result of Masser \cite{lettre}.…
In this paper, we obtain analogues of Zsigmondy's theorem and the primitive divisor results for the Lucas and Lehmer sequences in polynomial rings of several variables.
We consider the problem whether for a group G there exists a constant Lambda(G) > 1 such that for any (r,s)-matrix A over the integral group ring ZG the Fuglede-Kadison determinant of the G-equivariant bounded operator from L^2(G)^r to L^2(G)^s given by right multiplication with A is either one or greater or equal to Lambda(G). If G is the infinite cyclic group and we consider only r = s = 1, this is precisely Lehmer's problem.
In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in\mathbb{F}_q[x]$ and $φ(q,p(x))$ be the Euler's totient function of $p(x)$ over $\mathbb{F}_q[x],$ where $\mathbb{F}_q$ is a finite field with $q$ elements. We prove that $φ(q,p(x))|(q^{{\rm deg}(p(x))}-1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3, \; p(x)$ is the product of any $2$ non-associate irreducibes of degree $1;$ or (iii) $q=2,\; p(x)$ is the product of all irreducibles of degree $1,$ all irreducibles of degree $1$ and $2,$ and the product of any $3$ irreducibles one each…
We propose a lower bound estimate in Dobrowolski's style of the canonical height on a certain family of Drinfeld modules of characteristic 0, including under some hypothesis on their degree and their base field, the complex multiplication case, extending so a previous result of L. Denis on Carlitz modules. Our study is focused on the highest possible level of precision on the parameters involved with rapport to the main values which characterize the Drinfeld module (height, base field degree and rank) and it provides an estimate in function of both separable and inseparable degree of the algeb…
Lehmer's totient problem asks whether there exists any composite number $n$ such that $φ(n) \, \mid \, (n-1)$, where $φ$ is Euler totient function. It is known that if any such $n$ exists, it must be Carmichael and $n > 10^{30}$. In this paper, we develop a new approach to the problem via some recent results in group theory related to a function $ψ$ (the sum of order of elements of a group) and show that if $k φ(n) = n-1$ for some integer $k$, then $k$ must be $\geq 3$, and actually, if $5, 7, 11, 13 \not | n$, $k \geq 4$. This implies that any counterexample must be such that $n > 10^{8171}$ …
We prove a special case of the following conjecture of Zilber-Pink generalising the Manin-Mumford conjecture : let $X$ be a curve inside an Abelian variety $A$ over $\bar{\Q}$, provided $X$ is not contained in a torsion subvariety, the intersection of $X$ with the union of all subgroup schemes of codimension at least 2 is finite ; we settle the case where $A$ is a power of a simple Abelian variety of C.M. type. This generalises the previous known result, due to Viada and Rémond-Viada (who was able to prove the conjecture for power of an elliptic curve with complex multiplication). The proof is…
We obtain a lower bound for the normalised height of a non-torsion subvariety $V$ of a C.M. abelian variety. This lower bound is optimal in terms of the geometric degree of $V$, up to a power of a ``log''. We thus extend the results of F. Amoroso and S. David on the same problem on a multiplicative group $\mathbb{G}_m^n$. We prove furthermore that the optimal lower bound (conjectured by S. David and P. Philippon) is a corollary of the conjecture of S. David and M. Hindry on the abelian Lehmer's problem. We deduce these results from a density theorem on the non-torsion points of $V$.
We are concerned here with Lehmer's problem in dimension 2 ; we give a lower bound for the height of a non-torsion point of $G\_m^2$ on a non-torsion curve defined over $Q$, depending on the degree of the curve only. We have first been inspired by \cite{Am-Da3}; we develop a new approach, inherent in the dimension two (or more precisely the codimension two), and then obtain a better result where the error's term is improved significantly, moreover we give an explicit expression for the constant.
We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the $j$-invariant of the elliptic curve is constant. In more detail, given an elliptic curve $E$ with a point $P$ of infinite order, the sequence $D_1$, $D_2, \ldots$ of denominators of multiples $P$, $2P,\ldots$ of $P$ is a strong divisibility sequence in the sense that $\gcd(D_m, D_n) = D_{\gcd(m,n)}$. This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences. A number $N$ is called a Zsigmondy bound of the sequence if each term $D_{n}$ with $n>N$ presents a new p…
