Konstantinos Smpokos
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.
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A new characterization of prime numbers and solution to Lehmer's conjecture on Euler's totient function
利用初等对称多项式与数论结果解决 Lehmer 猜想，并由此得到素数的新刻画。
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.
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…
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.
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 $n > 10^{8171}$ …
A composite positive integer n is Lehmer if ϕ(n) divides n-1, where ϕ(n) is the Euler's totient function. No Lehmer number is known, nor has it been proved that they don't exist. In 2007, the second author [7] proved that there is no Lehmer number in the Fibonacci sequence. In this paper, we adapt the method from [7] to show that there is no Lehmer number in the companion Lucas sequence of the Fibonacci sequence $(L_n)_{n\geq 0}$ given by $L_0 = 2, L_1 = 1$ and $L_{n+2} = L_{n+1} + L_n$ for all $n\geq 0.$
A composite number $n$ is called Lehmer when $ϕ(n) | n - 1$, where $ϕ$ is the Euler totient function. In 1932, D.~H.~Lehmer conjectured that there are no composite Lehmer numbers and showed that Lehmer numbers must be odd and square-free. Although a number of additional constraints have been found since, the problem remains still open. For each odd number $m>1$, let $m^\star$ be the largest number such that $2^{m^\star}$ divides $m-1$. Using this notion we present some new necessary conditions and introduce a method to construct some new family of numbers $n$ which are not Lehmer number.
A composite positive integer $n$ has the Lehmer property if $ϕ(n)$ divides $n-1,$ where $ϕ$ is an Euler totient function. In this note we shall prove that if $n$ has the Lehmer property, then $n\leq 2^{2^{K}}-2^{2^{K-1}}$, where $K$ is the number of prime divisors of $n$. We apply this bound to repunit numbers and prove that there are at most finitely many numbers with the Lehmer property in the set $$ \left\{\frac{g^{n}-1}{g-1}\ \bigg|\ n,g\in\mathbb{N},\ ν_{2}(g)+ν_{2}(g+1)\leq L\ \right\}, $$ where $ν_{2}(g)$ denotes the highest power of $2$ that divides $g$, and $L\geq 1$ is a fixed real n…
Lehmer's totient problem asks if there exist composite integers n satisfying the condition phi(n)|(n-1), (where phi is the Euler-phi function) while Carmichael numbers satisfy the weaker condition lambda(n)|(n-1) (where lambda is the Carmichael universal exponent function). We weaken the condition further, looking at those composite n where each prime divisor of phi(n) also divides n-1. (So rad(phi(n))|(n-1).) While these numbers appear to be far more numerous than the Carmichael numbers, we show that their distribution has the same rough upper bound as that of the Carmichael numbers, a bound …
On Lehmer's problem and related problems
本文研究了Lehmer问题及相关问题，证明了合数N满足特定等式时M的上界，并对unitary totient function、Dedekind function及unitary divisors之和给出了类似结论。
In this paper, we prove that there is no number with the Lehmer property in the sequence of Pell numbers.
A Lehmer number modulo a prime $p$ is an integer $a$ with $1 \leq a \leq p-1$ whose inverse $\bar{a}$ within the same range has opposite parity. Lehmer numbers that are also primitive roots have been discussed by Wang and Wang in an endeavour to count the number of ways $1$ can be expressed as the sum of two primitive roots that are also Lehmer numbers (an extension of a question of S. Golomb). In this paper we give an explicit estimate for the number of Lehmer primitive roots modulo $p$ and prove that, for all primes $p \neq 2,3,7$, Lehmer primitive roots exist. We also make explicit the know…
This thesis is about the study of Diophantine equations involving binary recurrent sequences with arithmetic functions. Various Diophantine problems are investigated and new results are found out of this study. Firstly, we study several questions concerning the intersection between two classes of non-degenerate binary recurrence sequences and provide, whenever possible, effective bounds on the largest member of this intersection. Our main study concerns Diophantine equations of the form $φ(|au_n |)=|bv_m|,$ where $φ$ is the Euler totient function, $\{u_n\}_{n\geq 0}$ and $\{v_m\}_{m\geq 0}$ ar…
In this paper we study the distribution of totients $2$ mod. $4$. We prove that the asymptotic magnitude of such totients with multiplicity two is half of that of prime numbers. As a corollary we obtain that the relative asymptotic density of the number of those totients with multiplicity four over the number of totients with multiplicity two is zero. We also obtain that the set of totients which have, a bigger than one, power of a prime in their pre-images and multiplicity $k$ has relative asymptotic density over the number of all totients of multiplicity $k$ equals to zero. A result on the d…
On existence of generalised Lehmer numbers modulo primes
本文推广Lehmer数为模奇素数$q$的$q$-元组，计数并给出带显式误差项的渐近估计，利用特征和与Deligne界。
In this note, we correct an oversight from the paper mentioned in the title.
We shall give some results for an integer divisible by its unitary totient.
Some Bounds on the Lehmer Index
本文研究莱默数的莱默指数与其素因子倒数和的关系，提出莱默指数的上界、下界，以及莱默数以3为素因子时的指数约束。
For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by \[S_r(p^α)=\begin{cases} 0, & \mbox{if } p\leq r; \\ p^{α-1}(p-r), & \mbox{if } p>r \end{cases}\] for all primes $p$ and positive integers $α$. The function $S_1$ is simply Euler's totient function $ϕ$. Masser and Shiu have established several fascinating results concerning sparsely totient numbers, positive integers $n$ satisfying $ϕ(n)<ϕ(m)$ for all integers $m>n$. We define a sparsely Schemmel totient number of order $r$ to be a positive integer $n$ such th…
Here, we show that there is no positive integer $n$ such that the $n$th Cullen number $C_n=n2^n+1$ has the property that it is composite but $ϕ(C_n)\mid C_n-1$.
The image of Euler's totient function is composed of the number 1 and even numbers. However, many even numbers are not in the image. We consider the problem of finding those even numbers which are in the image and those which are not. If an even number is in the image, then its preimage can have at most half its elements odd. However, it may contain only even numbers. We consider the structure of the preimage of certain numbers in the image of the totient function.
In the paper, we generalize some congruences of Lehmer for general composite numbers.
In this paper, we prove that there are infinitely many $n$ for which $rad(φ(n))|n-1$ but $n$ is not a Carmichael number. Additionally, we prove that for any $k\geq 3$, there exist infinitely many $n$ such that $φ(n)|(n-1)^k$ but $φ(n)\nmid (n-1)^{k-1}$. The constructs that we consider here are generalizations of Carmichael and Lehmer numbers, respectively, that were first formulated by Grau and Oller-Marcén.
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.
In this paper we study the multiplicities and the asymptotic behaviour of the numbers of totients in the strata given by 2-adic valuation.
In 2009, Luca and Nicolae proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are $1,2$, and $3$. In 2015, Faye and Luca proved that the only Pell numbers whose Euler totient function is another Pell number are $1$ and $2$. Here we add to these two results and prove that for any fixed natural number $P\geq 3$, if we define the sequence $\left(u_n\right)_n$ as $u_0=0$, $u_1=1$, and $u_n=Pu_{n-1}+u_{n-2}$ for all $n\geq 2$, then the only solution to the Diophantine equation $φ\left(u_n\right)=u_m$ is $φ\left(u_1\right)=φ(1)=1=u_1$.
Set-theoretical entropies of Euler's totient function and other number theoretical special functions
本文计算 Euler totient 函数与 Dedekind psi 函数的集合论及反变集合论熵，并考察其诱导的 Alexandroff 拓扑及相关数论特殊函数。
Let u(n) be the n-th term of a Lucas sequence or a Lehmer sequence.In this article we shall establish an estimate from below for the greatest prime factor of u(n) which is of the form nexp(logn/104loglogn). In so doing we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdos from 1965.In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.
In this article, we investigate sparse subsets of the natural numbers and study the sparseness of some sets associated with the Euler's totient function $ϕ$ via the property of `Banach Density'. These sets related to the totient function are defined as follows: $V:=ϕ(\mathbb{N})$ and $N_i:=\{N_i(m)\colon m\in V \}$ for $i = 1, 2, 3,$ where $N_1(m)=\max\{x\in \mathbb{N}\colon ϕ(x)\leq m\}$, $N_2(m)=\max(ϕ^{-1}(m))$ and $N_3(m)=\min(ϕ^{-1}(m))$ for $ m\in V$. Masser and Shiu call the elements of $N_1$ as `sparsely totient numbers' and construct an infinite family of these numbers. Here we constr…
Explicit formulas for Euler's totient function and the number of divisors
该研究利用 Menon 关系及与 Pillai 算术函数的联系，给出 φ(n) 与 τ(n) 基于取整有限和或 gcd 的显式公式，可为算术函数新界提供工具。
We construct fields of algebraic numbers that have the Lehmer property but not the Bogomolov property. This answers a recent implicit question of Pengo and the first author.
On the representation of rational numbers via Euler's totient function
本文证明任意正有理数可表示为欧拉函数相关的两种形式，并探讨了进一步结果。
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... .
Waring's problem involving D.H. Lehmer numbers
本文证明每个充分大的整数，除非模16余15或16，可表示为14个D.H. Lehmer数的四次方之和；且每个充分大的整数可表示为16个D.H. Lehmer数的四次方之和。
Nombres de Pisot, nombres de Salem et la conjecture de Lehmer
本文研究Pisot数与Salem数的关系，证明两者并集为实半直线的闭子集，从而解决Boyd猜想并否定Lehmer猜想。
In this article we construct a new family of quintic non-Pólya fields with large Pólya groups. We study the upper bound of Pólya numbers of such fields and show that the Pólya numbers never exceed five times the size of its Pólya group. Finally we show that such non-Pólya fields are non-monogenic fields of field index one.
We fix a gap in our proof of an upper bound for the number of positive integers $n\le x$ for which the Euler function $φ(n)$ has all prime factors at most $y$. While doing this we obtain a stronger, likely best-possible result.
Let $k\geq 3$ and $n\geq 3$ be odd integers, and let $m\geq 0$ be any integer. For a prime number $\ell$, we prove that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{\ell^{2m}-2k^n})$ is either divisible by $n$ or by a specific divisor of $n$. Applying this result, we construct an infinite family of certain tuples of imaginary quadratic fields of the form $$\left(\mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{d+1}), \mathbb{Q}(\sqrt{4d+1}), \mathbb{Q}(\sqrt{2d+4}), \mathbb{Q}(\sqrt{2d+16}), \cdots, \mathbb{Q}(\sqrt{2d+4^t}) \right)$$ with $d\in \mathbb{Z}$ and $1\leq 4^t\leq 2|d|…
A number of authors have proven explicit versions of Lehmer's conjecture for polynomials whose coefficients are all congruent to 1 modulo m. We prove a similar result for polynomials f(X) that are divisible in (Z/mZ)[X] by a polynomial of the form 1+X+...+X^n for some n > ε*deg(f). We also formulate and prove an analogous statement for elliptic curves.
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.
Let $q>2$ be an odd integer. For each integer $x$ with $0<x<q$ and $(q,x)= 1$, we know that there exists one and only one $\bar{x}$ with $0<\bar{x}<q$ such that $x\bar{x}\equiv1(\bmod q)$. A Lehmer number is defined to be any integer $a$ with $2\dagger(a+\bar{a})$. For any nonnegative integer $k$, Let $$ M(x,q,k)=\displaystyle\mathop {\displaystyle\mathop{\sum{'}}_{a=1}^{q} \displaystyle\mathop{\sum{'}}_{b\leq xq}}_{\mbox{$\tiny\begin{array}{c} 2|a+b+1\\ ab\equiv1(\bmod q)\end{array}$}}(a-b)^{2k}.$$ The main purpose of this paper is to study the properties of $M(x,q,k)$, and give a sharp asymp…
Jacobsthal's function h(k) represents the smallest number m such that every sequence of m consecutive integers contains an integer coprime to P_k, the product of the first k primes. The best known bound on h(k) is h(k) < C (k ln k)^2 for some unknown constant C, due to Iwaniec. We use a generalisation of Euler's totient function to give a stronger bound on h(k).
We begin by introducing an interesting class of functions, known as the Schemmel totient functions, that generalizes the Euler totient function. For each Schemmel totient function $L_m$, we define two new functions, denoted $R_m$ and $H_m$, that arise from iterating $L_m$. Roughly speaking, $R_m$ counts the number of iterations of $L_m$ needed to reach either $0$ or $1$, and $H_m$ takes the value (either $0$ or $1$) that the iteration trajectory eventually reaches. Our first major result is a proof that, for any positive integer $m$, the function $H_m$ is completely multiplicative. We then int…
Normal integral bases of Lehmer's cyclic quintic fields
本研究针对Emma Lehmer参数多项式生成的温和分歧循环五次域$K_n$，仅用多项式根给出其所有正规整基，推广了Lehmer和Spearman-Williams在特定参数条件下的相关研究结果。
A problem of D. H. Lehmer in short intervals. I
研究与模q乘法逆元奇偶性不同的整数个数问题，利用算术指数对和素变量不完全克洛斯特曼和估计，在特定条件下得到短区间内此类整数个数渐近公式并证明素变量版本，改进了前人结果。
Let $q$ be a large prime number, $a$ be any integer, $ε$ be a fixed small positive quantity. Friedlander and Shparlinksi \cite{FSh} have shown that there exists a positive integer $n\ll q^{5/2+ε}$ such that $ϕ(n)$ falls into the residue class $a \pmod q.$ Here, $ϕ(n)$ denotes Euler's function. In the present paper we improve this bound to $n\ll q^{2+ε}.$
Let $N_1(m)=\max\{n \colon ϕ(n) \leq m\}$ and $N_1 = \{N_1(m) \colon m \in ϕ(\mathbb{N})\}$ where $ϕ(n)$ denotes the Euler's totient function. Masser and Shiu \cite{masser} call the elements of $N_1$ as `sparsely totient numbers' and initiated the study of these numbers. In this article, we establish several results for sparsely totient numbers. First, we show that a squarefree integer divides all sufficiently large sparsely totient numbers and a non-squarefree integer divides infinitely many sparsely totient numbers. Next, we construct explicit infinite families of sparsely totient numbers an…
Lehmer-Type bounds and counting rational points of bounded heights on Abelian varieties
本文研究了数域K上阿贝尔簇A的$\bar{K}$-点Néron-Tate高度的Lehmer型界，估计了高度$\leq \log B$的K-有理点数量，并针对椭圆曲线及其乘积将估计中的常数C变为显式可计算。
A problem of D. H. Lehmer in short intervals. II
研究与模\(q\)乘法逆元奇偶性不同的整数个数问题，通过对含克洛斯特曼分数的双线性形式估计，得到小于\(q^{1/2}\)的此类整数个数渐近公式及误差项二阶矩估计。
A note on square totients
本文研究了素数 $p$ 和 $q$ 的对数，其中 $(p - 1)(q - 1)$ 为完全平方数，并给出了相关数量的上界和下界。
In this paper we introduce and study a family $Φ_k$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equation $\gcd(x_1^2+\ldots +x_k^2, n)=1$ with $x_1,\ldots,x_k \in {\mathbb{Z}}/n{\mathbb{Z}}$ which, for $k=2,4$ and $8$ coincide, respectively, with the number of units in the rings of Gaussian integers, quaternions and octonions over ${\mathbb{Z}}/n{\mathbb{Z}}$. We prove that $Φ_k$ is multiplicative for every $k$, we obtain an explicit formula for $Φ_k(n)$ in terms of the prime-power decomposition of $n$ and derive an…
We characterize which residue classes contain infinitely many totients (values of Euler's function) and which do not. We show that the union of all residue classes that are totient-free has asymptotic density 3/4, that is, almost all numbers that are 2 mod 4 are in a residue class that is totient-free. In the other direction, we show the existence of a positive density of odd numbers m, such that for any $s\ge0$ and any even number $a$, the residue class $a\pmod{2^sm}$ contains infinitely many totients.
The conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{house(α)}(z)$ associated with the dynamical zeta functions $ζ_{house(α)}(z)$ of the Rényi--Parry arithmetical dynamical systems, for $α$ an algebraic integer $α$ of house "$house(α)$" greater than 1, (ii) the discovery of lenticuli of poles of $ζ_{house(α)}(z)$ which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when $house(α)$ tends to $1^+$, giving rise to a continuous lenticular minorant ${\rm M}_{r}(house(α))$ of the Mahler…
Congruence properties of Lehmer-Euler numbers
本文研究Lehmer在1935年利用三次单位根定义的广义欧拉数，给出其同余性质、递推公式和显式公式，并构造新的多项式序列，得到与欧拉数及中心阶乘数相关的恒等式。
A p-adic criterion for Lehmer's conjecture
本文建立小高度代数数共轭元分布的 p 进类似物，用位于 Q_p 有限扩张中的共轭元数量给出 Weil 高度下界，并据此证明一类代数数满足 Lehmer 猜想。
Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called "Beck-type" companions to other identities. In this paper, we establish…
Two inequalities involving the Euler totient function and the sum of the $k$-th powers of the divisors of balancing numbers are explored.
For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by \[S_r(p^α)=\begin{cases} 0, & \mbox{if } p\leq r; \\ p^{α-1}(p-r), & \mbox{if } p>r \end{cases}\] for all primes $p$ and positive integers $α$. The function $S_1$ is simply Euler's totient function $ϕ$. We define a Schemmel nontotient number of order $r$ to be a positive integer that is not in the range of the function $S_r$. In this paper, we modify several proofs due to Zhang in order to illustrate how many of the results currently known about nontotient num…
Let $V$ be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on $V$ is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime. Asymptotic results for the number of Lehmer points and visible points on $V$ are obtained, and the distribution of visible points into different congruence classes is investigated.
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