Euler Totient Function Research Articles

A Menon-type identity concerning Dirichlet characters and a generalization of the gcd function

Menon’s identity is a classical identity involving gcd sums and the Euler totient function \(\phi \). In a recent paper, Zhao and Cao (Int. J. Number Theory 13(9) (2017) 2373–2379) derived the Menon-type identity \(\sum _{\begin{array}{c} k=1 \end{array}}^{n}(k-1,n)\chi (k) = \phi (n)\tau (\frac{n}{d})\), where \(\chi \) is a Dirichlet character mod n with conductor d. We derive an identity similar to this replacing gcd with a generalization it. We also show that some of the arguments used in the derivation of Zhao–Cao identity can be improved if one uses the method we employ here.

A New Methodology to Find Private Key of RSA Based on Euler Totient Function

The aim of this paper is to present a new methodology to find the private key of RSA. A new initial value which is generated from a new equation is selected to speed up the process. In fact, after this value is found, brute force attack is chosen to discover the private key. In addition, for a proposed equation, the multiplier of Euler totient function to find both of the public key and the private key is assigned as 1. Then, it implies that an equation that estimates a new initial value is suitable for the small multiplier. The experimental results show that if all prime factors of the modulus are assigned larger than 3 and the multiplier is 1, the distance between an initial value and the private key is decreased about 66%. On the other hand, the distance is decreased less than 1% when the multiplier is larger than 66. Therefore, to avoid attacking by using the proposed method, the multiplier which is larger than 66 should be chosen. Furthermore, it is shown that if the public key equals 3, the multiplier always equals 2.

English

Menon’s identity is a classical identity involving gcd sums and the Euler totient function φ. A natural generalization of φ is the Klee’s function Φs. We derive a Menon-type identity using Klee’s function and a generalization of the gcd function. This identity generalizes an identity given by Y. Li and D. Kim (2017).

Some Pythagorean type equations concerning arithmetic functions

We investigate some equations involving the number of divisors d(n); the sum of divisors σ(n); Euler's totient function ϕ(n); the number of distinct prime factors ω(n); and the number of all prime factors (counted with multiplicity) Ω(n). The first part deals with equation f(xy) + f(xz) = f(yz). In the second part, as an analogy to x2 + y2 = z2, we study equation f(x2) + f(y2) = f(z2) and its generalization to higher degrees and more terms. We use just elementary methods and basic facts about the above functions and indicate why and how to discuss this topic in group study sessions or special maths classes of secondary schools in the framework of inquiry based learning.
 Subject Classification: 97F60, 11A25

On a generalization of the Euler totient function

In a recent paper, J. Kaczorowski defined the generalized Euler totient function $\varphi(n, F)$ corresponding to a polynomial Euler product $F$. Let $E(x, F)$ denote the error term in the asymptotic formula of the summatory function of $\varphi(n, F).$ J.~Kaczorowski proved that the mean square of $E(x, F)$ has an asymptotic formula if $F$ satisfies GRH. In this paper we shall prove a sharper asymptotic formula under GRH.

Totient quotient and small gaps between primes

For each $$m\ge 1$$ , there exists $$H=H(m)>0$$ such that the set $$\begin{aligned} \bigg \{\frac{\phi (p+1)}{\phi (p-1)}:\,p\text { is prime and }[p+1,p+H]\text { contains at least }m\text { primes}\bigg \} \end{aligned}$$ is dense in $$[0,+\infty )$$ , where $$\phi $$ denotes the Euler totient function. This gives a unconditional weak form of a recent result of Garcia, Luca, Shi and Udell, which was proved under the assumption of a conjecture of Dickson.

On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function

Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk≡0(modn). In this paper, we shall find all positive solutions of the Diophantine equation z(φ(n))=n, where φ is the Euler totient function.

Analogues of Alladi's formula

In this note, we mainly show the analogue of one of Alladi's formulas over Q with respect to the Dirichlet convolutions involving the Möbius function μ(n), which is related to the natural densities of sets of primes by recent work of Dawsey, Sweeting and Woo, and Kural et al. This would give us several new analogues. In particular, we get that if (k,ℓ)=1, then−∑n⩾2p(n)≡ℓ(modk)μ(n)φ(n)=1φ(k), where p(n) is the smallest prime divisor of n, and φ(n) is Euler's totient function. This refines one of Hardy's formulas in 1921. At the end, we give some examples for the φ(n) replaced by functions “near n”, which include the sum-of-divisors function.

Repdigits in Euler functions of associated pell numbers

In this paper, we discuss some properties of Euler totient function of associated Pell numbers which are repdigits in base 10.

Permutations with Orders Coprime to a Given Integer

Let $m$ be a positive integer and let $\rho(m,n)$ be the proportion of permutations of the symmetric group $\mathrm{Sym}(n)$ whose order is coprime to $m$. In 2002, Pouyanne proved that $\rho(n,m)n^{1-\frac{\phi(m)}{m}}\sim \kappa_m$ where $\kappa_m$ is a complicated (unbounded) function of $m$. We show that there exists a positive constant $C(m)$ such that, for all $n \geq m$,\[C(m) \left(\frac{n}{m}\right)^{\frac{\phi(m)}{m}-1} \leq \rho(n,m) \leq \left(\frac{n}{m}\right)^{\frac{\phi(m)}{m}-1}\]where $\phi$ is Euler's totient function.

Shift-inequivalent decimations of the Sidelnikov–Lempel–Cohn–Eastman sequences

We consider the problem of finding maximal sets of shift-inequivalent decimations of Sidelnikov–Lempel–Cohn–Eastman (SLCE) sequences (as well as the equivalent problem of determining the multiplier groups of the almost difference sets associated with these sequences). This is an open problem that was originally posed in Cohn et al (IEEE Trans Inf Theory 23:38–42, 1977) and that was mentioned more recently as being open in Akiyama (Acta Arithmetica LXXV.2:97–104, 1996). We derive a numerical necessary condition for a residue to be a multiplier of an SLCE almost difference set. Using our necessary condition, we show that if p is an odd prime and S is an SLCE almost difference set in \({\mathbb {Z}}_p^*,\) then the multiplier group of S is trivial. Consequently, for each odd prime p, we obtain a family of \(\phi (p-1)\) shift-inequivalent balanced periodic sequences (where \(\phi \) is the Euler-Totient function) each having period \(p-1\) and nearly perfect autocorrelation.

Matrix Domain of a Regular Matrix Derived by Euler Totient Function in the Spaces $$c_0$$ and c

The main purpose of this manuscript is to introduce Banach spaces $$c_0^\Phi $$ and $$c^\Phi $$ as the matrix domain of a regular matrix $$\Phi $$ derived by the Euler totient function. These spaces consist of $$\varphi $$-convergent to zero and $$\varphi $$-convergent sequences, respectively. After determining $$\alpha $$-, $$\beta $$- and $$\gamma $$-duals of these spaces, some matrix classes are characterized. Finally, using the Hausdorff measure of noncompactness, the characterization of some classes of compact operators on the space $$c_0^\Phi $$ is given.

On a sum involving the Euler function

Let f be any arithmetic function and define Sf(x):=∑n⩽xf([x/n]). When f equals the Euler totient function φ, several authors studied the upper and lower bounds of Sφ(x). In this paper we shall prove that Sφ(x) has an asymptotic formula by the method of exponential sums. This result proves a conjecture proposed by Bordellés, Dai, Heyman, Pan and Shparlinski. Some other asymptotic formulas for arbitrary f are also given in this paper.

Iterating the Sum of Möbius Divisor Function and Euler Totient Function

In this paper, according to some numerical computational evidence, we investigate and prove certain identities and properties on the absolute Möbius divisor functions and Euler totient function when they are iterated. Subsequently, the relationship between the absolute Möbius divisor function with Fermat primes has been researched and some results have been obtained.

Diophantine equations involving the Euler totient function

We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.

The algebraic degree of spectra of circulant graphs

We investigate the algebraic degree of circulant graphs, i.e. the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals. Studying the algebraic degree of graphs seems more natural than characterizing graphs with integral spectra only. We prove that the algebraic degree of circulant graphs on n vertices is bounded above by φ(n)/2, where φ denotes Euler's totient function, and that the family of cycle graphs provides a family of maximum algebraic degree within the family of all circulant graphs. Moreover, we precisely determine the algebraic degree of circulant graphs on a prime number of vertices.

Bipartite graphs of small readability

We study a parameter of bipartite graphs called readability, introduced by Chikhi et al. (Discrete Applied Mathematics, 2016) and motivated by applications of overlap graphs in bioinformatics. The behavior of the parameter is poorly understood. The complexity of computing it is open and it is not known whether the decision version of the problem is in NP. The only known upper bound on the readability of a bipartite graph (following from a work of Braga and Meidanis, LATIN 2002) is exponential in the maximum degree of the graph.Graphs that arise in bioinformatics applications have low readability. In this paper, we focus on graph families with readability o(n), where n is the number of vertices. We show that the readability of n-vertex bipartite chain graphs is between Ω(log⁡n) and O(n). We give an efficiently testable characterization of bipartite graphs of readability at most 2 and completely determine the readability of grids, showing in particular that their readability never exceeds 3. As a consequence, we obtain a polynomial time algorithm to determine the readability of induced subgraphs of grids. One of the highlights of our techniques is the appearance of Euler's totient function in the analysis of the readability of bipartite chain graphs. We also develop a new technique for proving lower bounds on readability, which is applicable to dense graphs with a large number of distinct degrees.

A new upper bound for numbers with the Lehmer property and its application to repunit numbers

A composite positive integer [Formula: see text] has the Lehmer property if [Formula: see text] divides [Formula: see text] where [Formula: see text] is an Euler totient function. In this paper, we shall prove that if [Formula: see text] has the Lehmer property, then [Formula: see text], where [Formula: see text] is the number of prime divisors of [Formula: see text]. We apply this bound to repunit numbers and prove that there are at most finitely many numbers with the Lehmer property in the set [Formula: see text] where [Formula: see text] denotes the highest power of 2 that divides [Formula: see text], and [Formula: see text] is a fixed real number.

The classification of purely non-symplectic automorphisms of high order on K3 surfaces

An automorphism of order n of a K3 surface is called purely non-symplectic if it multiplies the holomorphic symplectic form by a primitive n-th root of unity. We give the classification of purely non-symplectic automorphisms with φ(n)≥12 where φ denotes the Euler totient function.

NOTE ON SUMS INVOLVING THE EULER FUNCTION

In this note, we provide refined estimates of two sums involving the Euler totient function, $$\begin{eqnarray}\mathop{\sum }_{n\leq x}\unicode[STIX]{x1D719}\biggl(\biggl[\frac{x}{n}\biggr]\biggr)\quad \text{and}\quad \mathop{\sum }_{n\leq x}\frac{\unicode[STIX]{x1D719}([x/n])}{[x/n]},\end{eqnarray}$$ where $[x]$ denotes the integral part of real $x$. The above summations were recently considered by Bordellès et al. [‘On a sum involving the Euler function’, Preprint, 2018, arXiv:1808.00188] and Wu [‘On a sum involving the Euler totient function’, Preprint, 2018, hal-01884018].