Numbers of the form k + f(k) Gabdullin, Mikhail R.; Iudelevich, Vitalii V.; Luca, Florian
Journal of number theory,
September 2024, 2024-09-00, Volume:
262
Journal Article
Peer reviewed
For a function f:N→N, letNf+(x)={n⩽x:n=k+f(k) for some k}. Let τ(n)=∑d|n1 be the divisor function, ω(n)=∑p|n1 be the prime divisor function, and φ(n)=#{1⩽k⩽n:gcd(k,n)=1} be Euler's totient function. ...We show that(1)x≪Nω+(x),(2)x≪Nτ+(x)⩽0.94x,(3)x≪Nφ+(x)⩽0.93x.
The objective of this paper is to answer the open problem proposed about the validity of phi-Euler’s theorem in the refined neutrosophic ring of integers 𝑍(𝐼1,𝐼2) . This work presents an algorithm ...to compute the values of Euler’s function on refined neutrosophic integers, and it prove that phi-Euler’s theorem is still true in 𝑍(𝐼1,𝐼2). On the other hand, we present a solution for another open question about the solutions of Fermat's Diophantine equation in refined neutrosophic ring of integers, where we determine the solutions of Fermat's Diophantine equation 𝑋 𝑛 + 𝑌 𝑛 = 𝑍 𝑛 ; 𝑛 ≥ 3 in 𝑍(𝐼1,𝐼2).
This paper is dedicated to study the properties of symbolic 5-plithogenic integers and number theory, where we present many number theoretical concepts such as symbolic 5-plithogenic Diophantine ...equations, symbolic 5-plithogenic congruencies, and symbolic 5-plithogenic Euler's function. Also, we present many examples to explain the validity and the scientific contribution of our work. Keywords: symbolic 5-plithogenic integer, symbolic 5-plithogenic Euler's function, symbolic 5-plithogenic Pythagoras triple
This paper is dedicated to study the properties of symbolic 5-plithogenic integers and number theory, where we present many number theoretical concepts such as symbolic 5-plithogenic Diophantine ...equations, symbolic 5-plithogenic congruencies, and symbolic 5-plithogenic Euler's function. Also, we present many examples to explain the validity and the scientific contribution of our work.
The homogeneous transform program is a function used to calculate the homogeneous transformation matrix at a specific position and orientation of a three-link manipulator. The homogeneous ...transformation matrix is a 4x4 matrix used to represent the position and orientation of an object in three-dimensional space. In the program, the rotation matrix R is calculated using the Euler formula and stored in a 4x4 matrix along with the position coordinates. The Jacobian matrix function calculates the Jacobian matrix at a specific position and orientation of a three-link manipulator using the homogeneous transformation matrix. The Euler formula used in the program is based on the rotation matrices for rotations around the x, y, and z-axes. The output of these functions can be useful for future research in developing advanced manipulators with improved accuracy and flexibility. Research gaps in exploring the limitations of these functions in real-world applications, particularly in scenarios involving complex manipulator configurations and environmental factors.
Fix an integer k≥2. We investigate the number of n≤x for which φ(n) is a perfect kth power. If we assume plausible conjectures on the distribution of smooth shifted primes, then the count of such n ...is at least x/L(x)1+o(1), as x→∞, where L(x)=exp(logx⋅logloglogx/loglogx). This lower bound is implicit in work of Banks–Friedlander–Pomerance–Shparlinski. We prove — unconditionally — that x/L(x)1+o(1) serves as an upper bound. In fact, we establish this same bound for the count of n≤x for which φ(n) is squarefull. The proof builds on methods recently introduced by the author to study “popular subsets” for Euler's function.
Abstract Menon’s identity states that for every positive integer n one has ∑ (a − 1, n) = φ (n)τ(n), where a runs through a reduced residue system (mod n), (a − 1, n) stands for the greatest common ...divisor of a − 1 and n, φ(n) is Euler’s totient function, and τ(n) is the number of divisors of n. Menon’s identity has been the subject of many research papers, also in the last years. We present detailed, self contained proofs of this identity by using different methods, and point out those that we could not identify in the literature. We survey the generalizations and analogs, and overview the results and proofs given by Menon in his original paper. Some historical remarks and an updated list of references are included as well.
In a recent work, Luca and Stănică examined quotients of the form φ(Cm)φ(Cn), where φ is Euler's totient function and C0,C1,C2… is the sequence of the Catalan numbers. They observed that the number 4 ...(and analogously 14) appears noticeably often as a value of these quotients. We give an explanation of this phenomenon, based on Dickson's conjecture. It turns out not only that the value 4 is (in a certain sense) special in relation to the quotients φ(Cn+1)φ(Cn), but also that the value 4k has similar “special” properties with respect to the quotients φ(Cn+k)φ(Cn), and in particular we show that Dickson's conjecture implies that, for each k, the number 4k appears infinitely often as a value of the quotients φ(Cn+k)φ(Cn).
Finite Q -groups have been recently studied and form a class of solvable groups, which satisfy interesting structural conditions. We survey some of their main properties and introduce the idea of Q ...-group for compact p-groups (p prime). A list of open questions is presented, along with several connections of arithmetic nature on a problem originally due to Frobenius.
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