In this paper, we show that the equation $φ(|x^{m}-y^{m}|)=|x^{n}-y^{n}|$ has no nontrivial solutions in integers $x,y,m,n$ with $xy\neq0, m>0, n>0$ except for the solutions $(x,y,m,n)=((2^{t-1}\pm1),-(2^{t-1}\mp1),2,1), (-(2^{t-1}\pm1),(2^{t-1}\mp1),2,1),$ where $t$ is a integer with $t\geq 2.$ The equation $φ(|\frac{x^{m}-y^{m}}{x-y}|)=|\frac{x^{n}-y^{n}}{x-y}|$ has no nontrivial solutions in integers $x,y,m,n$ with $xy\neq0, m>0, n>0$ except for the solutions $(x,y,m,n)=(a\pm1, -a, 1, 2), (a\pm i, -a, 2, 1),$ where $a$ is a integer with $i=1,2.$
We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.
In this paper we determine possible decompositions of Euler polynomials $E_k(x)$, i.e. possible ways of writing Euler polynomials as a functional composition of polynomials of lower degree. Using this result together with the well-known criterion of Bilu and Tichy, we prove that the Diophantine equation $$-1^k +2 ^k - \cdots + (-1)^{x} x^k=g(y),$$ with $g\in \mathbb{Q}[X]$ of degree at least $2$ and $k\geq 7$, has only finitely many integers solutions $x, y$ unless polynomial $g$ can be decomposed in ways that we list explicitly.
Quartic Rational Diophantine Quadruples and the Euler Surface
University of Zagreb, Faculty of Civil Engineering（萨格勒布大学土木工程学院）；University of Zagreb, Faculty of Science（萨格勒布大学理学院）；University of Zagreb Faculty of Electrical Engineering and Computing（萨格勒布大学电气工程和计算学院）
In this paper, we consider the equations involving Euler's totient function $ϕ$ and Lucas type sequences. In particular, we prove that the equation $ϕ(x^m-y^m)=x^n-y^n$ has no solutions in positive integers $x, y, m, n$ except for the trivial solutions $(x, y, m , n)=(a+1, a, 1, 1)$, where $a$ is a positive integer, and the equation $ϕ((x^m-y^m)/(x-y))=(x^n-y^n)/(x-y)$ has no solutions in positive integers $x, y, m, n$ except for the trivial solutions $(x, y, m , n)=(a, b, 1, 1)$, where $a, b$ are integers with $a>b\ge 1$.
In this paper we obtain a parametric solution of the hitherto unsolved diophantine equation $(x_1^5+x_2^5)(x_3^5+x_4^5)=(y_1^5+y_2^5)(y_3^5+y_4^5)$. Further, we show, using elliptic curves, that there exist infinitely many parametric solutions of the aforementioned diophantine equation, and they can be effectively computed.
By the theory of elliptic curves, we investigate the nontrivial rational parametric solutions of the Diophantine equation $f(x)f(y)=f(z)^n$, where $n=1,2$ and $f(X)$ are some simple Laurent polynomials.
By the theory of elliptic curves, we study the nontrivial rational parametric solutions and rational solutions of the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ for some simple Laurent polynomials $f$.
A perfect Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. Such cuboids are not yet discovered and their non-existence is also not proved. Perfect Euler cuboids are described by a system of four Diophantine equation possessing a natural $S_3$ symmetry. Recently these equations were factorized with respect to this $S_3$ symmetry and the factor equations were derived. In the present paper the factor equations are transformed to $E$-form and then reduced to a single biquadratic equation.
Many authors studied the problem that rational triangle pairs (triangle-parallelogram pairs) with the same area and the same perimeter. They investigated this problem by solving the rational solutions of the corresponding Diophantine equations. In this paper, we give a unified description of this problem by using the affine transformation in a rectangular coordinate system. According to the fact that two triangles with the same area and the same perimeter determine an affine transformation, this problem can be reduced to solving a specific Diophantine system. Moreover, we will give some ration…
In this article the Diophantine equations of the form $x^{2}-8C_{n}xy+ 16y^{2}=\pm2^{r}$, $x^{4}-8C_{n}x^{2}y+ 16y^{2}=\pm2^{r}$ and $x^{2}-8C_{n}xy^{2}+ 16y^{4}=\pm2^{r}$ are taken into consideration for the investigations of existence of any positive integer solutions. Furthermore, the solutions are manifested in terms of the balancing, Pell and Lucas-Pell numbers.
This paper is concerned with the diophantine equation $\sum_{i=1}^na_ix_i^4= \sum_{i=1}^na_iy_i^4$ where $n \geq 3$ and $a_i,\,i=1,\,2,\,\ldots,\,n$, are arbitrary integers. While a method of obtaining numerical solutions of such an equation has recently been given, it seems that an explicit parametric of this diophantine equation has not yet been published. We obtain a multi-parameter solution of this equation for arbitrary values of $a_i$ and for any positive integer $n \geq 3$, and deduce specific solutions when $n=3$ and $n=4$. The numerical solutions thus obtained are much smaller than th…
In this paper we consider Diophantine equation x4 + y4 = z4 + w4 (1)We construct some family of cubic curves.We prove that every rational point on Quar- tica x4 + y4 = z4 + w4 can be mapped to a point on some curve of this family. We also prove the opposite: each rational point belonging to our family of curves can be mapped to a rational point on the Quartica. (2) We find the point on our family of curves corresponding to a parametric solution of Leonard Euler. We construct several new parametric solutions of our Quartica, using a parametric solution of Leonard Euler and the algebraic operati…
Let $\bar{X}_{n}=(x_{1},\ldots,x_{n})$ and $σ_{i}(\bar{X}_{n})=\sum x_{k_{1}}\ldots x_{k_{i}}$ be $i$-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers $a, b, c$ such that for each $1\leq i\leq n$ the system of Diophantine equations \begin{equation*} σ_{i}(\bar{X}_{2n})=a, \quad σ_{2n-i}(\bar{X}_{2n})=b, \quad σ_{2n}(\bar{X}_{2n})=c \end{equation*} has infinitely many rational solutions. This result extend the recent results of Zhang and Cai, and the author. Moreover, we also consider some Diophantine systems involving sums of powers. …
Let $r\ge 1$ be an integer and ${\bf U}:=\{U_n\}_{n\ge 0}$ be the Lucas sequence given by $U_0=0,~U_1=1$, and $U_{n+2}=rU_{n+1}+U_n$ for $n\ge 0$. In this paper, we explain how to find all the solutions of the Diophantine equation, $AU_{n}+BU_{m}=CU_{n_1}+DU_{m_1}$, in integers $r\ge 1$, $0\le m<n,~0\le m_1<n_1$, $AU_n\ne CU_{n_1}$, where $A,B,C,D$ are given integers with $A\ne 0,~B\ne 0$, $m,n,m_1,n_1$ are nonnegative integer unknowns and $r$ is also unknown.
On the Diophantine Equation Involving Elementary Symmetric Polynomials and the Decomposition of Unity
Budapest University of Technology and Economics（布达佩斯技术与经济大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究与教育网络阿尔弗雷德·雷尼数学研究所）；Cracow University of Technology（克拉科夫理工大学）
We prove that there are infinitely many solutions of $$ |λ_0+λ_1p+λ_2P_r|<p^{-τ}, $$ where $r=3,$ $τ=\frac1{118}$, and $λ_0$ is an arbitrary real number and $λ_1,λ_2\in\BR$ with $λ_2\neq0$ and $0>\frac{λ_1}{λ_2}$ not in $\mathbb{Q}$. This improves a result by Harman. Moreover, we show that one can require the prime $p$ to be of the form $\floor{n^c}$ for some positive integer $n$, i.e. $p$ is a Piatetski-Shapiro prime, with $r=13$ and $τ=ρ(c),$ a constant explicitly determined by $c$ supported in $\left(1, 1+\frac1{149}\right].$
The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The third cuboid conjecture is the last of the three propositions suggested as intermediate stages in proving the non-existence of perfect Euler cuboids. It is associated with a certain Diophantine equation of the order 12. In this paper a structural theorem for the solutions of this Diophantine equation is proved.
We consider a variety of Euler's conjecture, i.e., whether the Diophantine system \[\begin{cases} n=a_{1}+a_{2}+\cdots+a_{s-1}, a_{1}a_{2}\cdots a_{s-1}(a_{1}+a_{2}+\cdots+a_{s-1})=b^{s} \end{cases}\] has solutions $n,b,a_i\in\mathbb{Z}^+,i=1,2,\ldots,s-1,s\geq 3.$ By using the theory of elliptic curves, we prove that it has no solutions $n,b,a_i\in\mathbb{Z}^+$ for $s=3$, but for $s=4$ it has infinitely many solutions $n,b,a_i\in\mathbb{Z}^+$ and for $s\geq 5$ there are infinitely many polynomial solutions $n,b,a_i\in\mathbb{Z}[t_1,t_2,\ldots,t_{s-3}]$ with positive value satisfying this Diop…
The above equation is also called as Euler(6,2,5) system. By computational aspect these systems are very interesting. And we can also apply these methods to other Diophantine equations. We give a brief history of these systems and how we searched for these big solutions on Boinc. Our two Boinc projects ran from April of 2010 to July of 2011.
The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The second cuboid conjecture is one of the three propositions suggested as intermediate stages in proving the non-existence of perfect Euler cuboids. It is associated with a certain polynomial Diophantine equation of the order 10. In this paper a structural theorem for the solutions of this Diophantine equation is proved and some examples of its application are considered.
In this paper we extend a result of Hirata-Kohno, Laishram, Shorey and Tijdeman on the Diophantine equation $n(n+d)...(n+(k-1)d)=by^2,$ where $n,d,k\geq 2$ and $y$ are positive integers such that $\gcd(n,d)=1.$
Let $σ_{i}(x_{1},\ldots, x_{n})=\sum_{1\leq k_{1}<k_{2}<\ldots <k_{i}\leq n}x_{k_{1}}\ldots x_{k_{i}}$ be the $i$-th elementary symmetric polynomial. In this note we generalize and extend the results obtained in a recent work of Zhang and Cai \cite{ZC,ZC2}. More precisely, we prove that for each $n\geq 4$ and rational numbers $a, b$ with $ab\neq 0$, the system of diophantine equations \begin{equation*} σ_{1}(x_{1},\ldots, x_{n})=a, \quad σ_{n}(x_{1},\ldots, x_{n})=b, \end{equation*} has infinitely many solutions depending on $n-3$ free parameters. A similar result is proved for the system \beg…
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…
A Pair of Diophantine Equations Involving the Fibonacci Numbers
Carnegie Mellon University（卡内基梅隆大学）；Texas A&M University（德克萨斯农工大学）；Institut Teknologi Bandung（万隆理工学院）；University of California, Berkeley（加州大学伯克利分校）；University of Virginia（弗吉尼亚大学）；Williams College（威廉姆斯学院）；University of Waterloo（滑铁卢大学）
For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello investigated the parity of $b_{t}(n)$ when $t\leq 28$. They discovered new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving every prime $p$ with $p\equiv 13, 17, 19, 23 \pmod{24}$. In this paper, we investigate the parity of $b_{21}(n)$ involving the primes $p$ with $p\equiv 1, 5, 7, 11 \pmod{24}$. We prove new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving the odd pr…
In this paper we give some relation involving values of q-Bernoulli, q-Euler and Bernstein polynomials. From these relations, we obtain some interesting identities on the q-Bernoulli, q-Euler and Bernstein polynomials.
In this paper we consider non-linear differential equations which are closely related to the generating functions of Frobenius-Euler polynomials. From our non-linear differential equations, we derive some new identities between the sums of products of Frobenius-Euler polynomials and Frobenius-Euler polynomials of higher order.
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 $φ\left(u_1\right)=φ(1)=1=u_1$.
In this paper, we will present several new congruences involving binomial coefficients under integer moduli, which are the continuation of the previous two work by Cai \textit{et al.} (2002, 2007).
This study investigates the existence of tuples $(k, \ell, m)$ of integers such that all of $k$, $\ell$, $m$, $k+\ell$, $\ell+m$, $m+k$, $k+\ell+m$ belong to $S(α)$, where $S(α)$ is the set of all integers of the form $\lfloor αn^2 \rfloor$ for $n\geq α^{-1/2}$ and $\lfloor x\rfloor$ denotes the integer part of $x$. We show that $T(α)$, the set of all such tuples, is infinite for all $α\in (0,1)\cap \mathbb{Q}$ and for almost all $α\in (0,1)$ in the sense of the Lebesgue measure. Furthermore, we show that if there exists $α>0$ such that $T(α)$ is finite, then there is no perfect Euler brick. W…
In this note we present some results concerning the unirationality of the algebraic variety $\cal{S}_{f}$ given by the equation \begin{equation*} N_{K/k}(X_{1}+αX_{2}+α^2 X_{3})=f(t), \end{equation*} where $k$ is a number field, $K=k(α)$, $α$ is a root of an irreducible polynomial $h(x)=x^3+ax+b\in k[x]$ and $f\in k[t]$. We are mainly interested in the case of pure cubic extensions, i.e. $a=0$ and $b\in k\setminus k^{3}$. We prove that if $\op{deg}f=4$ and the variety $\cal{S}_{f}$ contains a $k$-rational point $(x_{0},y_{0},z_{0},t_{0})$ with $f(t_{0})\neq 0$, then $\cal{S}_{f}$ is $k$-unirat…
In this paper, we deal with q-Euler numbers and q-Bernoulli numbers. We derive some interesting relations for q-Euler numbers and polynomials by using their generating function and derivative operator. Also, we show between the q-Euler numbers and q-Bernoulli numbers via the p-adic q-integral in the p-adic integer ring.
This paper has two parts. The first part surveys Euler's work on the constant gamma=0.57721... bearing his name, together with some of his related work on the gamma function, values of the zeta function and divergent series. The second part describes various mathematical developments involving Euler's constant, as well as another constant, the Euler-Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations and random matrix products. It includes recent results on Diophantine approximation and tran…
We prove that $$ \sum_{n \leq x} φ([x/n])\leq\bigg(\frac{1380}{4009}+\frac{2629}{4009}\cdot\frac1{ζ(2)}+o(1)\bigg)x\log x $$ as $x\to\infty$, where $φ$ denotes the Euler totient function and $[x]$ denotes the integer part of $x$.
In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum_{n\le x} ϕ\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qquad \sum_{n\le x} \frac{ϕ([x/n])}{[x/n]}$$ where $[x]$ denotes the integral part of real $x$. The above summations were recently considered by Bordellès et al. and Wu.
This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree $ θ,$ where $ θ>2$ is real and non-integral. For fixed non-zero real numbers $ λ_i $ not all of the same sign we write \begin{equation*} \mathcal F (\textbf{x}) = λ_1 x_1^θ+ \cdots + λ_s x_s^θ. \end{equation*} For a fixed positive real number $ τ$ we give an asymptotic formula for the number of positive integer solutions of the inequality $ | \mathcal F (\textbf{x}) | < τ$ inside a box of side length $P.$ Moreover, we investigate the problem of representing a large positive real number by a positiv…
An Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals. An Euler cuboid is called perfect if its space diagonal is also integer. Some Euler cuboids are already discovered. As for perfect cuboids, none of them is currently known and their non-existence is not yet proved. Euler cuboids and perfect cuboids are described by certain systems of Diophantine equations. These equations possess an intrinsic $S_3$ symmetry. Recently they were factorized with respect to this $S_3$ symmetry and the factor equations were derived. In the present paper the factor equatio…
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 obtain reasonably tight upper and lower bounds on the sum $\sum_{n \leqslant x} φ\left( \left\lfloor{x/n}\right\rfloor\right)$, involving the Euler functions $φ$ and the integer parts $\left\lfloor{x/n}\right\rfloor$ of the reciprocals of integers.
In this paper, we will deal with some new formulae for two product Genocchi polynomials together with both Euler polynomials and Bernoulli polynomials. We get some applications for Genocchi polynomials. Our applications possess a number of interesting properties to study in Theory of Analytic numbers which we express in the present paper.
A perfect Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. The problem of finding such parallelepipeds or proving their non-existence is an old unsolved mathematical problem. The Diophantine equations of a perfect Euler cuboid have an explicit $S_3$ symmetry. In this paper the cuboid equations are factorized with respect to their $S_3$ symmetry in terms of multisymmetric polynomials. Some factor equations are calculated explicitly.
In this paper we examine the diophantine equation $x^k-y^k=x-y$ where $k$ is a positive integer $\geq 2$, and consider its applications. While the complete solution of the equation $x^k-y^k=x-y$ in positive rational numbers is already known when $k=2$ or $3$, till now only one numerical solution of the equation in positive rational numbers has been published when $k=4$, and no nontrivial solution is known when $k \geq 5$. We describe a method of generating infinitely many positive rational solutions of the equation when $k=4$. We use the positive rational solutions of the equation with $k=2,\,…
Let $b_3(n)$ be the number of $3$-regular partitions of $n$. Recently, W. J. Keith and F. Zanello discovered infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p$ with $p \equiv 13, 17, 19, 23 \pmod {24}$, and O. X. M. Yao provided new infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p\geqslant 5$. In this paper, we introduce new infinite Ramanujan type congruences modulo $2$ for $b_3(2n)$. They complement naturally the results of Keith-Zanello and Yao and involve primes in $\mathcal P=\{p \text{ prime …
