Related papers: Furthur developements regarding Euler equation xyz…

This note proves two theorems regarding Fermat-type equation $x^r + y^r = dz^p$ where $r \geq 5$ is a prime. Our main result shows that, for infinitely many integers~$d$, the previous equation has no non-trivial primitive solutions such…

We present results for infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-, two- and three-dimensional series. All these series can be expressed in terms of…

We give conditions on the rational numbers a,b,c which imply that there are infinitely many triples (x,y,z) of rational numbers such that x+y+z=a+b+c and xyz=abc. We do the same for the equations x+y+z=a+b+c and x^3+y^3+z^3=a^3+b^3+c^3.…

[This is an older version of the paper, which will be updated soon.] In the present paper, we continue our research on the generalized Fermat equation $x^r + y^s = z^t$ with signature $(r, s, t)$, where $r, s, t \ge 2$ are positive integers…

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.

In this paper, we give a specific way of describing positive integer solutions of a Diophantine equation $(x+y)^2+(y+z)^2+(z+x)^2=12xyz$ and introduce a generalized cluster pattern behind it.

In this paper we present a new family of identities for Euler sums and integrals of polylogarithms by using the methods of generating function and integral representations of series. Then we apply it to obtain the closed forms of all…

This paper deals with the evaluation of some definite Euler-type integrals in terms of the Wright hypergeometric function. We obtain a theorem on the Wright hypergeometric function and then use this theorem to evaluate some definite…

We discuss finding large integer solutions of $d=2x^3+y^3+z^3$ by using Elsenhans and Jahnel's adaptation of Elkies' LLL-reduction method. We find 28 first solutions for $|d|<10000$.

In this paper we present a new method of solving the classical diophantine equation $A^4+B^4=C^4+D^4$. Two methods of solving this equation, given by Euler, yield parametric solutions given by polynomials of degrees 7 and 13. Several other…

We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes,…

In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We…

We prove the conjectural relations between Mahler measures and $L$-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for $L$-values of CM elliptic curves of conductors 27 and 36. Furthermore,…

We establish rigorously the existence of a three-parameter family of self-similar,globally bounded, and continuous weak solutions in two space dimensions to the compressible Euler equations with axisymmetry for gamma-law polytropic gases…

We present an infinite dimensional family of of exact solutions of the incompressible three-dimensional Euler equations. These solutions, proposed by Gibbon and Ohkitani, have infinite kinetic energy and blow up in finite time.

In this paper, by using the method of Contour Integral Representations and the Theorem of Residues and integral representations of series, we discuss the analytic representa- tions of parametric Euler sums that involve harmonic numbers…

: We establish existence of an infinite family of exponentially-decaying non-radial $C^2$ solutions to the equation $\Delta u + f(u) = 0$ on $R^2$ for a large class of nonlinearities $f$. These solutions have the form $u(r,\theta )=e^{i…

We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still…

Let $k,\ell\geq2$ be fixed integers and $C$ be an effectively computable constant depending only on $k$ and $\ell$. In this paper, we prove that all solutions of the equation $(x+1)^{k}+(x+2)^{k}+...+(\ell x)^{k}=y^{n}$ in integers $x,y,n$…

We determine the rational integers x,y,z such that x^3+y^9=z^2 and gcd(x,y,z)=1. First we determine a finite set of curves of genus 10 such that any primitive solution to x^3+y^9=z^2 corresponds to a rational point on one of those curves.…