Related papers: Cubic Thue Equations

We solve the Fermat-type equation \[ x^{13} + y^{13} = 3 z^7, \qquad \gcd(x,y,z) = 1 \] combining a unit sieve, the multi-Frey modular method, level raising, computations of systems of eigenvalues modulo 7 over a totally real field, and…

Considering the equation $$a^x+b^y = c^z,$$ Le gives an upper bound on the number of solutions $(x,y,z)$, and a bound on $z$. Here we give slight improvements to these results, also removing the restriction that $a$, $b$, $c$ be squarefree.

We prove that the focusing cubic wave equation in three spatial dimensions has a countable family of self-similar solutions which are smooth inside the past light cone of the singularity. These solutions are labeled by an integer index $n$…

We show that for an irreducible cubic $f\in\mathbb Z[x]$ and a full norm form $\mathbf N(x_1,\ldots,x_k)$ for a number field $K/\mathbb Q$ satisfying certain hypotheses the variety $f(t)=\mathbf N(x_1,\ldots,x_k)\ne 0$ satisfies the Hasse…

We present a method for tabulating all cubic function fields over $\mathbb{F}_q(t)$ whose discriminant $D$ has either odd degree or even degree and the leading coefficient of $-3D$ is a non-square in $\mathbb{F}_{q}^*$, up to a given bound…

Stiller proved that the Diophantine equation $x^2+119=15\cdot 2^{n}$ has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type…

A long-standing conjecture states that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, 454 is a sum of at most seven positive cubes. This was first observed by Jacobi in 1851 on…

This paper presents the non-linear generalization of a previous work on matrix differential models. It focusses on the construction of approximate solutions of first-order matrix differential equations Y'(x)=f(x,Y(x)) using matrix-cubic…

In this paper we derive an explicit formula for the number of representations of an integer by the sextenary form x^2+y^2+z^2+ 7s^2+7t^2+ 7u^2. We establish the following intriguing inequalities 2b(n)>=a_7(n)>=b(n) for n not equal to…

We prove that $|x-y|\ge 800X^{-4}$, where $x$ and $y$ are distinct singular moduli of discriminants not exceeding $X$. We apply this result to the "primitive element problem" for two singular moduli. In a previous article Faye and Riffaut…

The discriminant of a polynomial of the form $\pm x^n \pm x^m \pm 1$ has the form $n^n \pm m^m(n-m)^{n-m}$ when $n,m$ are relatively prime. We investigate when these discriminants have prime power divisors. We explain several symmetries…

We prove existence and uniqueness of distributional, bounded, nonnegative solutions to a fractional filtration equation in ${\mathbb R}^d$. With regards to uniqueness, it was shown even for more general equations in [19] that if two bounded…

In this paper we prove a correspondence between a canonical degree six covariant of binary quartic forms $F$ and a cubic covariant of a pair of ternary quadratic forms $(f_A, f_B)$. In the process we obtain a canonical way to diagonalize a…

In this paper we establish some results about the existence and precise forms of finite order entire solutions of some systems of quadratic trinomial functional equations one of which in $\mathbb{C}^n$, $n\in\mathbb{N}$ and other two in…

The resolvability of equations in integers containing truncated Newton's binomial, is determined by the divisibility of the binomial by the characteristic parameters of the equation, which most often is the binomial exponent. Two types of…

Let $f$ be a positive definite integral ternary quadratic form and let $r(k,f)$ be the number of representations of an integer $k$ by $f$. In this article we study the number of representations of squares by $f$. We say the genus of $f$,…

For $g\geq1$ denote by $F_{2g}=\langle x_1, y_1,\dots,x_g,y_g\rangle$ the free group on $2g$ generators and by $B_g=[x_1,y_1]\dots[x_g,y_g]$. For $l,c\geq 1$ and elements $w_1,\dots,w_l\in F_{2g}$ we study orientable quadratic equations of…

We investigate the average order of the divisor function at values of totally reducible binary cubic forms and discuss some applications.

An early result in the theory of Natural Dualities is that an algebra with a near unanimity (NU) term is dualizable. A converse to this is also true: if V(A) is congruence distributive and A is dualizable, then A has an NU term. An…

By studying the n-dimensional cube, the author finds a new way, and use it to proves that the indefinite equation $x^n+y^n=z^n(n=>3)$ has no any solutions of positive integers.