Related papers: Solving Fermat-type equations $x^4 + d y^2 = z^p$ …

In the following we consider Diophantine equations of the form $x^2+ zxy + y^2 = M$ for given $M,z \in \mathbb{Z}$ and discuss the number of its (primitive) solutions as well as the construction of them. To reach this goal we introduce…

In this short note we present a method of solving this Diophantine equation, method which is different from Ljunggren's, Mordell's, and R.K.Guy's.

Fix a rational prime $r \geq 5$. In this article, we study the integer solutions of the generalized Fermat equation of signature $(2p,2q,r)$, namely $x^{2p}+y^{2q}=z^r$, where the primes $p,q \geq 5$ are varying. For each rational prime $r…

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…

In this work, we investigate hyperelliptic curves of type $C: y^2 = x^{2g+1} + ax^{g+1} + bx$ over the finite field $\mathbb{F}_q, q = p^n, p > 2$. For the case of $g = 3$ and $4$ we propose algorithms to compute the number of points on the…

We show that the mod p Galois representations attached to a Q-curve E of degree d over an imaginary quadratic number field K are surjective for all p larger than some constant M_{K,d}, if E has potentially multiplicative reduction at any…

In this paper, we characterize meromorphic solutions $f(z_1,z_2),g(z_1,z_2)$ to the generalized Fermat Diophantine functional equations $h(z_1,z_2)f^m+k(z_1,z_2)g^n=1$ in $\mathbf{C}^2$ for integers $m,n\geq2$ and nonzero meromorphic…

Let $K$ be a totally real number field and $ \mathcal{O}_K$ be the ring of integers of $K$. This manuscript examines the asymptotic solutions of the Fermat equation of signature $(r, r, p)$, specifically $x^r+y^r=dz^p$ over $K$, where $r,p…

We will be presenting two theorems in this paper. The first theorem, which is a new result, is about the non-existence of integer solutions of the cubic diophantine equation. In the proof of this theorem we have used some known results from…

We deeply investigate the Diophantine equation $cx^2+d^{2m+1}=2y^n$ in integers $x, y\geq 1, m\geq 0$ and $n\geq 3$, where $c$ and $d$ are given coprime positive integers such that $cd\not\equiv 3 \pmod 4$. We first solve this equation for…

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves, for the first we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of…

In 2016 Izadi and Nabardi (b) showed (4-2-4) has infinitely many integer solutions. They used a specific congruent number elliptic curve.In 2019 Janfada and Nabardi,item C, showed that a necessary condition for n to have an integral…

An important unsolved problem in Diophantine number theory is to establish a general method to effectively find all solutions to any given $S$-unit equation with at least four terms. Although there are many works contributing to this…

In this paper, we consider some hybrid Diophantine equations of addition and multiplication. We first improve a result on new Hilbert-Waring problem. Then we consider the equation \begin{equation} \begin{cases} A+B=C ABC=D^n \end{cases}…

We solve completely the Lebesgue-Nagell equation x^2+D=y^n, in integers x, y, n>2, for D in the range 1 =< D =< 100.

We determine the set of primitive integral solutions to the generalised Fermat equation x^2 + y^3 = z^15. As expected, the only solutions are the trivial ones with xyz = 0 and the non-trivial pair (x,y,z) = (+-3, -2, 1).

Let $n$ be a positive integer. We discuss pairs of distinct odd primes $p$ and $q$ not dividing $n$ for which the Diophantine equations $pq=x^2+ny^2$ have integer solutions in $x$ and $y$. As its examples we classify all such pairs of $p$…

Different authors have done analysis regarding sums of powers References number 1,2 and 3, but systematic approach for solving Diophantine equations having sums of many biquadratics equal to a quartic has not been done before. In this paper…

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.

Integer solutions of the diophantine equation $A^4+hB^4=C^4+hD^4$ are known for all positive integer values of $h < 1000$. While a solution of the aforementioned diophantine equation for any arbitrary positive integer value of $h$ is not…