相关论文: The equation $(x-d)^5 + x^5 + (x+d)^5 = y^n$

In this paper we study equation $$(x-dr)^5+\cdots+x^5+\cdots+(x+dr)^5=y^p$$ under the condition $\gcd(x,r)=1$. We present a recipe for proving the non-existence of non-trivial integer solutions of the above equation, and as an application…

Let $5 \leq k \leq 11$ and $0\leq i \leq k-1$ be integers. We determine all solutions to the equation \begin{align*} n(n+d)(n+2d)\cdots(n+(i-1)d)(n+(i+1)d) \cdots (n+(k-1)d) = y^3 \end{align*} in integers $n,d,y$ with $ny \neq 0$, $d\geq…

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.…

In this paper we are interested in solving the Fermat-type equations x^5+y^5=dz^p where d is a positive integer and p a prime number $\ge 7$. We describe a new method based on modularity theorems which allows us to improve all the results…

In this paper, we give a resolution of the generalized Fermat equations $$x^5 + y^5 = 3 z^n \text{ and } x^{13} + y^{13} = 3 z^n,$$ for all integers $n \ge 2$, and all integers $n \ge 2$ which are not a multiple of $7$, respectively, using…

In this part we show that the diophantine equation $X^5+Y^5=\varepsilon^5(1-X^5Y^5)$, where $\varepsilon=\frac{-1+\sqrt{5}}{2}$, has solutions in specific abelian extensions of quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ in which $-d \equiv…

We solve the Diophantine equations $x^5 + y^5 = dz^p$ with $d=2, 3$ for a set of prime numbers of density 1/4, 1/2, respectively. The method consists in relating a possible solution to another Diophantine equation and solving the later by…

We give the complete solution in integers $(n,a,b,x,y)$ of the title equation when $\gcd(x,y)=1$, except for the case when $xab$ is odd.

In this paper we carry out the steps of Darmon's program for the generalized Fermat equation $$ x^n + y^n = z^5. $$ In particular, we develop the machinery necessary to prove an optimal bound on the exponent $n$ for solutions satisfying…

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…

If $k$ is a sufficiently large positive integer, we show that the Diophantine equation $$n (n+d) \cdots (n+ (k-1)d) = y^{\ell}$$ has at most finitely many solutions in positive integers $n, d, y$ and $\ell$, with $\operatorname{gcd}(n,d)=1$…

We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curves defined over a number field. This method is used to solve some arithmetic problems that remained open.

We report the fourth known primitive solution to the Diophantine equation $a^5 + b^5 + c^5 + d^5 = e^5$, extending the list of solutions from 1966, 1996, and 2004. This result was obtained via a large-scale computational search based on an…

We give an infinite number of integer solutions to the Diophantine equation x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n, and some solutions to some similar equations.

In this paper, we refine the method introduced by Izadi and Baghalaghdam to search integer solutions to the Diophantine equation $X_1^5+X_2^5+X_3^5=Y_1^3+Y_2^3+Y_3^3$. We show that the Diophantine equation has infinitely many positive…

In this paper, we give a new method answer to a recent conjecture proposed by Budaghyan, Calderini, Carlet, Davidova and Kaleyski about the equation $x^d+(x+1)^d=b$ in $\mathbb{F}_{q^4}$, where $n$ is a positive integer, $q=2^n$ and…

The famous open problem of finding positive integer solutions to $a^5 + b^5 = c^5 + d^5$ is considered, and related solutions are found in two distinct settings: firstly, where $a$ and $b$ are both positive integers with $c$ and $d$ both…

In this paper we determine the perfect powers that are sums of three fifth powers in an arithmetic progression. More precisely, we completely solve the Diophantine equation $$ (x-d)^5 + x^5 + (x + d)^5 = z^n,~n\geq 2, $$ where $d,x,z \in…

In this paper, we develop the modular method for the generalized Fermat equation appearing in the title, within the framework of Darmon's program and using Frey hyperelliptic curves. As an application, we study a conjecture of Laradji,…

We study genus $g$ hyperelliptic curves with reduced automorphism group $A_5$ and give equations $y^2=f(x)$ for such curves in both cases where $f(x)$ is a decomposable polynomial in $x^2$ or $x^5$. For any fixed genus the locus of such…