Related papers: Generalized Fruit Diophantine equation and Hyperel…

In this paper, we consider the equation $(a^n-2^{m})(b^n-2^{m})=x^2$. By assuming the abc conjecture is true, in [8], Luca and Walsh gave a theorem, which implies that the above equation has only finitely many solutions $n,x$ if a and b are…

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…

The main aim of this article is to find all solutions of the Diophantine equation $x^2 + p^k=y^n$ where $p \equiv 1 \pmod 4$, $\frac{p-1}{3}$ is a perfect square and the class number of $\mathbb{Z}[\sqrt{-p}]$ is $2$. In this article, I…

For any sufficiently strong theory of arithmetic, the set of Diophantine equations provably unsolvable in the theory is algorithmically undecidable, as a consequence of the MRDP theorem. In contrast, we show decidability of Diophantine…

The purpose of this paper is to exhibit infinite families of conjugate projective curves in a number field whose complement have the same abelian fundamental group, but are non-homeomorphic. In particular, for any $d>3$ we find Zariski…

For a finite field $\mathbb{F}_q$ of characteristic $p\geq 5$ and $K=\mathbb{F}_q(t)$, we consider the family of elliptic curves $E_d$ over $K$ given by $y^2+xy - t^dy=x^3$ for all integers $d$ coprime to $q$. We provide an explicit…

N.Garc\'ia-Fritz and H.Pasten showed that Hilbert's 10th problem is unsolvable in the ring of integers of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and $q$. We improve their…

Using elementary number theory we study Diophantine equations over the rational integers of the following form, $y^2=(x+a)(x+a+k)(x+b)(x+b+k)$, $y^2=c^2x^4+ax^2+b$ and $y^2=(x^2-1)(x^2-\alpha^2)(x^2-(\alpha+1)^2).$ We express their integer…

We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(\zeta_3, \sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free integers such tha Hilbert's 10-th Problem is unsolvable…

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.

Given a Diophantine triple $\{c_1(t),c_2(t),c_3(t)\}$, the elliptic curve over Q(t) induced by this triple, i.e. $y^2=(c_1(t) x+1) (c_2(t) x+1) (c_3(t) x+1)$, can have as torsion group one of the non-cyclic groups in Mazur's theorem, i.e.…

In this paper, we prove results about solutions of the Diophantine equation $x^p+y^p=z^3$ over various number fields using the modular method. Firstly, by assuming some standard modularity conjecture we prove an asymptotic result for…

In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $n\geq 2$ and $d=p^b$, $p$ a prime and $p\leq 10^4$. The main ingredients are the characterization of primitive divisors on…

The object of this paper is to give a new proof of all the solutions of the Diophantine equation x^2+11^m=y^n; in positive integers x, y with odd m>1 and n>=3.

In a series of articles published in the C.R. Paris more than a century ago, T. P\'epin announced a list of ``theorems'' concerning the solvability of diophantine equations of the type $ax^4 + by^4 = z^2$. In this article, we show how to…

In this note we find all the solutions of the Diophantine equation $x^4\pm y^4=iz^2$ using elliptic curves over $\mathbb Q(i)$. Also, using the same method we give a new proof of Hilbert's result that the equation $x^4\pm y^4=z^2$ has only…

We prove that the Diophantine equation N X^2 + 2^L 3^M = Y^N has no solutions (N,X,Y,L,M) in positive integers with N > 1 and gcd(NX,Y) = 1, generalizing results of Luca, Wang and Wang, and Luca and Soydan. Our proofs use results of Bilu,…

In this paper we investigate Diophantine equations of the form $T^2=G(\overline{X}),\; \overline{X}=(X_{1},\ldots,X_{m})$, where $m=3$ or $m=4$ and $G$ is specific homogenous quintic form. First, we prove that if…

For two relatively prime positive integers $a, b\in \mathbb{N}$, it is known that exactly one of the two Diophantine equations $$ax + by \ =\ \frac{(a-1)(b-1)}{2}\ \mbox{ and }\ 1 + ax + by \ =\ \frac{(a-1)(b-1)}{2}$$ has a nonnegative…

In this paper, we gave solutions of the Diophantine equations 16^{x}+p^{y}=z^{2}, 64^{x}+p^{y}=z^{2} where p is an odd prime, n is a positive integer and x,y,z are non-negative integers. Finally we gave a generalization of the Diophantine…