Related papers: Diophantine equations with sum of cubes and cube o…

In this paper, we solve the simultaneous Diophantine equations m.(x_1^k+....+x_{t_1}^k)=n.(y_1^k+....+y_{t_2}^k); k=1,3, where t_1, t_2>3, and m, n are fixed arbitrary and relatively prime positive integers. This is done by choosing two…

A rational Diophantine $m$-tuple is a set $\{a_1,\ldots,a_m\}$ of distinct nonzero rational numbers such that $a_i a_j+1$ is a square for all $1\leq i < j\leq m$. Similarly, we may ask when $a_ia_j+1$ is a $k$-th power. Here, we study the…

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 study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this…

By the theory of elliptic curves, we study the nontrivial rational parametric solutions and rational solutions of the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ for some simple Laurent polynomials $f$.

We obtain two parametric solutions of the diophantine equation $\phi(x_1, x_2, x_3)=\phi(y_1, y_2, y_3)$ where $\phi(x_1, x_2, x_3)$ is the octic form defined by $\phi(x_1, x_2, x_3)=x_1^8+ x_2^8 + x_3^8 - 2x_1^4x_2^4 - 2x_1^4x_3^4 -…

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…

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…

By finding all integral points on certain elliptic and hyperelliptic curves we completely solve the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$ for $-3\leq d\leq 3$ and $(k,l)\in\{(2,3),\; (2,4),\;(2,5),\; (2,6),\; (2,8),\; (3,4),\;…

In this paper, by using the elliptic curves theory, we study the fourth power Diophantine equation ${ \sum_{i=1}^n a_ix_{i} ^4= \sum_{j=1}^na_j y_{j}^4 }$, where $a_i$ and $n\geq3$ are fixed arbitrary integers. We solve the equation for…

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.

The solvability of Diophantine quartic equations is a contemporary area of interest due to its connection with generalized Fermat's equation. In this work, we are interested in the integer solutions of a similar Diophantine equation p u^2 =…

In this paper, first, we prove that the Diophantine system \[f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)\] has infinitely many integer solutions for $f(X)=X(X+a)$ with nonzero integers $a\equiv 0,1,4\pmod{5}$. Second, we show that the above…

In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups…

Diophantine equations are a popular and active area of research in number theory. In this paper we consider Mordell equations, which are of the form $y^2=x^3+d$, where $d$ is a (given) nonzero integer number and all solutions in integers…

In this paper, we use some extension of the Cayley-Hamilton theorem to find a family of matrices with integer entries that satisfy the non-linear Diophantine equation $ x^{n}+y^{p}=z^{q}$ where $n,p$ and $q$ are arbitrary positive integers.

For any given positive definite binary quadratic form $Q$ with integer coefficients, we establish two results on Diophantine approximation with integers represented by $Q$. Firstly, we show that for every irrational number $\alpha$, there…

A nontrivial solution of the equation A!B! = C! is a triple of positive integers (A, B, C) with A $\le$ B $\le$ C -- 2. It is conjectured that the only nontrivial solution is (6, 7, 10), and this conjecture has been checked up to C = 10 6.…

We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc), which satisfy some natural…

In this article, we show that the quartic Diophantine equations $x^4 \pm pqy^4=\pm z^2$ and $ x^4 \pm pq y^4= \pm iz^2$ have only trivial solutions for some primes $p$ and $q$ satisfying conditions $ p \equiv 3 \pmod 8, ~ q \equiv 1 \pmod 8…