Related papers: A new diophantine equation involving fifth powers
Let $\{x_{n}\}_{n \geq 0}$ be the balancing-like sequence defined by $x_{n+1} = A x_{n} - x_{n-1}$, for $A>2$, where $x_0 = 0$ and $x_1 = 1$. In this paper, we demonstrate how to find all the solutions of the Diophantine equation,…
In this paper we obtain four new parametric ideal solutions of the Tarry-Escott problem of degree 7, that is, of the simultaneous diophantine equations, $\sum_{i=1}^8x_i^r=\sum_{i=1}^8y_i^r,\;r=1,\,2,\,\dots,\,7$. While all the known…
In this study we find all solutions of the Diophantine equation $B_{n_{1}}+B_{n_{2}}=2^{a_{1}}+2^{a_{2}}+2^{a_{3}}$ in positive integer variables $(n_{1},n_{2},a_{1},a_{2},a_{3}),$ where $B_{n}$ denotes the $n$-th balancing number.
From a known result of diophantine equations of the first degree with 2 unknowns we simply find the results of the distribution function of the sequences of positive integers generated by the functions at the origin of the 3x+1 and 5x+1…
Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_0 = 0, F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for $n \geq 0$. In this paper, we solve all powers of two which are sums of four Fibonacci numbers with a few exceptions that we…
It is a generalization of Pell's equation $x^2-Dy^2=0$. Here, we show that: if our Diophantine equation has a particular integer solution and $ab$ is not a perfect square, then the equation has an infinite number of solutions; in this case…
We observe that the computation $5^2 = 25$ has the digital property of the result being equal to the exponent concatenated directly to the left of the base. The generalization to a Diophantine equation and inequality in number bases has…
The arithmetic-digital anomaly of $5\div 2 = 2.5$ has been observed several times in the past. We generalize it to an exponential Diophantine equation and inequality in the general number base, which is the object of our analysis. First, we…
The objective of the paper is to determine the complete solutions for the Diophantine equation $x^2 + 3^{\alpha}113^{\beta} = y^{\mathfrak{n}}$ in positive integers $x$ and $y$ (where $x, y \geq 1$), non-negative exponents $\alpha$ and…
We investigate the solvability of the Diophantine equation in the title, where $d>1$ is a square-free integer, $p, q$ are distinct odd primes and $x,y,a,b$ are unknown positive integers with $\gcd(x,y)=1$. We describe all the integer…
The present work includes some of the author's original researches on integer solutions of Diophantine liner equations and systems. The notion of "general integer solution" of a Diophantine linear equation with two unknowns is extended to…
In this paper, we deal with the quartic Diophantine equation $X^4-Y^4=R^2-S^2$ to present its infinitely many integer solutions.
We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in…
We give conditions on the rational numbers a,b,c which imply that there are infinitely many triples (x,y,z) of rational numbers such that x+y+z=a+b+c and xyz=abc. We do the same for the equations x+y+z=a+b+c and x^3+y^3+z^3=a^3+b^3+c^3.…
In this paper, we proved that there are infinitely many integer solutions of $X^6 - Y^6 = W^n - Z^n,\ n=2,3,4$.
Using a recent result of Salberger, we establish the paucity of non-trivial positive integer solutions to a certain system of diagonal Diophantine equations.
We prove that for each odd prime p, positive integer alpha, and non-negative integers beta and gamma, the Diophantine equation X^{2N} + 2^{2 alpha} 5^{2 beta} p^{2 gamma} = Z^5 has no solution with X, Z, N in Z^+, N > 1, and gcd(X,Z) = 1.
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 $f$.
The Diophantine equation $A^4+hB^4=C^4+hD^4$, where $h$ is a fixed arbitrary positive integer, has been investigated by some authors. Currently, by computer search, the integer solutions of this equation are known for all positive integer…
In this paper we completely solve the Diophantine equation $F_n+F_m=2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$, where $F_k$ denotes the $k$-th Fibonacci number. In addition to complex linear forms in logarithms and the Baker-Davenport…