Integral Solutions of the Ternary Cubic Equation 6(x^2+y^2 )-11xy=288z^3

The Ternary cubic Diophantine Equation represented by૟(࢞ ࢟ + ૛ ࢠૡૡ = ૛࢟࢞૚૚ − (૛ ૜ is analyzed for its infinite number of non-zero integral solutions. A few interesting among the solutions are also discussed. Keywords: Diophantine equation, Integral solutions, cubic equation with three unknowns, Ternary equation.

Sufficient condition for infinite solutions of Diophantine Equation: n! = P(s)

In this article I have used method which tells that number of solutions of Diophantine equation: n! = P(s) is infinite if some condition is satisfied. I have applied Inverse Laplace Transform to n! = P(s) and got function f(t) which is easier to deal with. The condition is given in section below contains zero of f(t) or zero of some modified function of f(t): g(t) = f(t) - h(t).

The Primitive-Solutions of Diophantine Equation x^2+pqy^2=z^2, for primes p,q

In this paper, we determine the primitive solutions of diophantine equations x^2+pqy^2=z^2, for positive integers x, y, z, and primes p,q. This work is based on the development of the previous results, namely using the solutions of the Diophantine equation x^2+y^2=z^2, and looking at characteristics of the solutions of the Diophantine equation x^2+3y^2=z^2 and x^2+9y^2=z^2.

Fermat and Mersenne numbers in $k$-Pell sequence

For an integer $k\geq 2$, let $(P_n^{(k)})_{n\geq 2-k}$ be the $k$-generalized Pell sequence, which starts with $0,\ldots,0,1$ ($k$ terms) and each term afterwards is defined by the recurrence$P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)},\quad \text{for all }n \geq 2.$For any positive integer $n$, a number of the form $2^n+1$ is referred to as a Fermat number, while a number of the form $2^n-1$ is referred to as a Mersenne number. The goal of this paper is to determine Fermat and Mersenne numbers which are members of the $k$-generalized Pell sequence. More precisely, we solve the Diophantine equation $P^{(k)}_n=2^a\pm 1$ in positive integers $n, k, a$ with $k \geq 2$, $a\geq 1$. We prove a theorem which asserts that, if the Diophantine equation $P^{(k)}_n=2^a\pm 1$ has a solution $(n,a,k)$ in positive integers $n, k, a$ with $k \geq 2$, $a\geq 1$, then we must have that $(n,a,k)\in \{(1,1,k),(3,2,k),(5,5,3)\}$. As a result of our theorem, we deduce that the number $1$ is the only Mersenne number and the number $5$ is the only Fermat number in the $k$-Pell sequence.

On the Ramanujan-Nagell type Diophantine equation \(Dx^2+k^n=B\)

In this paper, we prove that the Ramanujan-Nagell type Diophantine equation \(Dx^2+k^n=B\) has at most three nonnegative integer solutions \((x, n)\) for \(k\) a prime and \(B, D\) positive integers.

A note on the Diophantine Equation x^2-kxy+ky^2+ly=0

We investigate the Diophantine equation x^2 −kxy + ky^2 + ly = 0 for integers k and l with k even. We give a characterization of the positive solutions of this equation in terms of k and l. We also consider the same equation for other values of k and l.

On exponential Diophantine equation 17x +83y = z2 and 29x +71y = z2

This Diophantine is an equation that many researchers are interested in and studied in many form such 3x +5y · 7z = u2, (x+1)k + (x+2)k + … + (2x)k = yn and kax + lby = cz. The extensively studied form is ax + by = cz. In this paper we show that the Diophantine equations 17x +83y = z2 and 29x +71y = z2 has a unique non – negative integer solution (x, y, z) = (1,1,10)

On the exponential Diophantine equation mx +(m+1) y =(1+m+m 2) z

Let m > 1 be a positive integer. We show that the exponential Diophantine equation mx + (m + 1) y = (1 + m + m 2) z has only the positive integer solution (x, y, z) = (2, 1, 1) when m ≥ 2.