Related papers: On Diophantine Equation $x^2 = 2y^4-1$

Binary quadratic Diophantine equations are of interest from the viewpoint of computational complexity theory. They contain as special cases many examples of natural problems apparantly occupying intermediate stages in the P-NP hierarchy,…

In 2014, T. Komatsu and L. Szalay studied the balancing binomial coefficients. In this paper, we focus on the following Diophantine equation $$\binom{1}{5}+\binom{2}{5}+...+\binom{x-1}{5}=\binom{x+1}{5}+...+\binom{y}{5}$$ where $y>x>5$ are…

Using the modular method, we study solutions to the Diophantine equation $$Aa^p+Bb^p=Cc^2$$ over number fields. We first prove an asymptotic result for general number fields satisfying an appropriate $S$-unit condition by assuming some…

In this work, we prove the following result(Theorem 1): Suppose that n is a positive integer, p an odd prime, and such that either n is congruent to 0 modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2 modulo4 and p…

In this article, I study and solve the exponential Diophantine equation $M_p^{x} + (M_q + 1)^{y}= (lz)^2$ where $M_p$ and $M_q$ are Mersenne primes, $l$ is a prime number, and $x,y$, and $z$ are non-negative integers. Several illustrations…

In this note we consider the title Diophantine equation from both theoretical as well as experimental point of view. In particular, we prove that for $k=4, 6$ and each choice of the signs our equation has infinitely many co-prime positive…

This paper provides asymptotics with a sharp error term for the Dirichlet summatory function of a certain class of arithmetic functions. The result applies, e.g., to the sums over r^2(n) and r(n^3), where r(m) denotes the number of ways to…

We provide a method, using essentially elementary arithmetic, to solve the exponential diophantine equation 7x^2 + 59y^2 = 3^m, which leads to a simple algorithm, with no use of radicals or complex numbers, that generates all the…

Let $n$ be a positive integer. We discuss pairs of distinct odd primes $p$ and $q$ not dividing $n$ for which the Diophantine equations $pq=x^2+ny^2$ have integer solutions in $x$ and $y$. As its examples we classify all such pairs of $p$…

In this paper we first show that, under certain conditions, the solution of a single quadratic diophantine equation in four variables $Q(x_1,\,x_2,\,x_3,\,x_4)=0$ can be expressed in terms of bilinear forms in four parameters. We use this…

We collect a number of open questions concerning Diophantine equations, Diophantine Approximation and transcendental numbers. Revised version: corrected typos and added references.

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…

Recursive formulas are derived for the number of solutions of linear and quadratic Diophantine equations with positive coefficients. This result is further extended to general non-linear additive Diophantine equations. It is shown that all…

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 shall show that, for any given primes $\ell\geq 17$ and $p, q\equiv 1\pmod{\ell}$, the diophantine equation $(x^\ell-1)/(x-1)=p^m q$ has at most four positive integral solutions $(x, m)$ and give its application to odd perfect number…

Different authors have done analysis regarding sums of powers References number 1,2 and 3, but systematic approach for solving Diophantine equations having sums of many biquadratics equal to a quartic has not been done before. In this paper…

We give a survey on some results covering the last 60 years concerning Je\'smanowicz' conjecture. Moreover, we conclude the survey with a new result by showing that the special Diophantine equation $$(20k)^x+(99k)^y=(101k)^z$$ has no…

In this paper, the elliptic curves theory is used for solving the Diophantine equations $\sum_{i=1}^n a_ix_{i} ^6+\sum_{i=1}^m b_iy_{i} ^3= \sum_{i=1}^na_iX_{i}^6\pm\sum_{i=1}^m b_iY_{i} ^3$, where $n$, $m$ $\geq 1$ and $a_i$, $b_i$, are…

We develop an algebraic method of studying of Diophantine quadratic equations in three variables over the ring of Gaussian integers.

We study the problem of Diophantine approximation on lines in R^2 with prime numerator and denominator.