G. Soydan

activity

20102023

(x + 1)^{k} + (x + 2)^{k} + ... + (2 x)^{k} = y^{n}

13 citations · 50 across the 23 of their papers we have counts for

collaborators

28 papers

math.NT2023

On the Ternary Purely Exponential Diophantine Equation $(ak)^{x} + (bk)^{y} = ((a + b) k)^{z}$ with Prime Powers $a$ and $b$

Maohua Le, Gökhan Soydan

Let $k$ be a positive integer, and let $a, b$ be coprime positive integers with $min {a, b} > 1$ . In this paper, using a combination of some elementary number theory techniques with…

math.NT2023

An elementary approach to the generalized Ramanujan-Nagell equation

Elif Kızıldere Mutlu, Maohua Le, Gökhan Soydan

Let $k$ be a fixed positive integer with $k > 1$ . In this paper, using various elementary methods in number theory, we give criteria under which the equation $x^{2} + (2 k - 1)^{y} = k^{z}$ has n…

math.NT2021

A modular approach to the generalized Ramanujan-Nagell equation

Elif Kızıldere Mutlu, Maohua Le, Gökhan Soydan

Let $k$ be a positive integer. In this paper, using the modular approach, we prove that if $k \equiv 0 (mod 4)$ , $30 < k < 724$ and $2 k - 1$ is an odd prime power, then under the GRH, t…

math.NT2021

The Shuffle Variant of a Diophantine equation of Miyazaki and Togbé

Elif Kızıldere, Gökhan Soydan, Qing Han +1

In 2012, T. Miyazaki and A. Togbé gave all of the solutions of the Diophantine equations $(2 am - 1)^{x} + (2 m)^{y} = (2 am + 1)^{z}$ and $b^{x} + 2^{y} = (b + 2)^{z}$ in positive integers $x, y, z,$ $a > 1$ and…

math.NT2021★ 3 cited

On elliptic curves induced by rational Diophantine quadruples

Andrej Dujella, Gökhan Soydan

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 perf…

math.NT2021

On a class of generalized Fermat equations of signature $(2, 2 n, 3)$

Karolina Chałupka, Andrzej Dąbrowski, Gökhan Soydan

We consider the Diophantine equation $7 x^{2} + y^{2 n} = 4 z^{3}$ . We determine all solutions to this equation for $n = 2, 3, 4$ and $5$ . We formulate a Kraus type criterion for show…

math.NT2021

On the power values of the sum of three squares in arithmetic progression

Maohua Le, Gökhan Soydan

In this paper, using a deep result on the existence of primitive divisors of Lehmer numbers due to Y. Bilu, G. Hanrot and P. M. Voutier, we first give an explicit formula for all p…

math.NT2020

Rational Points in Geometric Progression on the Unit Circle

Gamze Savaş Çelik, Mohammad Sadek, Gökhan Soydan

A sequence of rational points on an algebraic planar curve is said to form an $r$ -geometric progression sequence if either the abscissae or the ordinates of these points form a geo…

math.NT2020

A note on the ternary purely exponential Diophantine equation $A^{x} + B^{y} = C^{z}$ with $A + B = C^{2}$

Elif Kızıldere, Maohua Le, Gökhan Soydan

Let $ℓ, m, r$ be fixed positive integers such that $2 ∤ ℓ$ , $3 ∤ ℓ m$ , $ℓ > r$ and $3 ∣ r$ . In this paper, using the BHV theorem on the existence of primitive div…

math.NT2020★ 2 cited

On the Diophantine equation $(5 p n^{2} - 1)^{x} + (p (p - 5) n^{2} + 1)^{y} = (p n)^{z}$

Elif Kızıldere, Gökhan Soydan

Let $p$ be a prime number with $p > 3$ , $p \equiv 3 (mod 4)$ and let $n$ be a positive integer. In this paper, we prove that the Diophantine equation $(5pn^{2}-1)^{x}+(p(p-5)n^{2}+1)^…

math.NT2020

Resolution of the equation $(3^{x_{1}} - 1) (3^{x_{2}} - 1) = (5^{y_{1}} - 1) (5^{y_{2}} - 1)$

Kálmán Liptai, László Németh, Gökhan Soydan +1

Consider the diophantine equation $(3^{x_{1}} - 1) (3^{x_{2}} - 1) = (5^{y_{1}} - 1) (5^{y_{2}} - 1)$ in positive integers $x_{1} \leq x_{2}$ , and $y_{1} \leq y_{2}$ . Each side of the equation is a product of two…

math.NT2020★ 10 cited

A brief survey on the generalized Lebesgue-Ramanujan-Nagell equation

Maohua Le, Gökhan Soydan

The generalized Lebesgue-Ramanujan-Nagell equation is an important type of polynomial-exponential Diophantine equation in number theory. In this survey, the recent results and some…

math.NT2020

On a class of Lebesgue-Ljunggren-Nagell type equations

Andrzej Dąbrowski, Nursena Günhan, Gökhan Soydan

Given odd, coprime integers $a$ , $b$ ( $a > 0$ ), we consider the Diophantine equation $a x^{2} + b^{2 l} = 4 y^{n}$ , $x, y \in Z$ , $l \in N$ , $n$ odd prime, $g cd (x, y) = 1$ . We completely…

math.NT2019★ 4 cited

The Diophantine Equation $(x + 1)^{k} + (x + 2)^{k} + \dots + (ℓ x)^{k} = y^{n}$ Revisited

Daniele Bartoli, Gökhan Soydan

Let $k, ℓ \geq 2$ be fixed integers and $C$ be an effectively computable constant depending only on $k$ and $ℓ$ . In this paper, we prove that all solutions of the equation $(x+1…

math.NT2019

Rational sequences on different models of elliptic curves

Gamze Savaş ÇELİK, Mohammad Sadek, Gökhan Soydan

Given a set $S$ of elements in a number field $k$ , we discuss the existence of planar algebraic curves over $k$ which possess rational points whose $x$ -coordinates are exactly the…

math.NT2018

Elliptic Curves Containing Sequences of Consecutive Cubes

Gamze Savaş Çelik, Gökhan Soydan

Let $E$ be an elliptic curve over $Q$ described by $y^{2} = x^{3} + K x + L$ where $K, L \in Q$ . A set of rational points $(x_{i}, y_{i}) \in E (Q)$ for $i=1, 2, \cdot…

math.NT2017★ 13 cited

On the Diophantine equation $(x + 1)^{k} + (x + 2)^{k} + ... + (2 x)^{k} = y^{n}$

Attila Bérczes, István Pink, Gamze SavaŞ +1

In this work, we give upper bounds for $n$ on the title equation. Our results depend on assertions describing the precise exponents of $2$ and $3$ appearing in the prime factorizat…

math.NT2017

A note on the Diophantine equations $x^{2} \pm 5^{α} \cdot p^{n} = y^{n}$

Gökhan Soydan

Suppose that $x$ is odd, $n \geq 7$ and $p \in / {2, 5}$ are primes. In this paper, we prove that the Diophantine equations $x^{2} \pm 5^{α} p^{n} = y^{n}$ have no solutions in positive int…

math.NT2017★ 4 cited

On the Diophantine equation $\sum_{j = 1}^{k} j F_{j}^{p} = F_{n}^{q}$

Gökhan Soydan, László Németh, László Szalay

Let $F_{n}$ denote the $n^{t h}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_{1}^{p} + 2 F_{2}^{p} + \dots + k F_{k}^{p} = F_{n}^{q}$ in the positive integers…

math.NT2017

Complete solution of the Diophantine Equation $x^{2} + 5^{a} \cdot 1 1^{b} = y^{n}$

Gökhan Soydan, Nikos Tzanakis

The title equation is completely solved in integers $(n, x, y, a, b)$ , where $n \geq 3$ , $g cd (x, y) = 1$ and $a, b \geq 0$ . The most difficult stage of the resolution is the explicit resolu…

math.NT2017★ 2 cited

On the Diophantine equation $(x + 1)^{k} + (x + 2)^{k} + ... + (l x)^{k} = y^{n}$

Gökhan Soydan

Let $k, l \geq 2$ be fixed integers. In this paper, firstly, we prove that all solutions of the equation $(x + 1)^{k} + (x + 2)^{k} + ... + (l x)^{k} = y^{n}$ in integers $x, y, n$ with $x,y\geq1, n…

math.NT2012★ 1 cited

On the Diophantine equation x^2+7^{alpha}.11^{beta}=y^n

Gokhan Soydan

In this paper, we give all the solutions of the Diophantine equation x^2+7^{alpha}.11^{beta}=y^n, in nonnegative integers x, y, n>=3 with x and y coprime, except for the case when…

math.NT2012★ 3 cited

On the Diophantine equation x^2+2^a.3^b.11^c=y^n

Ismail Naci Cangul, Musa Demirci, Ilker Inam +2

In this note, we find all the solutions of the Diophantine equation x^2 +2^a.3^b.11^c=y^n in nonnegative integers a, b, c, x, y, n>= 3 with x and y coprime.

math.NT2011

The Diophantine Equation x^{2}+11^{m}=y^{n}

Gokhan Soydan, Musa Demirci, Ismail Naci Cangul

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.

On the Ternary Purely Exponential Diophantine Equation (ak)x+(bk)y=((a+b)k)z with Prime Powers a and b

An elementary approach to the generalized Ramanujan-Nagell equation

A modular approach to the generalized Ramanujan-Nagell equation

The Shuffle Variant of a Diophantine equation of Miyazaki and Togbé

On elliptic curves induced by rational Diophantine quadruples

On a class of generalized Fermat equations of signature (2,2n,3)

On the power values of the sum of three squares in arithmetic progression

Rational Points in Geometric Progression on the Unit Circle

A note on the ternary purely exponential Diophantine equation Ax+By=Cz with A+B=C2

On the Diophantine equation (5pn2−1)x+(p(p−5)n2+1)y=(pn)z

Resolution of the equation (3x1​−1)(3x2​−1)=(5y1​−1)(5y2​−1)

A brief survey on the generalized Lebesgue-Ramanujan-Nagell equation

On a class of Lebesgue-Ljunggren-Nagell type equations

The Diophantine Equation (x+1)k+(x+2)k+⋯+(ℓx)k=yn Revisited

Rational sequences on different models of elliptic curves

Elliptic Curves Containing Sequences of Consecutive Cubes

On the Diophantine equation (x+1)k+(x+2)k+...+(2x)k=yn

A note on the Diophantine equations x2±5α⋅pn=yn

On the Diophantine equation ∑j=1k​jFjp​=Fnq​

Complete solution of the Diophantine Equation x2+5a⋅11b=yn

On the Diophantine equation (x+1)k+(x+2)k+...+(lx)k=yn

On the Diophantine equation x^2+7^{alpha}.11^{beta}=y^n

On the Diophantine equation x^2+2^a.3^b.11^c=y^n

The Diophantine Equation x^{2}+11^{m}=y^{n}

On the Ternary Purely Exponential Diophantine Equation $(ak)^{x} + (bk)^{y} = ((a + b) k)^{z}$ with Prime Powers $a$ and $b$

On a class of generalized Fermat equations of signature $(2, 2 n, 3)$

A note on the ternary purely exponential Diophantine equation $A^{x} + B^{y} = C^{z}$ with $A + B = C^{2}$

On the Diophantine equation $(5 p n^{2} - 1)^{x} + (p (p - 5) n^{2} + 1)^{y} = (p n)^{z}$

Resolution of the equation $(3^{x_{1}} - 1) (3^{x_{2}} - 1) = (5^{y_{1}} - 1) (5^{y_{2}} - 1)$

The Diophantine Equation $(x + 1)^{k} + (x + 2)^{k} + \dots + (ℓ x)^{k} = y^{n}$ Revisited

On the Diophantine equation $(x + 1)^{k} + (x + 2)^{k} + ... + (2 x)^{k} = y^{n}$

A note on the Diophantine equations $x^{2} \pm 5^{α} \cdot p^{n} = y^{n}$

On the Diophantine equation $\sum_{j = 1}^{k} j F_{j}^{p} = F_{n}^{q}$

Complete solution of the Diophantine Equation $x^{2} + 5^{a} \cdot 1 1^{b} = y^{n}$

On the Diophantine equation $(x + 1)^{k} + (x + 2)^{k} + ... + (l x)^{k} = y^{n}$