The exponential diophantine equation xy + yx = z2 via a generalization of the Ankeny–Artin–Chowla conjecture

2018 ◽  
Vol 14 (05) ◽  
pp. 1223-1228
Author(s):  
Hai Yang ◽  
Ruiqin Fu
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Class Number ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Pell Equation ◽  
Positive Integer Solution ◽  
Primitive Forms ◽  
Integer Solutions

Let [Formula: see text] be a positive integer which is not a square. Further, let [Formula: see text] be the least positive integer solution of the Pell equation [Formula: see text], and let [Formula: see text] denote the class number of binary quadratic primitive forms of discriminant [Formula: see text]. If [Formula: see text] satisfies [Formula: see text] and [Formula: see text], then [Formula: see text] is called an exceptional number. In this paper, under the assumption that there have no exceptional numbers, we prove that the equation [Formula: see text] has no positive integer solutions [Formula: see text] satisfy [Formula: see text] and [Formula: see text].

An upper bound for least solutions of the exponential Diophantine equation D1x2 - D2y2 = λkz

2015 ◽  
Vol 11 (04) ◽  
pp. 1107-1114 ◽  
Author(s):  
Hai Yang ◽  
Ruiqin Fu
Keyword(s):  
Positive Integer ◽  
Fundamental Solution ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Pell’S Equation ◽  
Integer Solutions

Let D1, D2, D, k, λ be fixed integers such that D1 ≥ 1, D2 ≥ 1, gcd (D1, D2) = 1, D = D1D2 is not a square, ∣k∣ > 1, gcd (D, k) = 1 and λ = 1 or 4 according as 2 ∤ k or not. In this paper, we prove that every solution class S(l) of the equation D1x2-D2y2 = λkz, gcd (x, y) = 1, z > 0, has a unique positive integer solution [Formula: see text] satisfying [Formula: see text] and [Formula: see text], where z runs over all integer solutions (x,y,z) of S(l),(u1,v1) is the fundamental solution of Pell's equation u2 - Dv2 = 1. This result corrects and improves some previous results given by M. H. Le.

scholarly journals A note on the Diophantine equation $|a^x-b^y|=c$

2010 ◽  
Vol 107 (2) ◽  
pp. 161
Author(s):  
Bo He ◽  
Alain Togbé ◽  
Shichun Yang
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Integer Solutions ◽  
Positive Integers

Let $a,b,$ and $c$ be positive integers. We show that if $(a,b) =(N^k-1,N)$, where $N,k\geq 2$, then there is at most one positive integer solution $(x,y)$ to the exponential Diophantine equation $|a^x-b^y|=c$, unless $(N,k)=(2,2)$. Combining this with results of Bennett [3] and the first author [6], we stated all cases for which the equation $|(N^k \pm 1)^x - N^y|=c$ has more than one positive integer solutions $(x,y)$.

ON THE EXPONENTIAL DIOPHANTINE EQUATION

2014 ◽  
Vol 90 (1) ◽  
pp. 9-19 ◽  
Author(s):  
TAKAFUMI MIYAZAKI ◽  
NOBUHIRO TERAI
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Baker’S Method ◽  
Elementary Methods ◽  
Positive Integers

AbstractLet $m$, $a$, $c$ be positive integers with $a\equiv 3, 5~({\rm mod} \hspace{0.334em} 8)$. We show that when $1+ c= {a}^{2} $, the exponential Diophantine equation $\mathop{({m}^{2} + 1)}\nolimits ^{x} + \mathop{(c{m}^{2} - 1)}\nolimits ^{y} = \mathop{(am)}\nolimits ^{z} $ has only the positive integer solution $(x, y, z)= (1, 1, 2)$ under the condition $m\equiv \pm 1~({\rm mod} \hspace{0.334em} a)$, except for the case $(m, a, c)= (1, 3, 8)$, where there are only two solutions: $(x, y, z)= (1, 1, 2), ~(5, 2, 4). $ In particular, when $a= 3$, the equation $\mathop{({m}^{2} + 1)}\nolimits ^{x} + \mathop{(8{m}^{2} - 1)}\nolimits ^{y} = \mathop{(3m)}\nolimits ^{z} $ has only the positive integer solution $(x, y, z)= (1, 1, 2)$, except if $m= 1$. The proof is based on elementary methods and Baker’s method.

scholarly journals A Diophantine equation about polygonal numbers

2021 ◽  
Vol 27 (3) ◽  
pp. 113-118
Author(s):  
Yangcheng Li ◽  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Nonnegative Integer ◽  
Diophantine Equations ◽  
Integer Solution ◽  
Pell Equation ◽  
Integer Solutions ◽  
Polygonal Numbers ◽  
Alpha Beta

It is well known that the number P_k(x)=\frac{x((k-2)(x-1)+2)}{2} is called the x-th k-gonal number, where x\geq1,k\geq3. Many Diophantine equations about polygonal numbers have been studied. By the theory of Pell equation, we show that if G(k-2)(A(p-2)a^2+2Cab+B(q-2)b^2) is a positive integer but not a perfect square, (2A(p-2)\alpha-(p-4)A + 2C\beta+2D)a + (2B(q-2)\beta-(q-4)B+2C\alpha+2E)b>0, 2G(k-2)\gamma-(k-4)G+2H>0 and the Diophantine equation \[AP_p(x)+BP_q(y)+Cxy+Dx+Ey+F=GP_k(z)+Hz\] has a nonnegative integer solution (\alpha,\beta,\gamma), then it has infinitely many positive integer solutions of the form (at + \alpha,bt + \beta,z), where p, q, k \geq 3 and p,q,k,a,b,t,A,B,G\in\mathbb{Z^+}, C,D,E,F,H\in\mathbb{Z}.

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

2020 ◽  
Vol 57 (2) ◽  
pp. 200-206
Author(s):  
Elif kizildere ◽  
Maohua le ◽  
Gökhan Soydan
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Primitive Divisors ◽  
Lehmer Numbers ◽  
Positive Integers

AbstractLet l,m,r be fixed positive integers such that 2| l, 3lm, l > r and 3 | r. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if min{rlm2 − 1,(l − r)lm2 + 1} > 30, then the equation (rlm2 − 1)x + ((l − r)lm2 + 1)y = (lm)z has only the positive integer solution (x,y,z) = (1,1,2).

The Diophantine equation (m2 + n2)x + (2mn)y = (m + n)2z

2020 ◽  
Vol 16 (08) ◽  
pp. 1701-1708
Author(s):  
Xiao-Hui Yan
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Positive Integers ◽  
Coprime Positive Integers

For fixed coprime positive integers [Formula: see text], [Formula: see text], [Formula: see text] with [Formula: see text] and [Formula: see text], there is a conjecture that the exponential Diophantine equation [Formula: see text] has only the positive integer solution [Formula: see text] for any positive integer [Formula: see text]. This is the analogue of Jésmanowicz conjecture. In this paper, we consider the equation [Formula: see text], where [Formula: see text] are coprime positive integers, and prove that the equation has no positive integer solution if [Formula: see text] and [Formula: see text].

scholarly journals A Note on Integer Solutions of the Diophantine Equation x2-dy2=1

1956 ◽  
Vol 3 (1) ◽  
pp. 55-56
Author(s):  
John Hunter
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Positive Integer Solution ◽  
Integer Solutions ◽  
Image Position

In the equationdis any positive integer which is not a perfect square. For convenience we shall consider only those solutions of (1) for which x and yare both positive. All the others can be obtained from these. In fact, it is well known that if (x0, y0) is the minimum positive integer solution of (1), then all integer solutions (x, y) are given byand, in particular, all positive integer solutions are given by

A Note on the Exponential Diophantine Equation x2+q2m=2yp

2012 ◽  
Vol 241-244 ◽  
pp. 2650-2653
Author(s):  
Jian Ping Wang
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Primitive Divisors ◽  
Lehmer Numbers

ln this paper, using a deep result on the existence of primitive divisors of Lehmer numbers given by Y. Bilu, G. Hanrot and P. M. Voutier, we prove that the equation has no positive integer solution (x, y, m, p, q), where p and q are odd primes with p>3, gcd(x, y)=1 and y is not the sum of two consecutive squares.

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

2021 ◽  
Vol 29 (3) ◽  
pp. 23-32
Author(s):  
Murat Alan
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution

Abstract 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.

A NOTE ON THE DIOPHANTINE EQUATION

2018 ◽  
Vol 98 (2) ◽  
pp. 188-195 ◽  
Author(s):  
MOU-JIE DENG ◽  
JIN GUO ◽  
AI-JUAN XU
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Terai’S Conjecture

Let $c\geq 2$ be a positive integer. Terai [‘A note on the Diophantine equation $x^{2}+q^{m}=c^{n}$’, Bull. Aust. Math. Soc.90 (2014), 20–27] conjectured that the exponential Diophantine equation $x^{2}+(2c-1)^{m}=c^{n}$ has only the positive integer solution $(x,m,n)=(c-1,1,2)$. He proved his conjecture under various conditions on $c$ and $2c-1$. In this paper, we prove Terai’s conjecture under a wider range of conditions on $c$ and $2c-1$. In particular, we show that the conjecture is true if $c\equiv 3\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$ and $3\leq c\leq 499$.

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