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

2021 ◽  
Vol 2070 (1) ◽  
pp. 012015
Author(s):  
Komon Paisal ◽  
Pailin Chayapham
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Integer Solution ◽  
Negative Integer ◽  
Exponential Diophantine Equation

Abstract 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)

Solutions for a Class of the Exponential Diophantine Equation

2013 ◽  
Vol 753-755 ◽  
pp. 3149-3152
Author(s):  
Yin Xia Ran
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Algebraic Number Theory ◽  
Necessary Condition ◽  
Exponential Diophantine Equation ◽  
Elementary Method ◽  
Integer Solutions

We studied the Diophantine equation x2+4n=y11. By using the elementary method and algebraic number theory, we obtain the following conclusions: (i) Let x be an odd number, one necessary condition which the equation has integer solutions is that 210n-1/11 contains some square factors. (ii) Let x be an even number, when n=11k(k≥1), all integer solutions for the equation are(x,y)=(0,4k) ; whenn=11k+5(k≥0) , all integer solutions are(x,y)=(±211k+5,22k+1); when n≡1,2,3,4,6,7,8,9,10 the equation has no integer solution.

scholarly journals On the Diophantine Equation 8^x+n^y=z^2

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Negative Integer

Let n be an positive integer with n = 10(mod15). In this paper, we prove that (1,0,3) is unique non negative integer solution (x,y,z) of the Diophantine equation 8^x+n^y=z^2 where x y, and z are non-negativeintegers.

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}.

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 $(x,y)$.

scholarly journals On the non-linear diophantine equation \({\boldsymbol{379}}^{\boldsymbol{x}}\boldsymbol{+}{\boldsymbol{397}}^{\boldsymbol{y}}\boldsymbol{=}{\boldsymbol{z}}^{\boldsymbol{2}}\)

2020 ◽  
Vol 4 (1) ◽  
pp. 397-399
Author(s):  
Sudhanshu Aggarwal ◽  
◽  
Nidhi Sharma ◽  
Keyword(s):  
Diophantine Equation ◽  
Existence Of Solution ◽  
Integer Solution ◽  
Negative Integer ◽  
Linear Diophantine Equation ◽  
Non Linear

In this article, authors discussed the existence of solution of non-linear diophantine equation \({379}^x+{397}^y=z^2,\) where \(x,y,z\) are non-negative integers. Results show that the considered non-linear diophantine equation has no non-negative integer solution.

scholarly journals On the Diophantine equations z2 = k(k2 + 3) and z2 = k(k2 + 12)

2021 ◽  
Vol 45 (1) ◽  
pp. 127-129
Author(s):  
Shah Mohammad Shahidul Islam ◽  
Abdullah Al Kafi Majumdar
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Analytical Method ◽  
Diophantine Equations ◽  
Integer Solution ◽  
Positive Integer Solution ◽  
Integral Solutions

This paper provides an analytical method of finding all the (positive, integral) solutions of the Diophantine equation z2 = k(k2+3). We also prove analytically that the Diophantine equation z2 = k(k2+12) has no positive, integer solution. J. Bangladesh Acad. Sci. 45(1); 127-129: June 2021

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].

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