scholarly journals The Diophantine Equation 8x2-y2+8x(1+t)+(2t+1)2=0 and t-Balancing Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Ahmet Tekcan ◽  
Merve Tayat ◽  
Meltem E. Özbek
Keyword(s):  
Diophantine Equation ◽  
Balancing Numbers ◽  
General Terms ◽  
Integer Solutions

Let t≥2 be an odd integer such that 2t2-1 is a prime. In this work, we determine all integer solutions of the Diophantine equation Dt:8x2-y2+8x(1+t)+(2t+1)2=0 and then we deduce the general terms of all t-balancing numbers.

scholarly journals A GENERALIZATION OF THE RAMANUJAN–NAGELL EQUATION

2018 ◽  
Vol 61 (03) ◽  
pp. 535-544
Author(s):  
TOMOHIRO YAMADA
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solutions

AbstractWe shall show that, for any positive integer D > 0 and any primes p1, p2, the diophantine equation x2 + D = 2sp1kp2l has at most 63 integer solutions (x, k, l, s) with x, k, l ≥ 0 and s ∈ {0, 2}.

On D. J. Lewis's Equation x3+117y3 = 5

1971 ◽  
Vol 14 (1) ◽  
pp. 111-111 ◽  
Author(s):  
R. Finkelstein ◽  
H. London
Keyword(s):  
Diophantine Equation ◽  
Exact Number ◽  
Integer Solutions

In a recent publication [2], D. J. Lewis stated that the Diophantine equation x3+117y3 = 5 has at most 18 integer solutions, but the exact number is unknown. In this paper we shall solve this problem by proving the followingTheorem. The equationx3+117y3 = 5 has no integer solutions.

scholarly journals Quadratic ideals, indefinite quadratic forms and some specific diophantine equations

2018 ◽  
Vol 36 (3) ◽  
pp. 173-192
Author(s):  
Ahmet Tekcan ◽  
Seyma Kutlu
Keyword(s):  
Diophantine Equation ◽  
Quadratic Forms ◽  
Diophantine Equations ◽  
Algebraic Properties ◽  
Integer Solutions ◽  
Indefinite Quadratic

Let $k\geq 1$ be an integer and let $P=k+2,Q=k$ and $D=k^{2}+4$. In this paper, we derived some algebraic properties of quadratic ideals $I_{\gamma}$ and indefinite quadratic forms $F_{\gamma }$ for quadratic irrationals $\gamma$, and then we determine the set of all integer solutions of the Diophantine equation $F_{\gamma }^{\pm k}(x,y)=\pm Q$.

The Positive Integer Solutions of a Diophantine Equation

2015 ◽  
Vol 713-715 ◽  
pp. 1483-1486
Author(s):  
Yi Wu ◽  
Zheng Ping Zhang
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Pell Equation ◽  
Integer Solutions ◽  
Basic Equations

In this paper, we studied the positive integer solutions of a typical Diophantine equation starting from two basic equations including a Diophantine equation and a Pell equation, and we will prove all the positive integer solutions of the typical Diophantine equation.

scholarly journals ON THE NUMBER OF SOLUTIONS OF THE DIOPHANTINE EQUATION axm−byn=c

2010 ◽  
Vol 81 (2) ◽  
pp. 177-185 ◽  
Author(s):  
BO HE ◽  
ALAIN TOGBÉ
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Number Of Solutions ◽  
Integer Solutions ◽  
Image Position ◽  
Positive Integers

AbstractLet a, b, c, x and y be positive integers. In this paper we sharpen a result of Le by showing that the Diophantine equation has at most two positive integer solutions (m,n) satisfying min (m,n)>1.

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 A Hypothetical Upper Bound on the Heights of the Solutions of a Diophantine Equation with a Finite Number of Solutions

2018 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Rational Solution ◽  
Infinitely Many Solutions ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Integer Solutions

Let f ( 1 ) = 1 , and let f ( n + 1 ) = 2 2 f ( n ) for every positive integer n. We consider the following hypothesis: if a system S ⊆ {xi · xj = xk : i, j, k ∈ {1, . . . , n}} ∪ {xi + 1 = xk : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x1, . . . , xn, then each such solution (x1, . . . , xn) satisfies x1, . . . , xn ≤ f (2n). We prove:   (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, then M is computable.

scholarly journals ON THE TERNARY QUADRATIC DIOPHANTINE EQUATION 6z2 = 6x2 – 5y2

2020 ◽  
Vol 1 (1) ◽  
pp. 14-22
Author(s):  
M. Gopalan ◽  
S. Nandhini ◽  
J. Shanthi
Keyword(s):  
Diophantine Equation ◽  
Quadratic Equation ◽  
Quadratic Diophantine Equation ◽  
Integer Solutions ◽  
Distinct Integer ◽  
The Given

The ternary homogeneous quadratic equation given by 6z2 = 6x2 -5y2 representing a cone is analyzed for its non-zero distinct integer solutions. A few interesting relations between the solutions and special polygonal and pyramided numbers are presented. Also, given a solution, formulas for generating a sequence of solutions based on the given solutions are presented.

scholarly journals A Peer Search on Integer Solutions to Quadratic Diophantine Equation with Three Unknowns (Equation)

2020 ◽  
pp. 166-171
Author(s):  
A. Vijayasankar ◽  
Sharadha Kumar ◽  
M. A. Gopalan
Keyword(s):  
Diophantine Equation ◽  
Quadratic Diophantine Equation ◽  
Integer Solutions ◽  
Integral Solutions

The non- homogeneous ternary quadratic diophantine (Equation) is analyzed for its patterns of non-zero distinct integral solutions. Various interesting relations between the solutions and special numbers namely polygonal, Pronic and Gnomonic numbers are exhibited.

Sign in / Sign up

Export Citation Format

Share Document