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 $F_{\gamma }^{\pm k}(x,y)=\pm Q$.

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 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 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 GENERALIZATION OF A THEOREM OF BUMBY ON QUARTIC DIOPHANTINE EQUATIONS

2006 ◽  
Vol 02 (02) ◽  
pp. 195-206 ◽  
Author(s):  
MICHAEL A. BENNETT ◽  
ALAIN TOGBÉ ◽  
P. G. WALSH
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Integer Solutions ◽  
Positive Integers

Bumby proved that the only positive integer solutions to the quartic Diophantine equation 3X4 - 2Y2 = 1 are (X, Y) = (1, 1),(3, 11). In this paper, we use Thue's hypergeometric method to prove that, for each integer m ≥ 1, the only positive integers solutions to the Diophantine equation (m2 + m + 1)X4 - (m2 + m)Y2 = 1 are (X,Y) = (1, 1),(2m + 1, 4m2 + 4m + 3).

The Integer Solutions of Diophantine Equation Χ2 − 21 = 4y5

2014 ◽  
Vol 687-691 ◽  
pp. 1182-1185
Author(s):  
Yi Wu ◽  
Lan Long ◽  
Zheng Ping Zhang
Keyword(s):  
Diophantine Equation ◽  
Fundamental Theorem ◽  
Diophantine Equations ◽  
Quadratic Fields ◽  
Algebraic Integers ◽  
Integer Solutions ◽  
Fundamental Theorem Of Arithmetic

In this paper, we studied the integer solutions of the typical Diophantine equations with some important theories in quadratic fields and the fundamental theorem of arithmetic in the ring of quadratic algebraic integers. We proved all the integer solutions of the Diophantine equation Χ2 − 21 = 4y5.

Reducible Diophantine Equations and Their Parametric Representations

1955 ◽  
Vol 7 ◽  
pp. 328-336
Author(s):  
E. Rosenthall
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Parametric Representation ◽  
Rational Integer ◽  
Homogeneous Polynomials ◽  
Linear Forms ◽  
Rational Field ◽  
Integer Solutions ◽  
Image Position ◽  
General Method

1. Reducible diophantine equations. The present paper will provide a general method for obtaining the complete parametric representation for the rational integer solutions of the multiplicative diophantine equation1.1 for some specified range of k, where the aijk,bijk are non-negative integers and the fki, hki are decomposable forms, that is to say they are integral irreducible homogeneous polynomials over the rational field R of degree k in k variables which can be written as the product of k linear forms.

scholarly journals The Quadratic Diophantine Equations x^2− P(t)y^2− 2P′(t)x + 4P(t)y + (P′(t))^2− 4P(t) − 1 = 0

2019 ◽  
Vol 11 (2) ◽  
pp. 30
Author(s):  
Amara Chandoul ◽  
Diego Marques ◽  
Samira Shaban Albrbar
Keyword(s):  
Finite Fields ◽  
Diophantine Equation ◽  
Recurrence Relations ◽  
Diophantine Equations ◽  
Integer Solutions

Let P := P(t) be a non square polynomial. In this paper, we consider the number of integer solutions of Diophantine equation E : x2− P(t)y2− 2P′(t)x + 4P(t)y + (P′(t))2− 4P(t) − 1 = 0. We derive some recurrence relations on the integer solutions (xn,yn) of E. In the last section, we consider the same problem over finite fields Fpfor primes p ≥ 5. Our main results are generaliations of previous results given by Ozcok and Tekcan (Ozkoc and Tekcan, 2010).

The exponential Diophantine equation x2 + (3n 2 + 1)y = (4n 2 + 1)z

2014 ◽  
Vol 64 (5) ◽  
Author(s):  
Jianping Wang ◽  
Tingting Wang ◽  
Wenpeng Zhang
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Exponential Diophantine Equation ◽  
Lucas Numbers ◽  
Exponential Diophantine Equations ◽  
Primitive Divisors ◽  
Integer Solutions

AbstractLet n be a positive integer. In this paper, using the results on the existence of primitive divisors of Lucas numbers and some properties of quadratic and exponential diophantine equations, we prove that if n ≡ 3 (mod 6), then the equation x 2 + (3n 2 + 1)y = (4n 2 + 1)z has only the positive integer solutions (x, y, z) = (n, 1, 1) and (8n 3 + 3n, 1, 3).

scholarly journals A note on the exponential Diophantine equation (A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z

2020 ◽  
Vol 55 (2) ◽  
pp. 195-201
Author(s):  
Maohua Le ◽  
◽  
Gökhan Soydan ◽  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Exponential Diophantine Equations ◽  
Integer Solutions ◽  
Positive Integers ◽  
Upper Bound For Solutions

Let A, B be positive integers such that min{A,B}>1, gcd(A,B) = 1 and 2|B. In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer n, if A >B3/8, then the equation (A2 n)x + (B2 n)y = ((A2 + B2)n)z has no positive integer solutions (x,y,z) with x > z > y; if B>A3/6, then it has no solutions (x,y,z) with y>z>x. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer n, if B ≡ 2 (mod 4) and A >B3/8, then this equation has only the positive integer solution (x,y,z)=(1,1,1).

scholarly journals Diophantine equations of Erdös-Moser type

1996 ◽  
Vol 53 (2) ◽  
pp. 281-292 ◽  
Author(s):  
Pieter Moree
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Consecutive Integer ◽  
Elementary Method ◽  
Integer Solutions

Using an old result of Von Staudt on sums of consecutive integer powers, we shall show by an elementary method that the Diophantine equation 1k + 2k + … + (x − l)k = axk has no solutions (a, x, k) with k > 1, . For a = 1 this equation reduces to the Erdös-Moser equation and the result to a result of Moser. Our method can also be used to deal with variants of the equation of the title, and two examples will be given. For one of them there are no integer solutions with

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