On Some Properties Of xi+yi+zi=dXYZ, X3+Y3+Z3=dXYZ, And X6+Y6+Z6=dXYZ; And A Critique Of Solutions For The Markoff Equation x2+y2+z2 = aXYZ.

2019 ◽  
Author(s):  
Michael C. I. Nwogugu
Keyword(s):  
Markoff Equation

scholarly journals The Markoff equation X2 + Y2 + Z2 = aXYZ over quadratic imaginary fields

1990 ◽  
Vol 35 (1) ◽  
pp. 72-104 ◽  
Author(s):  
Joseph H. Silverman
Keyword(s):  
Quadratic Imaginary Fields ◽  
Markoff Equation

scholarly journals On the Unicity Conjecture for Generalized Markoff Equation

2018 ◽  
Vol 3 ◽  
pp. 137 ◽  
Author(s):  
Youb Raj Gaire
Keyword(s):  
Positive Integer ◽  
Complete Solution ◽  
Partial Solution ◽  
Famous Conjecture ◽  
Markoff Equation ◽  
College Of Engineering ◽  
Positive Integers

<p>The Markoff equation x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = 3xyz is introduced by A.A. Markoff in 1879. A famous conjecture on the Markoff equation, made by Frobinus in 1913, states that any Markoff triples (x, y, z) with x ≤ y ≤ z is uniquely determined by its largest number z. The complete solution of this equation is still open however the partial solution is given by Barager (1996), Button (2001), Zhang (2007), Srinivasan (2009), Chen and Chen (2013). In 1957, Mordell developed a generalization to the Markoff equation of the form x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = Axyz + B where, A and B are positive integers. In 2015, Donald McGinn take a particular form of above equation with A = 1 and B = A and gave a partial solution to the unicity conjecture to this equation. In this paper, the partial solution to the unicity conjecture to the equation of the form x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> =3xyz + A where A is positive integer with A ≤ 4(x<sup>2</sup> –1) is given. </p><p><strong>Journal of Advanced College of Engineering and Management,</strong> Vol. 3, 2017, Page : 137-145</p>

Products of consecutive integers and the Markoff equation

1994 ◽  
Vol 48 (1) ◽  
pp. 120-120
Author(s):  
Arthur Baragar
Keyword(s):  
Markoff Equation ◽  
Consecutive Integers

Products of consecutive integers and the Markoff equation

1996 ◽  
Vol 51 (1-2) ◽  
pp. 129-136 ◽  
Author(s):  
Arthur Baragar
Keyword(s):  
Markoff Equation ◽  
Consecutive Integers

scholarly journals The Markoff equation over polynomial rings

2021 ◽  
Author(s):  
Ricardo Conceição ◽  
Rachael Kelly ◽  
Samuel VanFossen
Keyword(s):  
Polynomial Rings ◽  
Markoff Equation

scholarly journals Solutions of a generalized markoff equation in Fibonacci numbers

2020 ◽  
Vol 70 (5) ◽  
pp. 1069-1078
Author(s):  
Hayder Raheem Hashim ◽  
Szabolcs Tengely
Keyword(s):  
Fibonacci Numbers ◽  
Markoff Equation

AbstractIn this paper, we find all the solutions (X, Y, Z) = (FI, FJ, FK), where FI, FJ, and FK represent nonzero Fibonacci numbers, satisfying a generalization of Markoff equation called the Jin-Schmidt equation: AX2 + BY2 + CZ2 = DXYZ + 1.

On the Unicity Conjecture for Markoff Numbers

1996 ◽  
Vol 39 (1) ◽  
pp. 3-9 ◽  
Author(s):  
Arthur Baragar
Keyword(s):  
Positive Integer ◽  
Markoff Equation ◽  
Markoff Numbers

AbstractIn 1913 Frobenius conjectured that for any positive integer m, there exists at most one pair of integers (x, y) with 0 ≤ x ≤ y ≤ m such that (x, y, m) is a solution to the Markoff equation: x2 + y2 + m2 = 3xym. We show this is true if either m, 3m — 2 or 3m + 2 is prime, twice a prime or four times a prime.

Markoff Numbers

2018 ◽  
pp. 21-28
Author(s):  
Christophe Reutenauer
Keyword(s):  
Diophantine Equation ◽  
Free Monoid ◽  
Special Representation ◽  
Markoff Equation ◽  
Markoff Numbers

The Markoff equation is the diophantine equation x2 +y2 +z2 = 3xyz. A solution is called a Markoff triple. The main result in this chapter is a bijection between lower Christoffel words and Markoff triples. The bijection uses several ingredients: a special representation of the free monoid into SL2(N), the so-called Fricke relations, which relate the traces of two matrices in SL2, their product and their commutator (an equation reminiscent of the Markoff equation, as noted first by Harvey Cohn). Another lemma describes the socalled Markoff moves: they relate Markoff triples each to another. The chapter ends with a statement of the famous Frobenius conjecture: it asks whether the parametrization of Markoff numbers (that is, components of a Markoff triple), which is surjective by the theorem, is also injective.

scholarly journals The asymptotic growth of integer solutions to the Rosenberger equations

2004 ◽  
Vol 69 (3) ◽  
pp. 481-497 ◽  
Author(s):  
Arthur Baragar ◽  
Kensaku Umeda
Keyword(s):  
Finite Number ◽  
Infinite Number ◽  
Asymptotic Growth ◽  
Markoff Equation ◽  
Integer Solutions

Zagier showed that the number of integer solutions to the Markoff equation with components bounded by T grows asymptotically like C(log T)2, where C is explicity computable. Rosenberger showed that there are only a finite number of equations ax2 + by2 + cz2 = dxyz with a, b, and c dividing d, and for which the equation admits an infinite number of integer solutions. In this paper, we generalise Zagier's techniques so that they may be applied to the Rosenberger equations. We also apply these techniques to the equations ax2 + by2 + cz2 = dxyz + 1.

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