Integer solutions to decomposable and semi-decomposable form inequalities

2011 ◽  
pp. 663-673 ◽  
Author(s):  
MIN RU
Keyword(s):  
Integer Solutions ◽  
Decomposable Form

ON THE PROBLEM OF INTEGER SOLUTIONS TO DECOMPOSABLE FORM INEQUALITIES

2008 ◽  
Vol 04 (05) ◽  
pp. 859-872 ◽  
Author(s):  
YUANCHENG LIU
Keyword(s):  
Real Number ◽  
Number Field ◽  
Finite Set ◽  
Integer Solutions ◽  
Decomposable Form

This paper proves a conjecture proposed by Chen and Ru in [1] on the finiteness of the number of integer solutions to decomposable form inequalities. Let k be a number field and let F(X1,…,Xm) be a non-degenerate decomposable form with coefficients in k. We show that for every finite set of places S of k containing the archimedean places of k, for each real number λ < 1 and each constant c > 0, the inequality [Formula: see text] has only finitely many [Formula: see text]-non-proportional solutions, where HS(x1,…,xm) = Πυ∈S max 1≤i≤m ||xi||υ is the S-height.

scholarly journals Integer solutions of a sequence of decomposable form inequalities

1998 ◽  
Vol 86 (3) ◽  
pp. 227-237 ◽  
Author(s):  
K. Győry ◽  
Min Ru
Keyword(s):  
Integer Solutions ◽  
Decomposable Form

scholarly journals Integer solutions to decomposable form inequalities

2005 ◽  
Vol 115 (1) ◽  
pp. 58-70 ◽  
Author(s):  
Zhihua Chen ◽  
Min Ru
Keyword(s):  
Integer Solutions ◽  
Decomposable Form

scholarly journals A closer look at some new identities

1998 ◽  
Vol 21 (3) ◽  
pp. 581-586
Author(s):  
Geoffrey B. Campbell
Keyword(s):  
Positive Integer ◽  
Infinite Products ◽  
Integer Solutions ◽  
Origin Point

We obtain infinite products related to the concept of visible from the origin point vectors. Among these is∏k=3∞(1−Z)φ,(k)/k=11−Zexp(Z32(1−Z)2−12Z−12Z(1−Z)),  |Z|<1,in whichφ3(k)denotes for fixedk, the number of positive integer solutions of(a,b,k)=1wherea<b<k, assuming(a,b,k)is thegcd(a,b,k).

scholarly journals Solution of certain Pell equations

2018 ◽  
Vol 11 (04) ◽  
pp. 1850056 ◽  
Author(s):  
Zahid Raza ◽  
Hafsa Masood Malik
Keyword(s):  
Positive Integer ◽  
Fundamental Solution ◽  
Positive Solutions ◽  
Continued Fraction ◽  
Pell Equation ◽  
Lucas Sequences ◽  
Pell Equations ◽  
Integer Solutions ◽  
Positive Integers

Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the Pell form [Formula: see text] including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation [Formula: see text] in terms of [Formula: see text] We extend all the results of the papers [3, 10, 27, 37].

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