Continued Fractions, Jacobi Symbols, and Quadratic Diophantine Equations

2000 ◽  
Vol 43 (2) ◽  
pp. 218-225 ◽  
Author(s):  
R. A. Mollin ◽  
A. J. van der Poorten
Keyword(s):  
Continued Fractions ◽  
Diophantine Equations ◽  
Norm Form

AbstractThe results herein continue observations on norm form equations and continued fractions begun and continued in the works [1]−[3], and [5]−[6].

References for Teachers: The Theorem of Pythagoras

1951 ◽  
Vol 44 (8) ◽  
pp. 585-598
Author(s):  
William L. Schaaf
Keyword(s):  
Continued Fractions ◽  
Diophantine Equations ◽  
Irrational Number ◽  
Golden Section ◽  
Dynamic Symmetry ◽  
Squaring The Circle ◽  
Logarithmic Spirals ◽  
The Rich ◽  
The One ◽  
Star Polygons

Justly called the most celebrated theorem of geometry, the Pythagorean proposition is probably the one bit of mathematics which millions of laymen will remember long after they have forgotten whatever other mathematics they may once have known. The theorem is notable first because of the rich historical associations with which it is attended; secondly, because of the amazing variety of proofs which have been given; and thirdly, because further exploration quickly leads to interesting and unexpected byways, such as the Golden Section, dynamic symmetry, logarithmic spirals, angle trisection, duplication of the cube, squaring the circle, determining the value of π, the concept of irrational number, regular and star polygons and polyhedra, theory of numbers, constructibility of angles and polygons, continued fractions, phyllotaxy, musical scales, Diophantine equations, Heronian triangles, and Pythagorean number lore.

scholarly journals MINOR ARC MOMENTS OF WEYL SUMS

2012 ◽  
Vol 55 (1) ◽  
pp. 97-113 ◽  
Author(s):  
M. P. HARVEY
Keyword(s):  
Diophantine Equations ◽  
Diagonal Form ◽  
Norm Form ◽  
Goldbach Problem ◽  
Weyl Sum ◽  
Diophantine Problems

AbstractWe obtain an improved bound for the 2k-th moment of a degree k Weyl sum, restricted to a set of minor arcs, when k is small. We then present some applications of this bound to some Diophantine problems, including a case of the Waring–Goldbach problem, and a particular family of Diophantine equations defined as the sum of a norm form and a diagonal form.

scholarly journals Continued fractions and diophantine equations in positive characteristic

2021 ◽  
Vol 109 (123) ◽  
pp. 143-151
Author(s):  
Khalil Ayadi ◽  
Awatef Azaza ◽  
Salah Beldi
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Diophantine Equations ◽  
Positive Characteristic ◽  
Continued Fraction Expansion ◽  
Ring Of Polynomials ◽  
Algebraic Power Series

We exhibit explicitly the continued fraction expansion of some algebraic power series over a finite field. We also discuss some Diophantine equations on the ring of polynomials, which are intimately related to these power series.

scholarly journals Lagrange, central norms, and quadratic Diophantine equations

2005 ◽  
Vol 2005 (7) ◽  
pp. 1039-1047 ◽  
Author(s):  
R. A. Mollin
Keyword(s):  
Diophantine Equation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Diophantine Equations ◽  
Continued Fraction Expansion ◽  
Basic Properties

We consider the Diophantine equation of the formx2−Dy2=c, wherec=±1,±2, and provide a generalization of results of Lagrange with elementary proofs using only basic properties of simple continued fractions. As a consequence, we achieve a completely general, simple, and elegant criterion for the central norm to be2in the simple continued fraction expansion ofD.

scholarly journals Repdigits as Product of Terms of k-Bonacci Sequences

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 682
Author(s):  
Petr Coufal ◽  
Pavel Trojovský
Keyword(s):  
Recent Work ◽  
Continued Fractions ◽  
Diophantine Equations ◽  
Reduction Method ◽  
Fibonacci Numbers ◽  
Decimal Expansion ◽  
Generalized Fibonacci Numbers ◽  
Initial Values ◽  
Transcendental Method

For any integer k≥2, the sequence of the k-generalized Fibonacci numbers (or k-bonacci numbers) is defined by the k initial values F−(k−2)(k)=⋯=F0(k)=0 and F1(k)=1 and such that each term afterwards is the sum of the k preceding ones. In this paper, we search for repdigits (i.e., a number whose decimal expansion is of the form aa…a, with a∈[1,9]) in the sequence (Fn(k)Fn(k+m))n, for m∈[1,9]. This result generalizes a recent work of Bednařík and Trojovská (the case in which (k,m)=(2,1)). Our main tools are the transcendental method (for Diophantine equations) together with the theory of continued fractions (reduction method).

scholarly journals On the finiteness of solutions for polynomial-factorial Diophantine equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Wataru Takeda
Keyword(s):  
Diophantine Equations ◽  
Field Extension ◽  
Infinite Subset ◽  
Norm Form ◽  
Factorial Function ◽  
Integer Solutions

AbstractWe study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exist only finitely many l such that {l!} is represented by {N_{A}(x)}, where {N_{A}} is a norm form constructed from the field norm of a field extension {K/\mathbf{Q}}. We also deal with the equation {N_{A}(x)=l!_{S}}, where {l!_{S}} is the Bhargava factorial. In this paper, we also show that the Oesterlé–Masser conjecture implies that for any infinite subset S of {\mathbf{Z}} and for any polynomial {P(x)\in\mathbf{Z}[x]} of degree 2 or more the equation {P(x)=l!_{S}} has only finitely many solutions {(x,l)}. For some special infinite subsets S of {\mathbf{Z}}, we can show the finiteness of solutions for the equation {P(x)=l!_{S}} unconditionally.

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