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.

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].

scholarly journals Some Diophantine Problems Related to k-Fibonacci Numbers

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1047
Author(s):  
Pavel Trojovský ◽  
Štěpán Hubálovský
Keyword(s):  
Recurrence Relation ◽  
Diophantine Equations ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Recursive Relation ◽  
Initial Values ◽  
Lucas Sequence ◽  
Diophantine Problems ◽  
Positive Integers

Let k ≥ 1 be an integer and denote ( F k , n ) n as the k-Fibonacci sequence whose terms satisfy the recurrence relation F k , n = k F k , n − 1 + F k , n − 2 , with initial conditions F k , 0 = 0 and F k , 1 = 1 . In the same way, the k-Lucas sequence ( L k , n ) n is defined by satisfying the same recursive relation with initial values L k , 0 = 2 and L k , 1 = k . The sequences ( F k , n ) n ≥ 0 and ( L k , n ) n ≥ 0 were introduced by Falcon and Plaza, who derived many of their properties. In particular, they proved that F k , n 2 + F k , n + 1 2 = F k , 2 n + 1 and F k , n + 1 2 − F k , n − 1 2 = k F k , 2 n , for all k ≥ 1 and n ≥ 0 . In this paper, we shall prove that if k > 1 and F k , n s + F k , n + 1 s ∈ ( F k , m ) m ≥ 1 for infinitely many positive integers n, then s = 2 . Similarly, that if F k , n + 1 s − F k , n − 1 s ∈ ( k F k , m ) m ≥ 1 holds for infinitely many positive integers n, then s = 1 or s = 2 . This generalizes a Marques and Togbé result related to the case k = 1 . Furthermore, we shall solve the Diophantine equations F k , n = L k , m , F k , n = F n , k and L k , n = L n , k .

Norm form equations. III: positive characteristic

1986 ◽  
Vol 99 (3) ◽  
pp. 409-423 ◽  
Author(s):  
R. C. Mason
Keyword(s):  
Complete Resolution ◽  
Positive Characteristic ◽  
Approximation Theorem ◽  
Function Fields ◽  
Zero Characteristic ◽  
Norm Form ◽  
Starting Point ◽  
Diophantine Problems ◽  
General Norm ◽  
Image Position

This paper aims to provide a complete resolution of the general norm form equation over function fields of positive characteristic. In a previous paper [4] we studied norm forms in the simpler case of zero characteristic; that study forms the starting point for the present investigations. Diophantine problems over function fields of positive characteristic were first investigated by Armitage in 1968 [1], who clamied to have established an analogue of the Thue–Siegel–Roth–Uchiyama theorem for such fields. This claim was refuted by Osgood in 1975 [6], who also derived a correct analogue of Thue's approximation theorem. in 1983 a different attack was made on Diophantine problems over function fields, the principal weapon being a bound [2] for the heights of the solutions of the unit equation

CUBIC DIOPHANTINE INEQUALITIES INVOLVING A NORM FORM

2011 ◽  
Vol 07 (08) ◽  
pp. 2219-2235 ◽  
Author(s):  
M. P. HARVEY
Keyword(s):  
Asymptotic Formula ◽  
Diagonal Form ◽  
Diophantine Inequalities ◽  
Norm Form ◽  
Cubic Forms

We apply Freeman's variant of the Davenport–Heilbronn method to provide an asymptotic formula for the number of small values taken by a certain family of cubic forms with real coefficients. The cubic forms in question arise as the sum of a diagonal form and a norm form and should have at least seven variables.

scholarly journals Some diophantine problems concerning equal sums of integers and their cubes

2010 ◽  
Vol Volume 33 ◽  
Author(s):  
Ajai Choudhry
Keyword(s):  
Symmetric Solution ◽  
Diophantine Equations ◽  
Simple Solution ◽  
Parametric Solution ◽  
Quadratic Polynomials ◽  
Simultaneous Diophantine Equations ◽  
International Audience ◽  
Diophantine Problems ◽  
The Given ◽  
Complete Solutions

International audience This paper gives a complete four-parameter solution of the simultaneous diophantine equations $x+y+z=u+v+w, x^3+y^3+z^3=u^3+v^3+w^3,$ in terms of quadratic polynomials in which each parameter occurs only in the first degree. This solution is much simpler than the complete solutions of these equations published earlier. This simple solution is used to obtain solutions of several related diophantine problems. For instance, the paper gives a parametric solution of the arbitrarily long simultaneous diophantine chains of the type $x^k_1+y^k_1+z^k_1=x^k_2+y^k_2+z^k_2=\ldots=x^k_n+y^k_n+z^k_n=\ldots,~~k=1,3.$ Further, the complete ideal symmetric solution of the Tarry-Escott problem of degree $4$ is obtained, and it is also shown that any arbitrarily given integer can be expressed as the sum of four distinct nonzero integers such that the sum of the cubes of these four integers is equal to the cube of the given integer.

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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