scholarly journals Fibonacci Numbers with a Prescribed Block of Digits

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 639 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Lower Bounds ◽  
Diophantine Approximation ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Algebraic Numbers ◽  
Decimal Expansion ◽  
Linear Forms

In this paper, we prove that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b … b c … c . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation.

Fibonacci numbers in generalized Pell sequences

2020 ◽  
Vol 70 (5) ◽  
pp. 1057-1068
Author(s):  
Jhon J. Bravo ◽  
Jose L. Herrera
Keyword(s):  
Lower Bounds ◽  
Continued Fractions ◽  
Fibonacci Numbers ◽  
Independent Interest ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Pell Numbers

AbstractIn this paper, by using lower bounds for linear forms in logarithms of algebraic numbers and the theory of continued fractions, we find all Fibonacci numbers that appear in generalized Pell sequences. Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for Fibonacci numbers in the Pell sequence.

On Some Exponential Equations of S. S. Pillai

2001 ◽  
Vol 53 (5) ◽  
pp. 897-922 ◽  
Author(s):  
Michael A. Bennett
Keyword(s):  
Lower Bounds ◽  
Diophantine Equation ◽  
Prior Work ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Sharp Result ◽  
Exponential Equations ◽  
Positive Integers

AbstractIn this paper, we establish a number of theorems on the classic Diophantine equation of S. S. Pillai, ax – by = c, where a, b and c are given nonzero integers with a, b ≥ 2. In particular, we obtain the sharp result that there are at most two solutions in positive integers x and y and deduce a variety of explicit conditions under which there exists at most a single such solution. These improve or generalize prior work of Le, Leveque, Pillai, Scott and Terai. The main tools used include lower bounds for linear forms in the logarithms of (two) algebraic numbers and various elementary arguments.

Simultaneous approximation of numbers connected with the exponential function

1978 ◽  
Vol 25 (4) ◽  
pp. 466-478 ◽  
Author(s):  
Michel Waldschmidt
Keyword(s):  
Lower Bounds ◽  
Exponential Function ◽  
Complex Number ◽  
Simultaneous Approximation ◽  
Complex Numbers ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms

AbstractWe give several results concerning the simultaneous approximation of certain complex numbers. For instance, we give lower bounds for |a–ξo |+ | ea – ξ1 |, where a is any non-zero complex number, and ξ are two algebraic numbers. We also improve the estimate of the so-called Franklin Schneider theorem concerning | b – ξ | + | a – ξ | + | ab – ξ. We deduce these results from an estimate for linear forms in logarithms.

scholarly journals Quelques remarques sur des questions d'approximation diophantienne

1999 ◽  
Vol 59 (2) ◽  
pp. 323-334 ◽  
Author(s):  
Patrice Philippon
Keyword(s):  
Lower Bounds ◽  
Approximation Theory ◽  
Diophantine Approximation ◽  
Rational Numbers ◽  
Product Theorem ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
Subspace Theorem

Hoping for a hand-shake between methods from diophantine approximation theory and transcendance theory, we show how zeros estimates from transcendance theory imply Roth's type lemmas (including the product theorem). We also formulate some strong conjectures on lower bounds for linear forms in logarithms of rational numbers with rational coefficients, inspired by the subspace theorem and which would imply, for example, the abc conjecture.

scholarly journals Constants for lower bounds for linear forms in the logarithms of algebraic numbers I. The general case

1990 ◽  
Vol 55 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Josef Blass ◽  
A. Glass ◽  
David Manski ◽  
David Meronk ◽  
Ray Steiner
Keyword(s):  
Lower Bounds ◽  
Algebraic Numbers ◽  
Linear Forms

scholarly journals Constants for lower bounds for linear forms in the logarithms of algebraic numbers II. The homogeneous rational case

1990 ◽  
Vol 55 (1) ◽  
pp. 15-22 ◽  
Author(s):  
Josef Blass ◽  
A. Glass ◽  
David Manski ◽  
David Meronk ◽  
Ray Steiner
Keyword(s):  
Lower Bounds ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Rational Case

scholarly journals Tribonacci numbers that are concatenations of two repdigits

2020 ◽  
Vol 114 (4) ◽  
Author(s):  
Mahadi Ddamulira
Keyword(s):  
Lower Bounds ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Reduction Procedure ◽  
Linear Forms ◽  
Tribonacci Numbers

Abstract Let $$ (T_{n})_{n\ge 0} $$ ( T n ) n ≥ 0 be the sequence of Tribonacci numbers defined by $$ T_0=0 $$ T 0 = 0 , $$ T_1=T_2=1$$ T 1 = T 2 = 1 , and $$ T_{n+3}= T_{n+2}+T_{n+1} +T_n$$ T n + 3 = T n + 2 + T n + 1 + T n for all $$ n\ge 0 $$ n ≥ 0 . In this note, we use of lower bounds for linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction procedure to find all Tribonacci numbers that are concatenations of two repdigits.

scholarly journals On Repdigits as Sums of Fibonacci and Tribonacci Numbers

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1774
Author(s):  
Pavel Trojovský
Keyword(s):  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Decimal Expansion ◽  
Reduction Procedure ◽  
Linear Forms ◽  
Tribonacci Numbers ◽  
Symmetrical Type

In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can be seen as the easiest case of palindromic numbers, which are a ”symmetrical” type of numbers) that can be written in the form Fn+Tn, for some n≥1, where (Fn)n≥0 and (Tn)n≥0 are the sequences of Fibonacci and Tribonacci numbers, respectively.

scholarly journals A NOTE ON WEIGHTED BADLY APPROXIMABLE LINEAR FORMS

2016 ◽  
Vol 59 (2) ◽  
pp. 349-357 ◽  
Author(s):  
STEPHEN HARRAP ◽  
NIKOLAY MOSHCHEVITIN
Keyword(s):  
Diophantine Approximation ◽  
Affirmative Answer ◽  
Famous Conjecture ◽  
Linear Forms ◽  
Badly Approximable

AbstractWe prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give an affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation.

Sets with large intersection and ubiquity

2008 ◽  
Vol 144 (1) ◽  
pp. 119-144 ◽  
Author(s):  
ARNAUD DURAND
Keyword(s):  
Diophantine Approximation ◽  
Nondecreasing Function ◽  
Gauge Function ◽  
Algebraic Numbers ◽  
Positive Real ◽  
Integer Polynomials ◽  
Large Intersection ◽  
Image Position ◽  
Full Lebesgue Measure ◽  
Arbitrary Norm

AbstractA central problem motivated by Diophantine approximation is to determine the size properties of subsets of$\R^d$ ($d\in\N$)of the formwhere ‖⋅‖ denotes an arbitrary norm,Ia denumerable set, (xi,ri)i∈ Ia family of elements of$\R^d\$× (0, ∞) and ϕ a nonnegative nondecreasing function defined on [0, ∞). We show that ifFId, where Id denotes the identity function, has full Lebesgue measure in a given nonempty open subsetVof$\R^d\$, the setFϕbelongs to a class Gh(V) of sets with large intersection inVwith respect to a given gauge functionh. We establish that this class is closed under countable intersections and that each of its members has infinite Hausdorffg-measure for every gauge functiongwhich increases faster thanhnear zero. In particular, this yields a sufficient condition on a gauge functiongsuch that a given countable intersection of sets of the formFϕhas infinite Hausdorffg-measure. In addition, we supply several applications of our results to Diophantine approximation. For any nonincreasing sequenceψof positive real numbers converging to zero, we investigate the size and large intersection properties of the sets of all points that areψ-approximable by rationals, by rationals with restricted numerator and denominator and by real algebraic numbers. This enables us to refine the analogs of Jarník's theorem for these sets. We also study the approximation of zero by values of integer polynomials and deduce several new results concerning Mahler's and Koksma's classifications of real transcendental numbers.

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