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 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 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.

scholarly journals Estimating Mahler's measure

1995 ◽  
Vol 51 (1) ◽  
pp. 145-151
Author(s):  
G.R. Everest
Keyword(s):  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Riemann Sums ◽  
Integer Polynomials ◽  
Several Variables ◽  
Logarithmic Integral ◽  
Baker's Theorem

In 1962, Mahler defined a measure for integer polynomials in several variables as the logarithmic integral over the torus. Many results exist about the values taken by the measure but many unsolved problems remain. In one variable, it is possible to express the measure as an effective limit of Riemann sums. We show that the same is true in several variables, using a non-obvious parametrisation of the torus together with Baker's Theorem on linear forms in logarithms of algebraic numbers.

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.

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.

Computing the effectively computable bound in Baker's inequality for linear forms in logarithms

1976 ◽  
Vol 15 (1) ◽  
pp. 33-57 ◽  
Author(s):  
A.J. van der Poorten ◽  
J.H. Loxton
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
Computable Bound ◽  
Image Position ◽  
Detailed Computation

For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α1, …, αn all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n0(D) which is explicitly computed, the inequalities (in the standard notation)have no solution in rational integers b1, …, bn (bn ≠ 0) of absolute value at most B, whenever n ≥ n0(D). The very favourable dependence on n is particularly useful.

On Baker's inequality for linear forms in logarithms

1976 ◽  
Vol 80 (2) ◽  
pp. 233-248 ◽  
Author(s):  
Alfred J. van der Poorten
Keyword(s):  
Prime Ideal ◽  
Complex Case ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Image Position

AbstractLet α1, …, αn an be non-zero algebraic numbers with degrees at most d and heights respectively Al, …, An (all Aj ≥ 4) and let b1, …, bn be rational integers with absolute values at most B (≥ 4). Denote by p a prime ideal of the field and suppose that p divides the rational prime p. WriteThen it is shown thatfor some effectively computable constant C > 0 depending only on n, d and p. The argument suffices to prove similarly that in the complex case, iffor any fixed determination of the logarithms, thenfor some effectively computable constant C′ > 0 depending only on n and d (and he determination of the logarithms).

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