Displaying similar documents to “Pell and Pell-Lucas numbers of the form - 2 a - 3 b + 5 c

Repdigits in generalized Pell sequences

Jhon J. Bravo, Jose L. Herrera (2020)

Archivum Mathematicum

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For an integer k 2 , let ( n ) n be the k - generalized Pell sequence which starts with 0 , ... , 0 , 1 ( k terms) and each term afterwards is given by the linear recurrence n = 2 n - 1 + n - 2 + + n - k . In this paper, we find all k -generalized Pell numbers with only one distinct digit (the so-called repdigits). 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 repdigits in the usual Pell sequence ( P n ( 2 ) ) n . ...

Lucas factoriangular numbers

Bir Kafle, Florian Luca, Alain Togbé (2020)

Mathematica Bohemica

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We show that the only Lucas numbers which are factoriangular are 1 and 2 .

On perfect powers in k -generalized Pell sequence

Zafer Şiar, Refik Keskin, Elif Segah Öztaş (2023)

Mathematica Bohemica

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Let k 2 and let ( P n ( k ) ) n 2 - k be the k -generalized Pell sequence defined by P n ( k ) = 2 P n - 1 ( k ) + P n - 2 ( k ) + + P n - k ( k ) for n 2 with initial conditions P - ( k - 2 ) ( k ) = P - ( k - 3 ) ( k ) = = P - 1 ( k ) = P 0 ( k ) = 0 , P 1 ( k ) = 1 . In this study, we handle the equation P n ( k ) = y m in positive integers n , m , y , k such that k , y 2 , and give an upper bound on n . Also, we will show that the equation P n ( k ) = y m with 2 y 1000 has only one solution given by P 7 ( 2 ) = 13 2 .

An approximation property of quadratic irrationals

Takao Komatsu (2002)

Bulletin de la Société Mathématique de France

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Let α > 1 be irrational. Several authors studied the numbers m ( α ) = inf { | y | : y Λ m , y 0 } , where m is a positive integer and Λ m denotes the set of all real numbers of the form y = ϵ 0 α n + ϵ 1 α n - 1 + + ϵ n - 1 α + ϵ n with restricted integer coefficients | ϵ i | m . The value of 1 ( α ) was determined for many particular Pisot numbers and m ( α ) for the golden number. In this paper the value of  m ( α ) is determined for irrational numbers  α , satisfying α 2 = a α ± 1 with a positive integer a .

Bartz-Marlewski equation with generalized Lucas components

Hayder R. Hashim (2022)

Archivum Mathematicum

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Let { U n } = { U n ( P , Q ) } and { V n } = { V n ( P , Q ) } be the Lucas sequences of the first and second kind respectively at the parameters P 1 and Q { - 1 , 1 } . In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation x 2 - 3 x y + y 2 + x = 0 , where ( x , y ) = ( U i , U j ) or ( V i , V j ) with i , j 1 . Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.

The number of solutions to the generalized Pillai equation ± r a x ± s b y = c .

Reese Scott, Robert Styer (2013)

Journal de Théorie des Nombres de Bordeaux

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We consider N , the number of solutions ( x , y , u , v ) to the equation ( - 1 ) u r a x + ( - 1 ) v s b y = c in nonnegative integers x , y and integers u , v { 0 , 1 } , for given integers a > 1 , b > 1 , c > 0 , r > 0 and s > 0 . When gcd ( r a , s b ) = 1 , we show that N 3 except for a finite number of cases all of which satisfy max ( a , b , r , s , x , y ) < 2 · 10 15 for each solution; when gcd ( a , b ) > 1 , we show that N 3 except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving N = 3 solutions.

Mersenne numbers as a difference of two Lucas numbers

Murat Alan (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let ( L n ) n 0 be the Lucas sequence. We show that the Diophantine equation L n - L m = M k has only the nonnegative integer solutions ( n , m , k ) = ( 2 , 0 , 1 ) , ( 3 , 1 , 2 ) , ( 3 , 2 , 1 ) , ( 4 , 3 , 2 ) , ( 5 , 3 , 3 ) , ( 6 , 2 , 4 ) , ( 6 , 5 , 3 ) where M k = 2 k - 1 is the k th Mersenne number and n > m .

On the distribution of ( k , r ) -integers in Piatetski-Shapiro sequences

Teerapat Srichan (2021)

Czechoslovak Mathematical Journal

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A natural number n is said to be a ( k , r ) -integer if n = a k b , where k > r > 1 and b is not divisible by the r th power of any prime. We study the distribution of such ( k , r ) -integers in the Piatetski-Shapiro sequence { n c } with c > 1 . As a corollary, we also obtain similar results for semi- r -free integers.

A localization property for B p q s and F p q s spaces

Hans Triebel (1994)

Studia Mathematica

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Let f j = k a k f ( 2 j + 1 x - 2 k ) , where the sum is taken over the lattice of all points k in n having integer-valued components, j∈ℕ and a k . Let A p q s be either B p q s or F p q s (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on n . The aim of the paper is to clarify under what conditions f j | A p q s is equivalent to 2 j ( s - n / p ) ( k | a k | p ) 1 / p f | A p q s .

Cobham's theorem for substitutions

Fabien Durand (2011)

Journal of the European Mathematical Society

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The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let α and β be two multiplicatively independent Perron numbers. Then a sequence x A , where A is a finite alphabet, is both α -substitutive and β -substitutive if and only if x is ultimately...

Towards Bauer's theorem for linear recurrence sequences

Mariusz Skałba (2003)

Colloquium Mathematicae

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Consider a recurrence sequence ( x k ) k of integers satisfying x k + n = a n - 1 x k + n - 1 + . . . + a x k + 1 + a x k , where a , a , . . . , a n - 1 are fixed and a₀ ∈ -1,1. Assume that x k > 0 for all sufficiently large k. If there exists k₀∈ ℤ such that x k < 0 then for each negative integer -D there exist infinitely many rational primes q such that q | x k for some k ∈ ℕ and (-D/q) = -1.

Repdigits in the base b as sums of four balancing numbers

Refik Keskin, Faticko Erduvan (2021)

Mathematica Bohemica

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The sequence of balancing numbers ( B n ) is defined by the recurrence relation B n = 6 B n - 1 - B n - 2 for n 2 with initial conditions B 0 = 0 and B 1 = 1 . B n is called the n th balancing number. In this paper, we find all repdigits in the base b , which are sums of four balancing numbers. As a result of our theorem, we state that if B n is repdigit in the base b and has at least two digits, then ( n , b ) = ( 2 , 5 ) , ( 3 , 6 ) . Namely, B 2 = 6 = ( 11 ) 5 and B 3 = 35 = ( 55 ) 6 .

Σ s -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that any Σ s -product of at most 𝔠 -many L Σ ( ω ) -spaces has the L Σ ( ω ) -property. This result generalizes some known results about L Σ ( ω ) -spaces. On the other hand, we prove that every Σ s -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every Σ s -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

A Diophantine inequality with four squares and one k th power of primes

Quanwu Mu, Minhui Zhu, Ping Li (2019)

Czechoslovak Mathematical Journal

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Let k 5 be an odd integer and η be any given real number. We prove that if λ 1 , λ 2 , λ 3 , λ 4 , μ are nonzero real numbers, not all of the same sign, and λ 1 / λ 2 is irrational, then for any real number σ with 0 < σ < 1 / ( 8 ϑ ( k ) ) , the inequality | λ 1 p 1 2 + λ 2 p 2 2 + λ 3 p 3 2 + λ 4 p 4 2 + μ p 5 k + η | < max 1 j 5 p j - σ has infinitely many solutions in prime variables p 1 , p 2 , , p 5 , where ϑ ( k ) = 3 × 2 ( k - 5 ) / 2 for k = 5 , 7 , 9 and ϑ ( k ) = [ ( k 2 + 2 k + 5 ) / 8 ] for odd integer k with k 11 . This improves a recent result in W. Ge, T. Wang (2018).