Displaying similar documents to “On the Diophantine equation j = 1 k j F j p = F n q

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 Diophantine equation ( 2 x - 1 ) ( p y - 1 ) = 2 z 2

Ruizhou Tong (2021)

Czechoslovak Mathematical Journal

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Let p be an odd prime. By using the elementary methods we prove that: (1) if 2 x , p ± 3 ( mod 8 ) , the Diophantine equation ( 2 x - 1 ) ( p y - 1 ) = 2 z 2 has no positive integer solution except when p = 3 or p is of the form p = 2 a 0 2 + 1 , where a 0 > 1 is an odd positive integer. (2) if 2 x , 2 y , y 2 , 4 , then the Diophantine equation ( 2 x - 1 ) ( p y - 1 ) = 2 z 2 has no positive integer solution.

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.

A remark on a Diophantine equation of S. S. Pillai

Azizul Hoque (2024)

Czechoslovak Mathematical Journal

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S. S. Pillai proved that for a fixed positive integer a , the exponential Diophantine equation x y - y x = a , min ( x , y ) > 1 , has only finitely many solutions in integers x and y . We prove that when a is of the form 2 z 2 , the above equation has no solution in integers x and y with gcd ( x , y ) = 1 .

A note on the article by F. Luca “On the system of Diophantine equations a ² + b ² = ( m ² + 1 ) r and a x + b y = ( m ² + 1 ) z ” (Acta Arith. 153 (2012), 373-392)

Takafumi Miyazaki (2014)

Acta Arithmetica

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Let r,m be positive integers with r > 1, m even, and A,B be integers satisfying A + B ( - 1 ) = ( m + ( - 1 ) ) r . We prove that the Diophantine equation | A | x + | B | y = ( m ² + 1 ) z has no positive integer solutions in (x,y,z) other than (x,y,z) = (2,2,r), whenever r > 10 74 or m > 10 34 . Our result is an explicit refinement of a theorem due to F. Luca.

The exceptional set for Diophantine inequality with unlike powers of prime variables

Wenxu Ge, Feng Zhao (2018)

Czechoslovak Mathematical Journal

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Suppose that λ 1 , λ 2 , λ 3 , λ 4 are nonzero real numbers, not all negative, δ > 0 , 𝒱 is a well-spaced set, and the ratio λ 1 / λ 2 is algebraic and irrational. Denote by E ( 𝒱 , N , δ ) the number of v 𝒱 with v N such that the inequality | λ 1 p 1 2 + λ 2 p 2 3 + λ 3 p 3 4 + λ 4 p 4 5 - v | < v - δ has no solution in primes p 1 , p 2 , p 3 , p 4 . We show that E ( 𝒱 , N , δ ) N 1 + 2 δ - 1 / 72 + ε for any ε > 0 .

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.

Padovan and Perrin numbers as products of two generalized Lucas numbers

Kouèssi Norbert Adédji, Japhet Odjoumani, Alain Togbé (2023)

Archivum Mathematicum

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Let P m and E m be the m -th Padovan and Perrin numbers respectively. Let r , s be non-zero integers with r 1 and s { - 1 , 1 } , let { U n } n 0 be the generalized Lucas sequence given by U n + 2 = r U n + 1 + s U n , with U 0 = 0 and U 1 = 1 . In this paper, we give effective bounds for the solutions of the following Diophantine equations P m = U n U k and E m = U n U k , where m , n and k are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing sequences.

A note on the weighted Khintchine-Groshev Theorem

Mumtaz Hussain, Tatiana Yusupova (2014)

Journal de Théorie des Nombres de Bordeaux

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Let W ( m , n ; ψ ̲ ) denote the set of ψ 1 , ... , ψ n –approximable points in m n . The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions ψ ̲ . Removing monotonicity from the Khintchine–Groshev theorem is attributed to different authors for different cases of m and n . It can not be removed for m = n = 1 as Duffin–Schaeffer provided the counter example. We deal with the only remaining case m = 2 and thereby remove all unnecessary conditions from the Khintchine–Groshev theorem. ...

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).

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 .

Lucas sequences and repdigits

Hayder Raheem Hashim, Szabolcs Tengely (2022)

Mathematica Bohemica

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Let ( G n ) n 1 be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are { U n } and { V n } , respectively. We show that the Diophantine equation G n = B · ( g l m - 1 ) / ( g l - 1 ) has only finitely many solutions in n , m + , where g 2 , l is even and 1 B g l - 1 . Furthermore, these solutions can be effectively determined by reducing such equation to biquadratic elliptic curves. Then, by a result of Baker (and its best improvement due to Hajdu and Herendi) related to the bounds of the integral...

Diophantine equations involving factorials

Horst Alzer, Florian Luca (2017)

Mathematica Bohemica

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We study the Diophantine equations ( k ! ) n - k n = ( n ! ) k - n k and ( k ! ) n + k n = ( n ! ) k + n k , where k and n are positive integers. We show that the first one holds if and only if k = n or ( k , n ) = ( 1 , 2 ) , ( 2 , 1 ) and that the second one holds if and only if k = n .

On the exponential diophantine equation x y + y x = z z

Xiaoying Du (2017)

Czechoslovak Mathematical Journal

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For any positive integer D which is not a square, let ( u 1 , v 1 ) be the least positive integer solution of the Pell equation u 2 - D v 2 = 1 , and let h ( 4 D ) denote the class number of binary quadratic primitive forms of discriminant 4 D . If D satisfies 2 D and v 1 h ( 4 D ) 0 ( mod D ) , then D is called a singular number. In this paper, we prove that if ( x , y , z ) is a positive integer solution of the equation x y + y x = z z with 2 z , then maximum max { x , y , z } < 480000 and both x , y are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions...

On some Diophantine equations involving balancing numbers

Euloge Tchammou, Alain Togbé (2021)

Archivum Mathematicum

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In this paper, we find all the solutions of the Diophantine equation B 1 p + 2 B 2 p + + k B k p = B n q in positive integer variables ( k , n ) , where B i is the i t h balancing number if the exponents p , q are included in the set { 1 , 2 } .